Paper Title: Entropy, Thermodynamics and the Geometrization of the Language Model.



plain


   
    Entropy, Thermodynamics and the Geometrization of the Language Model
        
    

    Wenzhe Yang
    
    

    
    
    

    
    <https://orcid.org/0000-0002-2456-7453>

    
    



§ ABSTRACT 

In this paper, we discuss how pure mathematics and theoretical physics can be applied to the study of language models. Using set theory and analysis, we formulate mathematically rigorous definitions of language models, and introduce the concept of the moduli space of distributions for a language model. We formulate a generalized distributional hypothesis using functional analysis and topology. We define the entropy function associated with a language model and show how it allows us to understand many interesting phenomena in languages. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. Using the entropy function, we formulate a conjecture about AGI. 

Then, we show how thermodynamics gives us an immediate interpretation to language models. In particular we will define the concepts of partition function, internal energy and free energy for a language model, which offer insights into how language models work. Based on these results, we introduce a general concept of the geometrization of language models and define what is called the Boltzmann manifold. While the current LLMs are the special cases of the Boltzmann manifold.








Keywords: Language Model, Moduli Space, Entropy Function, Thermodynamics, Geometrization of Language Model, Boltzmann Manifold.

 




§ INTRODUCTION


In November 2022, OpenAI surprised the world with the release of the highly “intelligent” ChatGPT, which possesses many striking abilities that far exceed any other available systems <cit.>. For example, it can answer questions with a very high accuracy and engage in human-like conversations. It is the first time that a computer performs well enough in human languages that makes people wonder whether it acquires true intelligence. Ever since then, the arm race in the area of Large Language Model (LLM) has become extremely fierce, with the announcements of new LLMs with hundreds of billions of parameters regularly. Now there are abundant Chatbots based on LLMs available in markets, many of which have astonishing new abilities and behaviors <cit.>. Especially, when the number of parameters (e.g., weights) are increased to a threshold, then some surprising behaviors emerges <cit.>, which is extremely intriguing. 

An LLM is a generative model, which means it can practically sample a new sentence given a prompt. More precisely, suppose we are given an arbitrary prompt w_1⋯ w_n, where each w_i is a word, an LLM can compute the probability distribution of the next word, i.e., P(w_n+1|w_1 ⋯ w_n). Next, we sample the next word w_n+1 according to this probability distribution <cit.>. After we obtain w_n+1, we repeat this process and sample w_n+2 according to the probability distribution P(w_n+2|w_1 ⋯ w_n w_n+1) computed by the LLM. We repeat this process until some conditions are met, e.g., the LLM outputs an end-of-sequence symbol.

Let us now briefly review the basic structures of the LLMs based on the transformer architecture, to the extent that will be needed in this paper. For a more detailed and thorough treatment, the readers are referred to the book <cit.>. First, in order for a computer to be able to compute words, words are first embedded into a vector space, which is called the Word-Embedding <cit.>. Suppose 𝒲 is the finite set of all words, a Word-Embedding is an injective map

    ι: 𝒲→ℝ^N,

where the standard inner product ⟨·, ·⟩ on ℝ^N measures the correlations between words. Given an arbitrary vector 𝐯∈ℝ^N, it defines a distribution on 𝒲 via

    P_𝐯(w) = exp( ⟨𝐯, ι(w) ⟩)/∑_w' ∈𝒲exp( ⟨𝐯, ι(w') ⟩),

which is called the Boltzmann distribution in statistical mechanics <cit.>. 


In particular, given a word w ∈𝒲, it also gives us a distribution on 𝒲 via

    P_ι(w)(w') = exp( ⟨ι(w), ι(w') ⟩)/∑_w”∈𝒲exp( ⟨ι(w), ι(w”) ⟩),

the value of which measures the correlations between w and w'. In practice, the embedding ι is chosen such that P_ι(w)(w') is the probability that w' is in an l-neighborhood of w <cit.>. One nice property of this embedding is that generally the embedded vectors {ι(w)| w ∈𝒲} lie in a lower dimensional vector subspace of ℝ^N as these vectors are not independent from each other. A famous linear relation is 

    ι(Queen) - ι(Woman) = ι(King) - ι(Man).

For more details, the readers are referred to the papers <cit.>.

After the Word-Embedding, a sentence w_1⋯ w_n is mapped to a vector

    (ι(w_1),⋯,ι(w_n)) ∈ (ℝ^N)^n.

As the name suggests, a transformer is a transformation

    T_n: (ℝ^N)^n → (ℝ^N)^n,

which is a differentiable often smooth map <cit.>. For the details of the construction of a transformer and some of the ideas behind it, the readers are referred to the paper <cit.> and the book <cit.>. In particular, each transformer depends on internal parameters which are tuned during the training process. Notice that transformers can be nested in the form

    𝒯_n = T_n,k∘⋯∘ T_n,1,

where k is the number of layers and different layers have different parameters. Let Proj be the projection of (ℝ^N)^n to the last ℝ^N in the direct product, hence Proj∘𝒯(ι(w_1),⋯,ι(w_n)) is a vector of ℝ^N, which defines a distribution on 𝒲 via Eq. (<ref>). The training of the transformers on a very large corpse is to make sure that this distribution is a good approximation of P(·|w_1⋯ w_n) <cit.>.

The LLMs have made rapid advance from a numerical and experimental point of view, but there have been very few papers that attempt to understand the properties and behaviors of LLMs from a analytical point of view. More precisely, we now desperately need a theory that can explain these surprising properties and behaviors, which is the main motivation of this paper. 


In this paper, we first use set theory, analysis and probability theory to rigorously define the concept of Causal Language Model (CLM), and show it is equivalent to the Predicative Language Model (PLM), which basically predicts the next word of a prompt. We formulate a generalized distributional hypothesis for a CLM based on functional analysis and topology. We introduce the concept of the moduli space of distributions on words, which is the key metric space to understand the power and properties of a language model.

The development of human language is for human to exchange information efficiently between individuals. In order to analytically study a language model, we must understand how information is quantified during conversations. In this paper, we introduce the entropy function for CLM and PLM, and show how it can play a crucial role in the study of language models. In particular, we analyze the zero points of the entropy function, and show how to use information theory to understand the meaning of them. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. We also formulate an extension conjecture about the properties of AGI using the entropy function.

Next, we use thermodynamics to give an physical interpretation of the language models. In this interpretation, a sentence is a microstate that has a potential energy. All the possible outputs of a prompt define a statistical ensemble, which is distributed according to the Boltzmann distribution <cit.>. We introduce the concept of the partition function, internal energy and Helmholtz free energy for this statistical ensemble associated. Using the Helmholtz free energy, we show how the game of word-predicting can be interpreted as the process of a molecule growing in physics.

Later, we formulate the concept of the geometrization of a language model, and show that the accuracy of a language model is determined by how precise the embedding of the moduli space in a geometrization is. We also define the Word-Embedding with respect to a general manifold with a pairing structure, and introduce the corresponding Boltzmann manifold with respect to it. At last, we show how the current LLMs can be viewed as special cases of the geometrization of language models that use the linear space together with its inner product. Based on our theory, we also pose several important open questions, e.g., which manifold together with a pairing is optimal for the geometrization of language models. Perhaps different languages would need different manifolds and pairings, namely different Boltzmann manifolds in the geometrization. Another important direction is whether the current researches on statistical mechanics and differential geometry can offer new tools in the study of LLMs.


The layout of this paper is as follows. In Section <ref>, we use set theory, analysis and probability theory to rigorously define language models, i.e., CLM and PLM, and we introduce the moduli space of distributions associated with a language model. In Section <ref>, we introduce the entropy function, and show how information theory can be applied to the study of language models. In Section <ref>, we present a thermodynamic interpretation of language models and define statistical functions such as partition function, internal energy and Helmholtz free energy for a language model. In Section <ref>, we formulate the concept of the geometrization of language models, and define the Boltzmann manifold. In Section <ref>, we show that the current LLMs are special cases of the geometrization of language models. In Section <ref>, we conclude this paper and propose several open problems.









§ THE LANGUAGE MODEL AND MODULI SPACE
 



In this section, we will use set theory, analysis and probability theory to give rigorous definitions of language models. We first define what is a general sentence in the sense of set theory <cit.>. Then we introduce the definitions of the General Language Model (GLM), the Causal Language Model (CLM) and the Predicative Language Model (PLM). At last, we define what is called the moduli space of distributions for a PLM, which is the crucial metric space to understand the properties and behaviors of language models.



 §.§ Sentence as a Free Mathematical Sequence?


Suppose 𝒲 is the finite set of all words in a language, the set of all sentences associated with 𝒲 is defined in the same way as in set theory and logic <cit.>. First, let ∅ be the unique sentence with no word, whose length is by definition 0. Let 𝒮_0 be the set that consists of a unique element ∅

    𝒮_0 = {∅}.


The inclusion of ∅ as a sentence will make our analysis simpler from a mathematical point of view. While later we will also see ∅ is a suitable symbol for the end (or even the beginning) of a sentence.

A sentence of length 1 is just a word of 𝒲. Let 𝒮_1 be the set of all sentences of length 1, namely

    𝒮_1 = 𝒲.

Given an positive integer n ≥ 2, let 𝒮_n be the set of all sentences of length n, i.e., an element of 𝒮_n is a sequence

    𝐬_n=w_1 w_2 ⋯ w_n with w_i ∈𝒲.

Here, the words in a sentence do not need to be different from each other, e.g., we allow the case where w_i=w_j, i ≠ j. The set of all sentences of finite lengths is by definition the union of all 𝒮_n

    𝒮 = ⋃_n=0^∞𝒮_n.

Notice that here we allow the length of a sentence to be arbitrarily large. 

In the definition of 𝒮, we have allowed the existences of arbitrary sentences that do not satisfy grammar rules. In this paper, the grammar is instead considered as part of the language model.



The cardinality of 𝒮 is countably infinite, i.e., there exists a bijective map between 𝒮 and ℕ.


Since 𝒲 is a finite set, this lemma is an immediate result of elementary set theory, see the book <cit.> for more details.



Even in this most general setting, we do not consider the case of sentence of infinite length, i.e., infinite sequence, which is very important from both practical and set-theoretic points of view. Practically, human brain has not evolved to understand infinity properly, so it makes no sense to a mortal that a sentence is infinite. Set-theoretically, the set of all infinite sequences has the same cardinality as the real numbers, i.e., its cardinality is ℵ_1 <cit.>. So we do not really want to walk into the deep water of axiomatic set theory.


There is a natural operation defined on 𝒮 that is called concatenation: given a sentence s_m ∈𝒮_m and a sentence s_n ∈𝒮_n, their concatenation is a new sentence s_m s_n that lies in 𝒮_m+n. As a convention, the concatenation between ∅ and s_n is s_n itself, i.e.,

    ∅𝐬_n = 𝐬_n ∅ = 𝐬_n.

This concatenation operation is associative, but not commutative, while the empty sentence ∅ serves as a two-sided identity. So the set 𝒮 has a monoid structure <cit.>.








 §.§ The Causal Language Model


Now we are ready to give a more mathematical and general definition of the language model.


 
Given an arbitrary sentence s ∈𝒮, a General Language Model (GLM) ℒ_G defines a joint probability distribution P_ℒ_G,𝐬 on 𝒮×𝒮, where the probability P_ℒ_G,𝐬(𝐬_1,𝐬_2) measures the likelihood of the concatenation 𝐬_1𝐬𝐬_2 among all possible choices. Namely, P_ℒ_G,𝐬(𝐬_1,𝐬_2) is the probability that there is a sentence 𝐬_1 before 𝐬 and a sentence 𝐬_2 after 𝐬.



Now, we consider the case where the sentence 𝐬_1 on the left side of 𝐬 is ∅, in which case the probability distribution in Definition <ref> defines a distribution on 𝒮. Hence it motivates the following definition.


Given an arbitrary sentence 𝐬∈𝒮, a Causal Language Model (CLM) ℒ defines a probability distribution on 𝒮. Namely, given a prompt 𝐬, we have a distribution P(·|𝐬) such that the probability that ℒ outputs a sentence 𝐬' is P_ℒ(𝐬'|𝐬).


In the definition, we allow the prompt 𝐬 to be ∅, in which case the CLM ℒ defines a probability distribution on 𝒮, i.e., P(𝐬_out|∅). It is called causal because 𝐬' is after 𝐬, so there is a causality between 𝐬 and 𝐬'. From the definition, the input to ℒ can be of arbitrary length, while the output sentence can also be of arbitrary length. In fact, such a definition of CLM is extremely powerful.




Let ℬ(𝒮,ℝ) be the set of all functions with finite absolute summation, i.e.,

    ℬ(𝒮,ℝ) = {f:𝒮→ℝ| ∑_𝐬∈𝒮|f(𝐬)| < ∞}.

The set ℬ(𝒮,ℝ) is a metric space with a metric given by

    ‖ f-g ‖  =  ∑_𝐬∈𝒮 | f(𝐬) - g(𝐬) |.

For more details about this space, the readers are referred to the book <cit.>. Clearly for every 𝐬, the distribution P_ℒ(·|𝐬) is a non-negative function that lies in ℬ(𝒮,ℝ), in fact ‖ f ‖ = 1. Therefore, there exists a map

    ρ_ℒ: 𝒮→ℬ(𝒮,ℝ).


In 1950s, linguists formulated the distributional hypothesis, which basically says that the meaning of words are determined by their contexts <cit.>. We generalize this hypothesis using real analysis.

If ℒ is an intelligent language model, then the map ρ_ℒ is an embedding that is also discrete. Namely, each point ρ_ℒ(𝐬) has a neighborhood that does not contain any other ρ_ℒ(𝐬') where 𝐬≠𝐬'. 

For more details of the meaning of “discrete” in the sense of topology, the readers are referred to the book <cit.>. As a result, the embedding ρ_ℒ induces a metric on 𝒮.



It should be stressed that in the definition of a CLM, we do not require it to have “intelligence” behaviors. For example, for a given prompt 𝐬, the probability distribution P(· | 𝐬) can be a “random” distribution over 𝒮, which can even outputs sentences that are quite pathological. Hence, intelligence is not incorporated in the definition of the language model. Instead, only highly special language models can exhibit intelligence behaviors.


 







 §.§ The Predicative Language Model
 

Now we give the definition of the Predicative Language Model.


Given a sentence 𝐬, a Predicative Language Model (PLM) ℒ_P defines a probability distribution P(·|𝐬) on 𝒲∪∅. Here, P(∅|𝐬) is the probability that ℒ_P outputs nothing and P(w|𝐬) is the probability that ℒ_P outputs the word w ∈𝒲.


We have the following lemma.

The definition of the PLM is equivalent to the definition of the CLM.


If we have a PLM ℒ_P, then by repetitively using it we can generate sentences. More precisely, given an input 𝐬∈𝒮, we can generate a sentence 𝐬'=w_1w_2⋯ w_n until ℒ_P outputs ∅ via

    ℒ_P(𝐬) = w_1,  ⋯,  ℒ_P(𝐬w_1 ⋯ w_n-1) = w_n,  ℒ_P(𝐬w_1 ⋯ w_n) = ∅.

The probability that ℒ_P outputs 𝐬'=w_1⋯ w_n given 𝐬 is 

    P(𝐬'|𝐬)=P(w_1|𝐬) · P(w_2|𝐬w_1) ·…· P(w_n|𝐬w_1 ⋯ w_n-1) ·  P(∅|𝐬𝐬').

In this way, the PLM ℒ_P defines a probability distribution on 𝒮 for any prompt 𝐬, hence it produces a CLM.

We now show the inverse construction from CLM to PLM. Namely, suppose we have a CLM ℒ, we now construct a PLM ℒ_P that reverses the process in the previous paragraph. Given a prompt 𝐬, let us construct a distribution on 𝒲∪∅. First, P(∅|𝐬) is straightforward to define, i.e., P_ℒ(∅|𝐬). For an arbitrary word w, let 
    P(w|𝐬) = ∑_𝐬' ∈𝒮 P_ℒ(w𝐬'|𝐬).

Notice that here the sum is just over all sentences whose first word is w. In this way, we obtain a PLM.



As shown in the proof of this lemma, the empty sentence ∅, which is the unit of the monoid 𝒮, plays the role of ending the sentence outputted by ℒ_P. Hence comes a crucial definition in our treatment of PLM.

 
The space of all possible distributions on 𝒲∪{∅} that come from a PLM ℒ_P is called the moduli space of distributions of ℒ_P, which is expressed as

    ℳ = { P(·| 𝐬)| 𝐬∈𝒮}.


In fact, this moduli space is a metric space. Let ℬ(𝒲∪{∅},ℝ) be the set of all functions with finite absolute summation

    ℬ(𝒲∪{∅},ℝ) = {f:𝒲∪{∅}→ℝ| ∑_w ∈𝒲∪{∅}|f(w)| < ∞}.

The set ℬ(𝒲∪{∅},ℝ) is a metric space with a metric given by

    ‖ f-g ‖  =  ∑_w∈𝒲∪{∅} | f(w) - g(w) |.

For more details, the readers are referred to the book <cit.>. Clearly, for every 𝐬∈𝒮, the distribution P(·|𝐬) ∈ℳ is a non-negative function that lies in ℬ(𝒲∪{∅},ℝ), in fact ‖ P(·|𝐬) ‖ = 1. Therefore, ‖·‖ induces a metric d on the moduli space ℳ

    d: ℳ×ℳ→ℝ.


The moduli space ℳ together with the metric d is the key object to understand a PLM.








§ THE ENTROPY FUNCTION
 


In this section, we introduce the concept of the entropy function and show how it quantifies the information contained in the output for a given prompt. We will study the zero-points, which are also the minimal points of the entropy function, and discuss their crucial properties. We argue that these zero points and the points where the entropy is close to 0 are the key obstacles for a LLM to approximate an intelligent language model. At last, we formulate a conjecture regarding AGI.




 §.§ Information Flow and Entropy


The entropy function for a CLM ℒ is defined just as in statistical mechanics and information theory <cit.>.


Given a prompt s ∈𝒮, the entropy of 𝐬 with respect to a CLM ℒ is given by

    S_ℒ(𝐬)= - ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|𝐬) log P_ℒ(𝐬'|𝐬),

where the sum is over all possible outputs 𝐬'.

We will see that the entropy function is a key mathematical concept to analyze the behaviors of language models. From the definition, the entropy function is a non-negative function defined on 𝒮

    S_ℒ: 𝒮→ℝ.

As a special example, if the prompt is ∅, i.e., no prompt, the entropy associated with it has a special name which is stated as a definition.

The background entropy of a CLM ℒ is

    S_ℒ(∅) = - ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|∅) log P_ℒ(𝐬'|∅).


The background entropy S_ℒ(∅) is a quantity that gives a zeroth order measurement about how complicated a language is. If S_ℒ(∅) is very large, this intuitively mean that the CLM is quite diverse. While if S_ℒ(∅) is very small, it intuitively means that the CLM is very rigid.


Since the development of language is for individuals to exchange information, so a language model itself must be able to demonstrate this. Here we argue that the definition of the entropy function exactly quantify the information during this process <cit.>. More precisely, given a prompt 𝐬, the entropy S_ℒ(𝐬) measures how vague are the answers to the prompt 𝐬 with respect to this CLM ℒ. If S(𝐬) is very large, this means that the answers can be very diverse. On the other hand, if S(𝐬) is very small, this means the answer to 𝐬 is very rigid. Therefore, after the CLM outputs 𝐬', we can say the information revealed by 𝐬' is S_ℒ(𝐬) <cit.>.


Just like statistical mechanics, we argue that the most important function for a CLM is the entropy function <cit.>.








 §.§ The Zero Points of the Entropy Function






Let us now look at the zero points of S_ℒ. If S_ℒ(𝐬)=0, then it means that the answer to the prompt 𝐬 is unique in the CLM ℒ, i.e., there exists a unique sentence 𝐬' ∈𝒮 such that P_ℒ(𝐬'|𝐬) = 1. Hence there is the interpretation that the CLM ℒ detects that the answer to the prompt 𝐬 is unique, and thus the CLM itself actually contain this information. For example, suppose the prompt 𝐬 is

    𝐬 =

If the entropy function of an intelligent CLM ℒ vanishes at 𝐬, i.e., S_ℒ(𝐬) = 0, then this means there is only one possible output of ℒ for the prompt 𝐬. Of course, an intelligent ℒ that knows about human history will output 

    𝐬' =

with probability 1. It should be noticed that at the point 𝐬, the vanishing of the entropy only implies there is only one possible output to 𝐬. Another CLM ℒ_1 could still have S_ℒ_1(𝐬)=0, but it instead outputs 

    𝐬” = .

We say the CLM ℒ_1 outputs a `wrong' answer because Gauss is a mathematician, so the information given by ℒ_1 is wrong. But `wrong' information is still information, the entropy function itself only quantifies the amount of information, but it cannot detect whether the information contained is right or wrong, which needs to be checked with what happens in the outside world. 

Given a CLM ℒ, a zero point 𝐬 of the entropy function S_ℒ is called a singularity of ℒ.

Hence the conclusion is that a CLM memorize information at its singularities. We argue that the existence of these singularities and the points where the entropy function is very small are the key obstacle for an LLM to approximate an intelligent CLM. 






 §.§ An AGI Conjecture
 


It is hard to define which CLM is Artificial General Intelligence (AGI) mathematically! Our idea is that if a CLM ℒ is an AGI, then it must satisfy certain mathematical properties. In this paper, we formulate a conjecture about this.








 
If a CLM is an AGI, then there exists a finite subset 𝒮_AGI⊂𝒮 such that the entropy function S_ℒ: 𝒮→ℝ is determined by its value on 𝒮_AGI.

The motivation of this conjecture comes from modern mathematics. In modern mathematics, we first give a finite set of definitions and axioms, and then mathematics is about proof, which uses these finite set of rules of prove a result that is not known a priori. The key motivation for Conjecture <ref> is that the value of S_ℒ on 𝒮 is determined using mathematical deductions by its value on the finite set 𝒮_AGI <cit.>. 




§ A THERMODYNAMIC INTERPRETATION OF LANGUAGE MODELS
 

In this section, we will give a thermodynamic interpretation of the CLM defined in Section <ref>. We show how the concepts and properties of the CLM naturally admit physical interpretations. Then we will define thermodynamic functions such as internal energy, Helmholtz free energy on CLM, and show how they help us understand phenomena in language models. At last, we give a physical interpretation to how a sentence “grows” in a PLM.




 §.§ The Boltzmann Distribution


First, let us introduce the Boltzman distribution in statistical mechanics <cit.>. Suppose we are given a physical system that can occupy different microstates, and suppose the energy of the microstate i is E_i. Let k be the Boltzmann constant and T be the temperature, then the probability that the system occupies  microstate i satisfies

    p_i ∝exp( - E_i/kT).

Let the partition function of the system be

    𝒵= ∑_iexp( - E_i/kT),

where the sum is over all possible microstates. Then p_i is just

    p_i = 1/𝒵·exp( - E_i/kT),

which is called the Boltzman distribution. The Boltzmann distribution is the distribution that maximizes the entropy

    S = - ∑_i p_i log p_i

subject to the constraint ∑ p_i =1 and ∑_i p_i E_i is a constant, i.e., the average energy is kept fixed.



 §.§ A Thermodynamic Interpretation
 

We now give a thermodynamic interpretation of CLM. 

In a CLM ℒ, a sentence is a microstate and for every sentence 𝐬∈𝒮, there is an energy E_ℒ(𝐬) associated with it.

The energy of ∅ is called the vacuum energy, and the energy of a word w ∈𝒲 is called its self-energy <cit.>.


As in physics, a word w ∈𝒲 itself can have self-interaction, it has an internal energy E(w), the meaning of this internal energy is that given a random sentence 𝐬, the probability that the first word of 𝐬 is w is proportional to exp(-β E(w) ). Here β is the inverse of temperature.


Given a length-2 sentence w_1w_2, its energy E_ℒ(w_1w_2) is the interaction between the two words w_1 and w_2, and E_ℒ(w_1w_2) is lower if the interaction between w_1 and w_2 is stronger. 

Since a language is essentially a discrete object, a word does not have a continuous degree of freedom. Therefore, there is no kinetic energy for a word, and only potential energy exists when words form a sentence. 

Here comes the definition of the statistical ensemble in a language model.

For a given prompt 𝐬, the statistical ensemble associated with it is 

    {𝐬𝐬'|𝐬' ∈𝒮}.



For later purpose, we introduce another quantity β which is the inverse of temperature <cit.>. For a fixed CLM, the value of β is usually fixed to be 1. 

In a CLM ℒ, for a given prompt 𝐬 the probability that ℒ outputs 𝐬' satisfies

    P_ℒ(𝐬'|𝐬) ∝exp(-β E_ℒ(𝐬𝐬') ),

where E_ℒ(𝐬𝐬') is the energy of the concatenation 𝐬𝐬'.   


More precisely, for a given prompt 𝐬, the probability that ℒ outputs 𝐬' is

    P_ℒ(𝐬'|𝐬) = 1/𝒵_ℒ·exp( - β E_ℒ(𝐬𝐬') ),

where E_ℒ(𝐬𝐬') is the energy of the concatenation 𝐬𝐬' and 𝒵_ℒ is the partition function associated with 𝐬

    𝒵_ℒ(𝐬) = ∑_𝐬' ∈𝒮exp( - β E_ℒ(𝐬𝐬')).

The partition function is in fact a thermodynamic function defined on 𝒮

    𝒵_ℒ:𝒮→ℝ.

 

The average energy of the statistical ensemble associated with 𝐬 is 

    U_ℒ (𝐬) = ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|𝐬) E_ℒ(𝐬𝐬'),

which is called the internal energy of the ensemble. The internal energy U_ℒ is also a well-defined function on 𝒮

    U_ℒ : 𝒮→ℝ.

But notice that U_ℒ is different from E_ℒ: E_ℒ(𝐬) is the energy of the microstate 𝐬, while U_ℒ(𝐬) is the average energy of the statistical ensemble associated with 𝐬. Recall that the entropy for the statistical ensemble associated to 𝐬 is

    S_ℒ(𝐬)=-  ∑_𝐬' ∈𝒮 P_ℒ(𝐬'|𝐬) log  P_ℒ(𝐬'|𝐬) ,

which is simplified to

    S_ℒ = β U_ℒ + log𝒵_ℒ.

The Helmholtz free energy is given by

    F_ℒ = U_ℒ - 1/β S_ℒ=- 1/βlog𝒵_ℒ.

The Helmholtz free energy determines whether a sentence 𝐬 is likely to be followed by another sentence or not.







 §.§ A Physical Picture of How a Sentence Grows



Let us now interpret the process of predicting the next word in a PLM using how molecule grows in statistical mechanics. First, because of the equivalence between CLM and PLM, the interpretation in Section <ref> admits immediate generalization to PLM. In this physical picture, a word is view as an atom, and a sentence is a sequence of atoms, whose energy depends on the order of these atoms. Suppose we are given a prompt 𝐬∈𝒮_n, which is composed of n atoms. To predict the next word is to add an atom to 𝐬. For a word w ∈𝒲∪{∅}, the energy of 𝐬w is E(𝐬w), and the internal energy is by definition

    ∑_w ∈𝒲∪{∅} P(w|𝐬) E(𝐬w).

We also have the entropy

    - ∑_w ∈𝒲∪{∅} P(w|𝐬) log P(w|𝐬).

There are two trends that fight each other

    
  * the internal energy (Eq. (<ref>)) wants to be as small as possible,
    
  * the entropy (Eq. (<ref>)) wants to be as large as possible.

As a result, the Helmholtz free energy (Eq. (<ref>)) is minimized, and the resulting distribution is the Boltzmann distribution. After sampling the next word, say w_n+1, we can continue this process with the new sentence 𝐬w_n+1 and sample its next one. This process is highly similar to the process of how a molecule grow in statistical mechanics!


But there remains the question of how to compute the energy of a sentence. A sentence can be very long, hence the computation of its energy is a typical many-body problem. The power of LLM is to approximately compute the interaction energy between the “atoms” in a sentence using machine learning.




§ THE GEOMETRIZATION OF LANGUAGE MODELS
 

In this section, we will introduce the concept of the geometrization of a PLM, which lays the foundation for studying language models using differential geometry. We will also define a more general version of the Word-Embedding, which leads to the concept of a Boltzmann manifold.





 §.§ The Geometrization of PLM


Suppose we have a PLM ℒ_P, whose moduli space of distributions is ℳ (Definition <ref>). Moreover, we have a map

    ρ: 𝒮→ℳ,

where ρ sends a sentence 𝐬∈𝒮 to the distribution P(·| 𝐬) ∈ℳ. Notice that ℳ is a metric space, the details of which can be found in Section <ref>.

Generally, the moduli space ℳ itself is extremely complicated, to study which we need to borrow ideas from both mathematics and physics. The first step is to geometrize this complicated metric space, which “prompts” the following definition.


The ϵ-geometrization of a PLM is a manifold M and a continuous map

    Ψ: ℳ→ M,

where each point of M represents a probability distribution on 𝒲∪∅ that is also computable. Moreover, for every point P(·| 𝐬) of ℳ, the norm of the difference ‖Ψ(P(·| 𝐬))-P(·| 𝐬) ‖ < ϵ(n). Here, the error ϵ(n) can depend on the length of the sentence 𝐬∈𝒮_n.

Notice that the norm ‖·‖ is defined in the metric space ℬ(𝒲∪{∅},ℝ) (Eq. (<ref>)). The composition Ψ∘ρ is a map from 𝒮 to M

    Ψ∘ρ: 𝒮→ M.

The motivation of the geometrization of the moduli space ℳ is that we try to find a finite dimensional manifold M to approximately describe the infinite set ℳ. Then we use the geometry of this manifold to study the bizarre metric space ℳ.




 §.§ The Word-Embedding and the Boltzmann Manifold
 

One way to construct an ϵ-geometrization is via an additional structure on a manifold called the pairing. Suppose M is a manifold, a symmetric pairing is a smooth function 

    ⟨·, ·⟩ : M × M →ℝ,

which maps any two points pt_1, pt_2 ∈ M to a number in ℝ. Here “symmetric” means

    ⟨pt_1, pt_2 ⟩ = ⟨pt_2, pt_1 ⟩.

A Word-Embedding is an injective map

    ι: 𝒲∪{∅}→ M.

Given an arbitrary point pt∈ M, it defines a distribution P_pt on 𝒲∪{∅} via

    P_pt(w) =  exp( - β⟨ pt,ι(w) ⟩)/∑_w' ∈𝒲∪∅exp( - β⟨ pt,ι(w') ⟩),

which is exactly from the Boltzmann distribution <cit.>. Hence every point of M defines a distribution on 𝒲∪{∅} and M is a space of distributions on 𝒲∪{∅}.


The manifold M together with the pairing ⟨·,·⟩ and the Word-Embedding ι is called a Boltzmann manifold. 


Since every point of the Boltzmann manifold M defines a distribution on 𝒲∪{∅}, M is a subspace of  ℬ(𝒲∪{∅},ℝ) (Eq. (<ref>)). Therefore, given two points pt and pt' of M, their distance is

    d(pt,pt') =   ‖ P_pt-P_pt'‖.

This distance function induces a metric g_B on M, which is called the Boltzmann metric, hence M is in fact a Riemannian manifold <cit.>.

Now suppose M gives us an ϵ-geometrization with the Moduli-Embedding Ψ: ℳ→ M. Since 𝒮_n is the direct product 𝒲^n, thus if we believe the map Ψ∘ρ has sufficiently nice properties, then there exists a diagram

    𝒮_n = 𝒲^n @>ι^n>> M^n
    
    @VVρV @VVΛ_nV
    ℳ @>Ψ>> M
      ,

which is commutative up to a small error that depends on n. More precisely, we assume there exists a smooth map Λ_n:M^n → M such that for every 𝐬=w_1w_2⋯ w_n, we have

    Ψ∘ρ (𝐬) = Λ_n( ι(w_1), ⋯, ι(w_n) ).

It means that the map ρ∘Ψ is determined by the map ι and {Λ_n:n ≥ 2 }. The point is that the map Ψ∘ρ is defined on the infinite set 𝒮, which can be highly complicated. While the map ι is defined on a finite set and Λ_n is defined on a smooth manifold, hence can be studied by the modern theory of differential geometry <cit.>.









§ LLM AS A SPECIAL GEOMETRIZATION
 

In this section, we show that the current LLMs based on the transformer architecture are special cases of geometrization in the sense of Section <ref>. We also introduce the fundamental thermodynamic relation of LLMs.




 §.§ The Geometrization and LLM



The Boltzmann manifold used in LLMs is the simplest one: ℝ^N together with the standard inner product ⟨· , ·⟩. In practice, first we obtain a Word-Embedding  <cit.>

    ι: 𝒲∪{∅}→ℝ^N.

Then a point 𝐯 of ℝ^N defines a distribution on 𝒲∪{∅} via 

    P_𝐯(w) = exp(-β⟨𝐯, ι(w)⟩)/∑_w' ∈𝒲∪{∅}exp(-β⟨𝐯, ι(w')⟩),

where β is the inverse of temperature. The Moduli-Embedding is a map

    Φ: ℳ→ℝ^N.

Given a sentence 𝐬∈𝒮, Ψ∘ρ(𝐬) is a vector of ℝ^N, which gives the distribution of the next word of 𝐬 via Eq. (<ref>), i.e.,

    P(w|𝐬) = exp(-β⟨Ψ∘ρ(𝐬), ι(w)⟩)/∑_w' ∈𝒲∪{∅}exp(-β⟨Ψ∘ρ(𝐬), ι(w')⟩).




For every n ≥ 1, there exists a map Λ_n: (ℝ^N)^n →ℝ^N that makes the following diagram commutative up to a small error

    𝒮_n = 𝒲^n @>ι^n>> (ℝ^N)^n
    
    @VVρV @VVΛ_nV
    ℳ @>Ψ>> ℝ^N
    .


Namely, given a sentence 𝐬=w_1⋯ w_n, the distribution of the word after 𝐬 is given by the vector Λ_n(ι(w_1),⋯,ι(w_n)) via Eq. (<ref>). In practice, both ι and Λ_n are determined by data and machine learning <cit.>. Currently, the most popular method to construct the map Λ_n is based on the transformer architecture <cit.>.





 §.§ The Fundamental Thermodynamic Relation of LLMs


Given a geometrization (ℝ^N, ⟨·,·⟩, ι), the properties of w ∈𝒲∪{∅} with respect to this geometrization are determined by the embedding vector ϕ(w) ∈ℝ^N. Borrowing language from physics <cit.>, the space ℝ^N is also called the phase space of a language model. Just as in CLM, in a PLM we also have the entropy function S, the internal energy U and Helmholtz free energy F defined on 𝒮.



Suppose we are given a geometrization (ℝ^N, ⟨·,·⟩, ι) and the maps {Λ_n| n≥ 1 } associated with a PLM. The finite set 𝒮_n is viewed as a subspace of (ℝ^N)^n via the embedding ι^n. Then all thermodynamic functions such as S, U and F defined over 𝒮_n admit smooth extensions to functions over (ℝ^N)^n up to small errors.

Let us now look at the entropy function S as an example. This conjecture implies that there exists a differentiable function S: (ℝ^N)^n →ℝ that makes the following diagram commutative up to a small error

    𝒮_n = 𝒲^n @>ι^n>> (ℝ^N)^n
    
    @VVρV @VVSV
    ℳ @>S>> ℝ.




In fact, from the construction of LLMs using the architecture of transformer, the extensions of thermodynamic functions always exist, but the error cannot be guaranteed to be small.


Suppose the coordinate of (ℝ^N)^n is x_i,j with 1 ≤ i ≤ N and 1 ≤ j ≤ n, then we have the fundamental thermodynamic relation of LLMs

    dU = 1/β dS -∑_i,jf_i,jdx_i,j,

where f_i,j is the generalized force given by

    f_i,j=∂S/∂ x_i,j.




As a function on (ℝ^N)^n, we conjecture that the entropy function S for a LLM is convex. 

Suppose 0 ≤λ≤ 1, given two points 𝐱=(x_i,j) and 𝐱'=(x'_i,j) of (ℝ^N)^n, we have

    S(λ𝐱 + (1 - λ) 𝐱' ) ≥λS( 𝐱) + (1 - λ ) S(𝐱' ).


It has an intuitive interpretation, if you mix the two vectors 𝐱=(x_i,j) and 𝐱'=(x'_i,j), the entropy is always greater than the average. Namely, if you mixed the meanings, it always becomes more vague. 








§ CONCLUSION AND OPEN PROBLEMS
 

In this paper, we have used set theory, analysis and probability theory to rigorously give the definitions of Causal Language Model (CLM) and Predicative Language Model (PLM). We propose a generalization of the distributional hypothesis. We introduce the concept of the moduli space of distributions for a CLM, which is the key metric space to understand the power and properties of a language model. Then we define the entropy function for a CLM, study its properties, and show how its zero points are crucial for a language model to incorporate information and logic. Using the entropy function, we formulate an extension conjecture about AGI.

We have used thermodynamics to show that language models admit thermodynamic interpretations. Namely, a sentence is a microstate and a prompt gives us a statistical ensemble, which is distributed according to the Boltzmann distributions. We also formulate the concept of the geometrization of language models and define the Boltzmann manifold. We show the embedding of the moduli space into a Boltzmann manifold determines how accurate can a geometrization be. At last, we discuss how the current LLMs can be viewed as special cases of geometrization with respect to the linear space together with its inner product.

There are many interesting open questions that deserve further study. Perhaps the most straightforward question is what is the best Boltzmann manifold for the geometrization of a language model. Current LLMs all uses the linear space together with its inner product. But it is likely that different languages would need different Boltzmann manifold for the geometrization. For example, some of the possible choices of Boltzmann manifold with non-trivial topology can be

    ℝ^N × S^M, ℝ^N × (S^1)^M,

where S^k is a k-dimensional sphere. For such manifold, it is very interesting to see whether the non-trivial topology has any effect on the properties of the resulting LLM.

Another important question is whether results in modern physics and mathematics can be applied to study language models. For example, whether statistical physics can help us understand the emergent abilities of LLMs. Whether the results in statistical mechanics and differential geometry can offer a better architecture than the transformer. There are numerous results in the study of many-body interaction systems, it is interesting to see whether these results can have applications to LLMs.







plain





 Paper Title: Persistent Topological Negativity in a High-Temperature Mixed-State.






Department of Physics, University of California, Santa Barbara, CA 93106, USA	


Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106


Department of Physics, University of California, Santa Barbara, CA 93106, USA	

§ ABSTRACT 
We study the entanglement structure of the Greenberger–Horne–Zeilinger (GHZ) state as it thermalizes under a strongly-symmetric quantum channel describing the Metropolis-Hastings dynamics for the d-dimensional classical Ising model at inverse temperature β. This channel outputs the classical Gibbs state when acting on a product state in the computational basis.  When applying this channel to a GHZ state in spatial dimension d>1, the resulting mixed state changes character at the Ising phase transition temperature from being long-range entangled to short-range-entangled as temperature increases. Nevertheless, we show that the topological entanglement negativity of a large region is insensitive to this transition and takes the same value as that of the pure GHZ state at any finite temperature β>0. We establish this result by devising a local operations and classical communication (LOCC) “decoder" that provides matching lower and upper bounds on the negativity in the thermodynamic limit which may be of independent interest. This perspective connects the negativity to an error-correction problem on the (d-1)-dimensional bipartitioning surface and explains the persistent negativity in certain correlated noise models found in previous studies. Numerical results confirm our analysis. 


 Persistent Topological Negativity in a High-Temperature Mixed-State
    Sagar Vijay
    February 2024
===================================================================



Introduction: The nature and fate of quantum entanglement in mixed-states of quantum many-body systems – arising in a description of thermal quantum matter  or after a pure state experiences local decoherence – have been recently investigated in order to understand the universal quantum correlations that can persist in open quantum systems <cit.>. Patterns of long-range-entanglement (LRE) in mixed quantum matter are intricately related to quantum many-body phases which can act as quantum error-correcting codes <cit.>.  Thermal states of quantum many-body systems are also now known to be efficiently classically-preparable at sufficiently high temperatures <cit.>.

The entanglement negativity provides a well-studied diagnostic of bipartite entanglement in a mixed-state, which vanishes if the state can be prepared by from a product state by local operations and classical communication (LOCC) across the entanglement bipartition <cit.>.  While the negativity of a region A grows with the area of its boundary in a generic Gibbs state <cit.>, contributions to the negativity which are independent of coarse-grained details of the bipartitioning surface signal the presence of LRE in a mixed-state and can probe the presence of mixed-state topological quantum order <cit.>.  Diagnostics of non-local mixed-state entanglement, such as the topological entanglement negativity <cit.>, are challenging to study analytically, and the subtle behavior of mixed-state LRE under a finite-depth local quantum channel (FDLC) remains to be fully understood <cit.>.  

In this work, we investigate the fate of mixed-state quantum entanglement under a Metropolis-Hastings <cit.> quantum channel describing the equilibration of a thermal classical system, specifically that of a d-dimensional Ising model at inverse temperature β. When acting on a pure state in the computational basis, this channel simply prepares a classical thermal state of the d-dimensional Ising model. 

The nature of the mixed quantum many-body state produced by the action of this channel on a generic input state is more striking. When the initial state is chosen to be a Greenberger–Horne–Zeilinger (GHZ) state, a symmetric, long-range-entangled ground-state of the ferromagnetic  phase, we show that the resulting quantum many-body mixed state at any finite temperature β > 0 has a negativity which is asymptotically exactly that of the pure GHZ state in the limit that the boundary of the sub-region becomes large. In spatial dimensions d>1, the negativity of a large, contiguous region in this mixed state ρ_β is thus a constant, independent of geometric details of the bipartitioning surface, and insensitive to the thermal phase transition, as summarized in Eq. (<ref>). Nevertheless, the entanglement properties of the mixed-state do change across this transition; we show that ρ_β cannot be expressed as a convex sum of short-range-entangled (SRE) pure states in the ordered phase but is preparable using an FDLC acting on a product initial state in the high-temperature, disordered phase.  This observed behavior is in stark contrast to the negativity of the Gibbs state of a local, quantum many-body Hamiltonian which necessarily vanishes above a finite temperature <cit.>.  

This result is established by using the fact that the entanglement negativity is an entanglement monotone <cit.>, so that the negativity of a subregion A of any state ρ, denoted (ρ), cannot increase under an LOCC operation  across the bipartitioning surface

    ([ρ])≤(ρ).

This property lends a new perspective on mixed-state negativity, which is of possibly broader interest outside of the specific mixed-states that we study. We explicitly construct an LOCC operation that recovers the GHZ state at any finite temperature and for any bipartition in the asymptotic limit that the boundary A becomes large.  We show that for a contiguous bipartition, this channel is the decoder for a (d-1)-dimensional repetition code, which the two parties in the LOCC protocol must use to determine the precise unitary circuit that must be applied to recover the GHZ state.  The well-known success of the (d-1)-dimensional repetition code up to a maximal error strength in dimensions d > 1 is intimately related to the recovery of the GHZ state, and thus the constant entanglement negativity, at any finite temperature. 

We comment on broader utility of this perspective for understanding mixed-state negativity, by demonstrating that this can provide a simple understanding of why the topological entanglement negativity can remain a constant in correlated noise models (e.g. in <cit.>).


Setup: Consider N qubits on a d-dimensional hypercubic lattice, in a state described by the density matrix

    ρ_β≡1/_β∑_σe^-β H|ψ_σ⟩⟨ψ_σ|.

Here, 

    H ≡ -J∑_⟨ i,j⟩Z_iZ_j

is the energy of a classical Ising model on the d-dimensional hypercubic lattice with ferromagnetic (J>0) nearest-neighbor interactions, while _β≡(e^-β H) is the corresponding partition function at inverse temperature β.  Furthermore, the state
|ψ_σ⟩≡ (|σ⟩ + U|σ⟩)/√(2), 
where |σ⟩≡|σ_1,…,σ_N⟩ is a product state in the Pauli Z basis, and U≡∏_jX_j is a unitary transformation which generates the Ising symmetry of Eq. (<ref>).

The mixed-state ρ_β is naturally obtained by starting with a GHZ state |ψ_+⟩ = (|↑⋯⟩ + |↓⋯⟩)/√(2), which is an equal-amplitude superposition of the macroscopically-distinct all-up and all-down qubit configurations, and then applying a channel Φ_β which describes a Metropolis-Hastings algorithm <cit.> in which the energy H is measured, and then a unitary operation X_j is applied on a random qubit with probability min{1, exp(-β Δ E_j)}, where Δ E_j≡ E_new - E_old is the energy difference between the new and old qubit configurations. 

Repeated measurements and feedback, starting from any product state in the Pauli Z basis, will eventually produce a Gibbs ensemble of the d-dimensional classical Ising model at inverse temperature β <cit.>.   We specifically define Φ_β to be  repeated application of the Metropolis-Hastings algorithm so as to produce a thermal steady-state when acting on the all-up state ρ_0≡|↑⋯⟩⟨↑⋯|, i.e.

    Φ_β[ρ_0] = 1/_β∑_σe^-β H|σ⟩⟨σ|.

Each operation that comprises the channel Φ_β manifestly commutes with the Ising symmetry transformation U.  Therefore, U is a strong symmetry of this channel <cit.>, i.e.    UΦ_β[ρ]=Φ_β[Uρ] for any state ρ, so that

    Φ_β[|ψ_+⟩⟨ψ_+|] = 1+U/√(2)Φ_β[ρ_0]1+U/√(2) = ρ_β.


Mixed-State Entanglement:
The state ρ_β witnesses a thermal phase transition with increasing temperature in spatial dimensions d>1 between a ferromagnetically-ordered and a disordered state. At any temperature, the state ρ_β has a strong Ising symmetry Uρ_β = ρ_β.  However, the ferromagnetically-ordered phase is also characterized by long-range-order in local operators which are charged under this symmetry (ρ_βZ_iZ_j) |i-j|→∞0.  These conditions are sufficient to establish  <cit.> that this mixed-state cannot be described as an ensemble of SRE pure-states.  

We may also show that in the disordered phase, ρ_β can be prepared by an FDLC, and is thus an SRE mixed-state [The ability to prepare ρ_β by an FDLC implies that this mixed-state describes an ensemble of SRE pure state.].  In the Supplemental Material <cit.>, we formally show that ρ_β = ℰ_ZZ[|ϕ(β)⟩⟨ϕ(β)|], 
where ℰ_ZZ is a quantum channel that describes maximal dephasing of Z_iZ_j along all nearest-neighbor bonds.  Furthermore, we show <cit.> that the wavefunction |ϕ(β)⟩≡_β^-1/2∑_μe^-β H/2|μ⟩, 
where |μ⟩ denotes a product state in the Pauli X basis, is the exact ground-state of the  Hamiltonian

    H_0(β) = ∑_j[∏_i∈NN(j)e^-β J Z_iZ_j -X_j],

which exhibits a phase transition at a critical value of β=β_c(d) which is in the d-dimensional Ising universality class.  When β<β_c(d),  |ϕ(β)⟩ is the Ising-symmetric ground-state in the disordered phase and can thus be prepared from a product state using a finite-depth local unitary circuit.  Thus in this range of temperatures, ρ_β can be prepared by an FDLC by composing this circuit with the quantum channel ℰ_ZZ.


Conditional Mutual Information:
We now investigate entanglement properties of ρ_β and their relation to the Ising phase transition. The conditional mutual information (CMI) is a measure of tripartite correlations in a state and has been used to study various aspects of entanglement in quantum systems<cit.>, including probing mixed state quantum phases of matter<cit.>. 
Given subsystems A, B, and C, the CMI is defined as

    I(A:C|B)
           =S(AB)+S(BC)-S(B)-S(ABC).

Consider the partitioning shown in Fig. <ref> with r>0, where C is taken to be the complement of AB. Note that for any proper subset A of qubits, ρ_β,A=_A̅(ρ_β)=ρ_β,A^Cl, where ρ_β^Cl=e^-β H/_β is the classical Gibbs state [This in particular shows that for mutual information we have I(A:C)=I^Cl(A:C) provided that r>0, where I^Cl is computed with respect to ρ_β^Cl.]. This shows that the first three terms in Eq. (<ref>) take the same value when computed for ρ_β as for ρ_β^Cl. Moreover, it is straightforward to use  Eq.(<ref>) to show S(ρ_β)=S(ρ_β^Cl)-log(2). Therefore I(A:B|C)=log(2)+I^Cl(A:C|B),
where I^Cl(A:C|B) is the CMI computed for ρ_β^Cl. On the other hand, since B is assumed to be non-empty, there is no interaction term in H with support on both A and C and hence I^Cl(A:C|B)=0 <cit.>, and

    I(A:B|C)=log(2),

independent of r, temperature β and the system size. It is worh mentioning that in contrast, for thermal Gibbs state of local quantum Hamiltonians, CMI decays exponentially with r <cit.>. 


Entanglement Negativity:
We now investigate the entanglement negativity of a subregion A of the mixed-state (<ref>), (ρ_β) ≡ (||ρ_β^T_A||_1 - 1)/2.  We will show that for a region A, 

    (ρ_β) = {[ 1/2 - O(exp(-|∂ A|))                  β>0;                     ;                    0                  β=0 ].

where |∂ A| is the number of qubits on the boundary of A. The second result is easily established. At infinite temperature, ρ_β = 0 = (1 + U)/2^N, so that ρ_β^T_A = ρ_β and (ρ_β=0)=0. 

The first result is determined as follows. The negativity is a convex function <cit.> so that 

    (ρ_β) ≤1/_β∑_σe^β J∑_⟨ i,j⟩σ_iσ_j(|ψ_σ⟩⟨ψ_σ|) = 1/2

for any region A, since |ψ_σ⟩ is related to the GHZ state |ψ_+⟩ by single qubit gates, so that (|ψ_σ⟩⟨ψ_σ|) = (|ψ_+⟩⟨ψ_+|).  The latter is exactly 1/2 for any region A. The negativity is also an entanglement monotone <cit.>, see Eq. (<ref>).  Here, we show that given the state ρ_β for any β > 0 and any region A, there is an LOCC operation  with respect to A and its complement A̅ that recovers the GHZ state with a fidelity that approaches unity exponentially quickly in the size of the boundary of A.  Combining this with (<ref>), allows us to establish Eq. (<ref>).  



We first construct the LOCC channel  for a contiguous bipartition, before generalizing this construction for a non-contiguous subregion A.  The state ρ_β describes the ensemble of pure-states |ψ_σ⟩ drawn with probability e^β J∑_⟨ i,j⟩σ_iσ_j/_β. Given a state from this ensemble, we measure the operators Z_iZ_j along all nearest-neighbor bonds lying exclusively within A or .  The resulting pattern of domain walls – an example of which is shown in the left panel of  Fig. <ref> in two spatial dimensions – is consistent with having drawn the state  |ψ_σ⟩ or the state  ∏_j∈ AX_j|ψ_σ⟩ and thus leaves one classical bit's worth of ambiguity in determining the precise state.  Equivalently, this ambiguity corresponds to the two possible ways that the observed domain walls can connect across the bipartition, as shown in Fig. <ref>.   

The preparation of the GHZ state through LOCC thus maps onto a decoding problem in a classical (d-1)-dimensional repetition code living on the interface between A and A.  The classical bit of information is the relative alignment of any two qubits across the bipartition in the Pauli Z basis (e.g. in Fig. <ref>, the classical bit can be taken to be τ_1≡σ_1σ_1).  The measured “syndromes" of the repetition code (e.g. τ_1τ_2 = (σ_1σ_2)(σ_1σ_2)) and the Boltzmann weights for the d-dimensional Ising model are used to perform maximum-likelihood decoding to guess the correct state.  Specifically, let w_1 and w_2 denote the Boltzmann weights corresponding to states (i) |ψ_σ⟩ and (ii) ∏_j∈ AX_j|ψ_σ⟩, respectively. We will guess that the state we have is (i) with probability p ≡ w_1/(w_1+w_2) and (ii) with probability 1-p.  We will subsequently apply the appropriate single-site unitary gates to turn this into the GHZ state.
The LOCC channel is thus 


  * Measure Z_iZ_j along all nearest-neighbor bonds lying exclusively within A or .

  * Let |σ⟩ be a product state in the Pauli Z basis which is consistent with the measurement outcomes. Apply the unitary U_1 = ∏_jX_j^(1-σ_j)/2 with probability p, or the unitary U_2 = ∏_j∈ AX_j∏_jX_j^(1-σ_j)/2 with probability 1-p. 


At any finite temperature, the relative probability p/(1-p) = w_1/w_2 scales exponentially in the size of the boundary of A, due to the fact that the energy of an Ising domain wall is extensive in its length.  Thus, the decoder for the repetition code should be successful at any finite temperature, and fail exactly at infinite temperature when w_1 = w_2 = 1/2.  This is consistent with the known behavior of the repetition code with single-qubit bit-flip errors with probability q, for which perfect decoding is possible for any q≤ 1/2. 

We may generalize this channel for the case where A is a non-contiguous region.  In this case, step 1 (above) is replaced by measurements of Z_iZ_j across all pairs of sites entirely within A and entirely within A̅.  As before, only one classical bit's worth of ambiguity thus remains after these measurements, in determining the precise state.  The quantum circuit that takes this state onto the GHZ state is then determined by a decoder for a repetition code with |∂ A| sites.  Thus, decoding the correct quantum circuit should again be successful at any β > 0, as |∂ A| becomes large, regardless of the geometry of the bipartition.  

Consistent with these expectations, we show <cit.> that for any bipartition, the fidelity of the recovered state R(ρ_β) with the GHZ state is exactly

    ℱ((ρ_β),|ψ_+⟩⟨ψ_+|) = 1/2[1 - ⟨tanh(β H_∂)⟩]

where H_∂ denotes terms in the Ising Hamiltonian acting exclusively along bonds that cross the bipartition between A and A and the expectation value is taken in the thermal state for the classical Ising model.  At any finite temperature, the energy H_∂ is negative, and extensively large in the boundary of A, so that ⟨tanh(β H_∂)⟩→ 1 as A becomes large.

Numerical Results: 
To numerically confirm this result, we obtain an alternate expression for the entanglement negativity of any bi-partition <cit.> 

    𝒩_A[ρ_β]=1/4|1-e^2β H_∂[σ]|.

One may now use classical Monte Carlo simulations to compute the thermal average in Eq.(<ref>).
However, care must be taken with sampling the spin configurations, since the expression inside the absolute value could be exponentially large for configurations with an exponentially small Boltzmann weight, in such a way that the overall contribution to the expectation value is O(1). 




The aforementioned issue can be avoided by using a technique known as importance sampling. Let q[σ] denote the probability distribution defined as q[σ_A,σ_]=1/2p[σ_A,σ_]+1/2p[σ_A,_], where p[σ]=e^β J∑_⟨ i,j⟩σ_iσ_j/𝒵_β. To sample from q[σ], one can simply sample from p[σ] and then with 1/2 probability, flip the spins in . We may thus write Eq.(<ref>) as,

    𝒩_A[ρ_β]
           =1/4∑_σ q[σ]p[σ]/q[σ]|1-e^2β H_∂[σ]|
       =1/2|tanh(β H_∂[σ])|_q.

where the final expectation value is taken with respect to the distribution q. The advantage of Eq.(<ref>) compared to Eq.(<ref>) is that the expression inside the expectation value is now bounded by unity everywhere, so that contributions from rare configurations are no longer a concern. 

In the following we present the numerical results for a 2D system of qubits placed on a L× L square lattice with periodic boundary conditions. 
Fig. <ref> shows the negativity 𝒩_A(ρ_β) when A is taken to be the L×L/2 cylinder. As is clear from Fig. <ref>, in the thermodynamic limit, the negativity is equal to 1/2 even above the Ising critical temperature (marked by the dashed line in Fig.<ref>), as expected from the expression in (<ref>). The fact that the negativity of subregions with extensive boundary is 1/2 irrespective of temperature, means that if one defines the topological negativity as the constant contribution to the negativity as in Ref. <cit.>, one would find that the resulting mixed state has 𝒩_topo=-1/2 even above the critical temperature, despite the fact that for T>T_c – as shown above – ρ_β is an SRE mixed-state. <cit.> (cf. Ref. <cit.>). 

Although the thermal phase transition at T_c has no effect in the negativity of subregions with extensive boundaries, the negativity of finite subregions (while continuous) becomes singular at T=T_c. Fig. <ref> shows the derivative of the negativity of a single site with respect to temperature which shows a singularity at T=T_c, which is consistent with the singularity of the heat capacity at the d=2 classical Ising phase transition, as to be expected from Eq. (<ref>). 




Discussion and Outlook: Using LOCC to recover states with a known pattern of LRE may provide another perspective that can be used to investigate mixed-state LRE under decoherence. For correlated decoherence, this is apparent in specific examples.  The two-dimensional toric code dephased by local operators Z_j+δX_j at each site j, with the vector δ given in Fig. <ref>, was studied in Ref. <cit.>.  Quantum state trajectories under this channel only contain patterns of ℤ_2 charge and flux which are separated by δ.  Therefore, for any dephasing strength, it is possible to use LOCC across the bipartition in Fig. <ref> to deterministically recover the toric code ground-state. The ℤ_2 charge and flux lying on the bipartitioning surface, which cannot be directly measured in the LOCC protocol, can be perfectly inferred from stabilizer measurements in A and A due to the noise correlations.  The negativity for this bipartition and for any dephasing strength <cit.> is thus exactly that of the toric-code ground-state <cit.>. Another example for which the LOCC perspective presented here could be insightful is the study of entanglement structure in the toric code with single-site Y decoherence <cit.>. In this case, while the noise model itself is not correlated, the LOCC protocol can benefit from the correlations that are present in the syndromes due to the extensively many strong symmetries of this channel <cit.>. The detailed study of this model and other correlated noise models is left to a future work. 

We identify other directions implied by our results.  First, it is interesting to consider the fate of mixed-state entanglement under thermalization by genuinely quantum channels which are strongly-symmetric.
More concretely, we may consider thermalizing the GHZ state via a strongly symmetric implementation of the quantum Metropolis-Hastings dynamics<cit.> for the d-dimensional transverse field quantum Ising model. The resulting mixed state would be the projection of the quantum Gibbs state onto the symmetric sector. A similar LOCC protocol to recover the GHZ state can be used to relate the entanglement negativity to the performance of a (d-1)-dimensional repetition code under a coherent correlated noise model. Energetic considerations can be used to argue about the success of the LOCC protocol in turning the resulting mixed state into the GHZ state. This lower bounds the negativity of the resulting mixed-state and may suggest that it 
would have a non-trivial negativity at high temperatures, though this remains to be confirmed. 

Second, while the persistence of topological negativity at high temperatures in our work can be understood as a consequence of the repetition code possessing maximum error threshold, it is interesting to search for settings where a similar line of argument would lead to a different code with a finite threshold. Such a system, would exhibit a finite temperature phase transition in negativity, which may not necessarily coincide with the standard order-disorder thermal phase transition. Furthermore, while we focused on the Metropolis-Hastings channel for the classical Ising model, a possibly fruitful future direction to pursue is to consider other classical systems with different symmetries and study the fate of entanglement in an initially symmetric state (e.g. continues symmetries, higher-form symmetries or non-Abelian symmetries <cit.>). 


Lastly, given that the topological negativity is unable to detect the phase transition in the complexity of the mixed-state ρ_β (from LRE to SRE), it is interesting to look for other quantities that would be able to capture the complexity of the quantum phase transition in our system as well as in more general settings. 

Acknowledgments:
We thank Yimu Bao, Matthew Fisher, Tarun Grover, Tibor Rakovszky, Subhayan Sahu, Shengqi Sang and Yaodong Li for useful discussions. This research was supported in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP). Use was made of computational facilities purchased with funds from the National Science Foundation (CNS-1725797) and administered by the Center for Scientific Computing (CSC). The CSC is supported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; NSF DMR 2308708) at UC Santa Barbara. SV is partly supported as an Alfred P. Sloan Research Fellow.








    Supplemental Material





§ ENTANGLEMENT OF Ρ_Β

Here, we show that the state (<ref>) is short-range-entangled (SRE) throughout the disordered phase at sufficiently high temperatures, and in any spatial dimension.  First, observe that this state may be written as

    ρ_β = ∑_μp_μ|ϕ_μ⟩⟨ϕ_μ|

where |ϕ_μ⟩≡ρ_β^1/2|μ⟩/ √(p_μ) and p_μ≡⟨μ|ρ_β|μ⟩, while |μ⟩ is a product state in the Pauli X basis.  Because of the strong symmetry of this state, (ρ_βU) = 1, where U ≡∏_jX_j.  As a result, each pure state appearing in (<ref>) must satisfy ∏_jX_j|ϕ_μ⟩ = |ϕ_μ⟩.  

Furthermore, observe that

    ρ_β^1/2 = 1/√(_β)∑_σe^-β H/2|ψ_σ⟩⟨ψ_σ|

It is evident from this expression that [Z_iZ_j,ρ_β^1/2]=0 for any i, j.  As a result, any state |ϕ_μ⟩ appearing in (<ref>) – for which U|ϕ_μ⟩ = |ϕ_μ⟩ – can be obtained by applying single-site unitary gates on the reference state |ϕ_+⟩ = ρ_β^1/2|+⋯ +⟩/√(p_+), where |+⋯ +⟩ is a product state in which X_j=+1 for each qubit:

    |ϕ_μ⟩ = ∏_jZ_j^1-μ_j/2|ϕ_+⟩

Finally, we note that p_μ = ⟨μ|ρ_β|μ⟩ = ⟨+⋯ +|∏_jZ_j^1-μ_j/2ρ_β∏_jZ_j^1-μ_j/2|+⋯ +⟩ = ⟨+⋯ +|ρ_β|+⋯ +⟩ = p_+.  As a result, p_μ = 1/2^N-1 where N is the total number of qubits.  To summarize, we may write (<ref>) as

    ρ_β   = 1/2^N-1∑_μ  s.t.  ∏_jμ_j= +1∏_jZ_j^1-μ_j/2|ϕ_+⟩⟨ϕ_+|∏_jZ_j^1-μ_j/2

We now re-cast the summation by introducing Ising degrees of freedom σ_ij along bonds of the hypercubic lattice, and letting μ_j≡ A_j[σ] ≡∏_i∈ NN(j)σ_ij, where the product is taken over sites that are nearest-neighbor to j.  With periodic boundary conditions, ∏_jA_j[σ] = +1, and as a result, the constraint on the summation ∏_jμ_j = +1 is naturally satisfied by this replacement.  With this replacement, we may write

    ρ_β   = 1/2^N_b∑_σ∏_jZ_j^1-A_j[σ]/2|ϕ_+⟩⟨ϕ_+|∏_jZ_j^1-A_j[σ]/2

Here, N_b = d· N is the total number of bonds on the d-dimensional hypercubic lattice.  The pre-factor ensures the normalization of the density matrix[Another way to derive this pre-factor is to observe that for a given configuration of μ_j satisfying the constraint ∏_jμ_j = 1, there are 2^N_b/2^N-1 different configurations of σ_ij such that μ_j = A_j[σ]. ]. Finally, using the fact that Z_j^1-A_j[σ]/2 = ∏_i∈NN(j)Z_j^1-σ_ij/2, we may write ∏_jZ_j^1-A_j[σ]/2 = ∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2, where the product in the second expression is taken over distinct, nearest-neighbor bonds.  As a result,

    ρ_β   = 1/2^N_b∑_σ∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2|ϕ_+⟩⟨ϕ_+|∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2

We conclude that ρ_β may be obtained by dephasing Z_iZ_j on each bond of the lattice, with maximal strength, a manifestly finite-depth, local quantum channel.

Finally, we may argue that the state |ϕ_+⟩ is SRE when the system is in a disordered phase.  We do this by first writing

    |ϕ_+⟩ = √(2^N-1)ρ_β^1/2|+⋯ +⟩ = √(2^N-1/_β)∑_σe^-β H/2|ψ_σ⟩⟨ψ_σ|+⋯ +⟩ = 1/√(_β)∑_σe^-β H/2|σ⟩

We now observe that (<ref>) is the ground-state of the Hamiltonian

    H = -∑_jQ_j

where

    Q_j≡ -X_j + ∏_i∈NN(j)e^-β J Z_iZ_j


 


§ LOCC OPERATIONS AND RECOVERING THE GHZ STATE


Here, we will show the existence of an LOCC protocol   with respect to bi-partition A and , which transforms state ρ_β back to the GHZ state in the thermodynamic limit. 
The LOCC protocol consists of two steps:


    
  * Measure Z_iZ_j on each link that is entirely in A or in  subregions, finding the outcome μ_ij=± 1. This does not include the links across the boundary, ∂ A.

    
  * For any given set of measurement outcomes μ={μ_i,j}, there exist two possible states which have domain wall configuration consistent with μ. 
     We pick one of the two states based on a coin toss weighted with their respective Boltzmann weight in ρ_β (Eq.(<ref>)) and
      apply a product of Pauli X_i operators to eliminate the domain walls in that assumed configuration.

In the following, we will derive Eq.(<ref>) for the fidelity between the GHZ state, |ψ_+⟩, and the output state (ρ_β).


 §.§ Fidelity


Let {M_μ} be the projective measurements that measure whether there is a domain wall or not on the links which are entirely
in region A or A̅,


    M_μ≡∏_<i,j>∈ A, A̅1+μ_i,jZ_iZ_j/2,

where μ_ij is the measurement outcome of measuring Z_iZ_j and the product only includes next nearest sites which are both in A or both in .
Note that,

    M_μ|ψ_σ⟩ = ∏_<i,j>∈ A, A̅δ_μ_i,j,σ_iσ_j|ψ_σ⟩.
 

A set of measurement outcomes μ specifies a certain domain wall configuration in A and , with some domain walls ending on the boundary ∂ A. Let 1∈ A and 1̅∈ be two adjacent sites across the boundary and define τ=± 1 to be a marker for whether or not there is a domain wall across the edge connecting 1 and 1̅, i.e. τ≡σ_1 σ_1̅ (see Fig. <ref>). Note that μ and τ together uniquely specify the global domain wall configuration, because there should be even number of domain walls across each plaquette of the square lattice. In particular if we label the sites along the boundary as shown in Fig.<ref>, we have,

    σ_i σ_ i̅=τ∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1.

Let X_μ,τ be the appropriate product of single qubit Pauli-Xs that corresponds to the domain wall configuration specified by μ and τ. In particular, if M_μ|ψ_σ⟩=|ψ_σ⟩, we have

    X_μ,τ|ψ_σ⟩≡|ψ_+⟩   if σ_1σ_1̅=τ
    ∏_j∈ AX_j|ψ_+⟩   if σ_1σ_1̅≠τ.

Given the measurement outcomes μ, the decoder  applies X_μ,τ with τ=± 1 chosen randomly with probability P_μ,τ. The probability P_μ,τ is chosen to be proportional to the Boltzmann weight of the spin configuration specified by μ and τ and according to Eq.(<ref>) can be written as,

    P_μ,τ = e^β J τ∑_i≥ 1∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1/∑_τ'=± 1 e^β J τ' ∑_i≥ 1∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1.

Putting everything together, we can compute the average “decoding" fidelity,


    F = ψ_+(ρ_β)ψ_+= ∑_σ,μ,τe^-β H[σ]/𝒵_βP_μ,τ⟨ψ_+|X_μ,τM_μ|ψ_σ⟩^2.


Using Eqs.(<ref>) and (<ref>) this simplifies to


    F = ∑_σ,μ,τe^-β H[σ]/𝒵_β P_μ,τ∏_<i,j>∈ A, A̅δ_μ_i,j,σ_iσ_jδ_σ_1σ_ 1̅,τ.

Plugging the expression for P_μ,τ into Eq. (<ref>), we get


    F =   ∑_σ,τe^-β H[σ]/𝒵_βe^β Jτ∑_i≥1σ_1σ_iσ_ 1̅σ_ i̅/∑_τ'=±1 e^β Jτ'∑_i≥1σ_1σ_iσ_ 1̅σ_ i̅δ_σ_1σ_ 1̅,τ
    
         =   ∑_σe^-β H[σ]/𝒵_βe^β J∑_i≥1σ_iσ_ i̅/∑_τ'=±1 e^βτ'σ_1 σ_ 1̅ J∑_i≥1σ_i σ_ i̅
    
         =   1/2∑_σe^-β H[σ]/𝒵_β(1-tanh(β J∑_i≥1σ_i σ_ i̅))
    
         =   1/2[1-⟨tanh(β H_∂)⟩]

where H_∂ is the boundary Hamiltonian between A and A̅. 



 §.§ Mapping to the d-1 repetition code


The problem of choosing which domain wall configuration to assume can be understood as a decoding problem for a (d-1)-dimensional classical repetition code. To see this, note that the decoding task in a classical error correcting code is basically figuring out what error has happened based on a given set of syndromes. In particular the decoding task in the classical repetition code on N bits can be described as follows. Consider an error vector e∈{1,-1}^N such that e_i=-1 if a bit flip has happened on the i'th bit and e_i=1 otherwise. 
The goal of the decoder is to figure out e, knowing only the syndrome values, i.e. e_i e_j for adjacent i and j on the appropriate graph, e.g. the d-dimensional square lattice in the standard d-dimensional repetition code. Moreover, the decoder assumes an underlying noise model, i.e. a random distribution from which the vector e is drawn. For example, in the independent and identically distributed (iid) noise model, it assumes each component e_i is -1 with probability p and 1 with probability 1-p, independent of other components. 

In our setting finding the domain wall configuration is clearly the same task, with e_i defined to be e_i≡σ_i σ_ i̅. Note that we can not know the value of e_i, but we know the value of e_i e_j=μ_i,j μ_i̅,j̅ for adjacent i and j. At low temperatures, the domain walls are rare so most e_is would be 1, and as one increases the temperature more e_i would be -1. The underlying noise model is given by P_μ,τ as described in the previous section. This noise model is not iid, but rather a correlated noise model. However, as we argue in the next section, the repetition code is able to guess e correctly with probability 1 in the thermodynamic limit unless β=0, at which point each e_i is ± 1 compeletly at random. 



§ THRESHOLD FOR THE REPETITION CODE

Here we argue that the repetition code decoding under the noise distribution P_μ,τ is successful with probability 1 for any β>0 in the thermodynamic limit. First we review the argument for maximum error correction threshold in the presence of the iid noise and then we argue that the same should remain true for more generic noise models. We use quantum notation for convenience although the problem we are considering is inherently classical. 

Consider an initial state |Ψ⟩ undergoing some errors, leaving it in the mixed state

    ρ = ∑_e p_e e|Ψ⟩⟨Ψ|e

where e is a product of X bit flips and p_e is the probability of having that error. We rewrite the sum to distinguish between the errors that leave ρ with fewer than N/2 bit flips and errors with higher weights:

    ρ = ∑_e: |e|<N/2p_e e|Ψ⟩⟨Ψ|e + ∑_e:|e|≥ N/2p_e e|Ψ⟩⟨Ψ|e.

The error correction channel, , measures the syndromes, and between two possible error configuration consistent with a syndrome, applies the one with the smaller weight. Therefore it always successfully corrects the errors when there are less than N/2 bit flips, hence,

    (ρ)=   ∑_e: |e|<N/2 p_e (e|Ψ⟩⟨Ψ|e) + ∑_e:|e|≥ N/2p_e  (e|Ψ⟩⟨Ψ|e)
    
        =   ∑_e: |e|<N/2 p_e|Ψ⟩⟨Ψ| + ∑_e:|e|≥ N/2p_e (e|Ψ⟩⟨Ψ|e).


The fidelity of this post-error correction state with the original state is lower bounded as,


    F =   ⟨Ψ|(ρ)|Ψ⟩
    
        =   ∑_e: |e|<N/2 p_e +  ∑_e:|e|≥ N/2p_e ⟨Ψ|(e|Ψ⟩⟨Ψ|e)|Ψ⟩≥∑_e: |e|<N/2 p_e


where the lower bound is because the second term in the expression above is non-negative. Now we will show that this lower bound is equal to one in the thermodynamic limit. First, we rearrange the sum to be over |e|=k, the number of bit flips error, and because  p_k is a binomial distribution,
p_k = Nkp^k(1-p)^N-p.

In the limit of large N, we can approximate the binomial distribution as a Gaussian distribution with mean Np and standard deviation √(Np(1-p)) 
via the central limit theorem. Since N/2 is N/2-Np/√(Np(1-p))=O( √(N)) standard deviations away from the peak of the Gaussian, we see that 

lim_N→∞∑_k=0^N/2-1 p_k = 1

when p<1/2. Since fidelity is upper-bounded by one, the fidelity is indeed equal to one in the thermodynamic limit. Thus, the repetition code will work successfully when p<1/2. 

For the more general case where e is drawn based on the P_μ,τ distribution, the bound in Eq.(<ref>) is still valid. Therefore, to show perfect recovery we only need to show that with probability one, less than half of the edges contain a domain wall. Intuitively, this is true because for β>0, not having a domain wall on an edge is energetically favorable. To make it rigorous, one can define indicator variables I_i=(1-e_i)/2 in terms of which the number of domain walls can be written as k=∑_iI_i. When β >0 it is more probable for I_i to be 0 rather than 1 and hence the expectation value of k/|∂ A| is less than 1/2. Noting that lim_|i-j|→∞Cov(I_i,I_j)=0, the weak law of large numbers for weakly dependent variables<cit.> shows that k/|∂ A| concentrates around its mean, which is less than 1/2 when β>0 in the thermodynamic limit. 



§ NEGATIVITY SPECTRUM

Here we calculate the entanglement negativity between a sub-region A and its complement A̅, in the state ρ_β. 
Let  denote the spin configuration that is related to σ by flipping every spin. Accordingly, the state |ψ_σ⟩ can be written as |ψ_σ⟩=|σ⟩+|⟩/√(√(2)) and Eq. (<ref>) can be rewritten as,


    ρ_β = 1/2𝒵_β∑_σ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅| +|σ⟩⟨σ̅| +|σ̅⟩⟨σ|),

where H[σ]=-J ∑_i,jσ_i σ_j is the classical Ising energy of the spin configuration σ. Taking the partial transpose of ρ_β with respect to sub-region A yields,

    ρ_β^T_A   = 1/2𝒵_β∑_σ[ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅|) +e^-β H[σ_A,σ_A](|σ_A,σ_A⟩⟨σ_A, σ̅_A| +|σ_A, σ̅_A⟩⟨σ_A,σ_A|)]
       =1/2𝒵_β∑_σ[ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅|) +e^-β H[σ_A,_A](|σ⟩⟨σ| +|σ⟩⟨σ|)]


where σ_A and σ_A̅ denote the spin configuration σ restericted to regions A and A̅ respectively. Thus, the eigenvalues and eigenstates of ρ_A^T_A are

    λ_σ;±=e^-β H[σ]± e^-β H[σ_A,σ̅_A]/_β, |ψ_σ^(±)⟩≡|σ⟩±|σ⟩/√(2).



We can now calculate the entanglement negativity  from the sum of all the negative eigenvalues. Firstly,
it is only the λ_σ;- eigenvalues which can be negative. 
We can also see that λ_σ;-=λ_σ̅;- from the ℤ_2 symmetry of the Hamiltonian and they correspond to the same eigenvector, up to some phase. 
Moreover, 
note that λ_σ;-=-λ_σ_A,σ̅_A̅;- so for a given spin configuration, either λ_σ;- or λ_σ_A,σ̅_A̅;- would be negative. Thus, we account for the two incidences of double counting with an additional 1/4 and get


    𝒩_A[ρ_β] = 1/4_β∑_σe^-β H[σ] - e^-β H[σ_A,σ_A].


Factoring out e^-β H[σ] we get


    𝒩_A[ρ_β]    = 1/4_β∑_σ e^-β H[σ]1 - e^-β( H[σ_A,σ_A]- H[σ])
       =1/4⟨1 - e^2β H_∂[σ])⟩.






 Paper Title: Exponential growth rate of lattice comb polymers.



Combs]Exponential growth rate of lattice comb polymers




^⋆ Department of Mathematics and Statistics, York University, Toronto,  M3J 1P3, Canada
 
† Department of Chemistry,  University of Toronto, Toronto,  M5S 3H6, Canada

^⋆rensburg@yorku.ca,
^†stuart.whittington@utoronto.ca




   August 5, 2024

§ ABSTRACT 
We investigate a lattice model of comb polymers and derive bounds on the exponential growth rate of the number of embeddings of
the comb.  A comb is composed of a backbone that is a self-avoiding walk and a set of t teeth, also modelled as mutually and self-avoiding 
walks, attached to the backbone at vertices or nodes of degree 3.  Each tooth of the comb has m_a edges and there are m_b edges in the backbone between adjacent 
degree 3 vertices and between the first and last nodes of degree 3 and the end vertices of degree 1 of the backbone.  We are interested in the exponential growth rate as 
t →∞ with m_a and m_b fixed.  We prove upper bounds on this growth rate and show that for small values of m_a the growth rate is strictly
less than that of self-avoiding walks.





82B41, 82B80, 65C05






Keywords: Self-avoiding walk, connective constant, lattice polymer, comb polymer





[
    E J Janse van Rensburg^⋆ & S G Whittington†
    
===============================================










§ INTRODUCTION



Comb polymers are a class of branched polymers with simple connectivity and yet useful and
interesting properties.  Generally, a comb is a long (linear) backbone polymer with side chains
(called teeth) substituted at regular intervals along the backbone.  Copolymeric combs are useful
as steric stabilizers of dispersions <cit.>.  Usually the backbone of the comb adsorbs
on a colloidal particle and the teeth extend in solution.  These teeth lose conformational 
degrees of freedom when another particle approaches, resulting in a repulsive entropic force
between particles.  

In this paper we consider self-avoiding walk models of square and cubic lattices combs.  In
figure <ref> a schematic drawing of a comb, and its representation in the square lattice,
are shown.  Teeth in the lattice comb are attached at nodes to the backbone, and
these nodes are uniformly spaced along the backbone.  The length of each tooth is m_a,
while the nodes are spaced a distance m_b apart along the backbone.  If there are t
teeth substituted along the backbone, then we say the degree of the comb is t,
and the backbone has length (t+1) m_b while the size of the comb is (t+1) m_b+t m_a.
This lattice model is of a comb in a dilute solution in a good solvent, and the 
conformational degrees of freedom in the lattice quantify its conformational entropy. 
For a general review of the properties of combs and related branched polymers see 
<cit.>.



Much of the theoretical work on lattice combs has focussed on their dimensions in a dilute 
solution (for example, see references <cit.> and other references 
therein).  A central question about lattice combs is the exponential rate of increase of the
number of conformations (or embeddings) of the comb as a function of its total size.  This
question was considered in various ways in references <cit.>.  We shall be concerned about this question as well,
but in the context explored in references <cit.> 
(see especially section 5 therein) and <cit.>. 

If the number of teeth is fixed at t=2 and m_a=m_b=m →∞ then <cit.> 
the exponential growth rate is the same as that of self-avoiding walks.  In 1993
Lipson <cit.> used Monte Carlo methods and combinatorial upper bounds 
to argue that, if m_a and m_b are fixed but t goes to infinity, then 
the exponential growth rate is strictly less than that of self-avoiding walks.  The 
main result in this paper is a proof of this when the teeth are very short.  
We shall use and extend some of the ideas that appeared 
in <cit.>, and also supplement our results with numerical data.

We denote the number of self-avoiding walks of length n steps from the origin 
in a lattice by c_n, and recall that the connective constant of 
self-avoiding walks is defined by <cit.>

    κ = lim_n→∞1/nlog c_n = inf_n>01/nlog c_n.

The growth constant μ of self-avoiding walks is defined by 
μ = e^κ. 

The connective constant is lattice dependent, and 
we shall sometimes add a subscript to denote the lattice, for example, 
κ_d for the connective constant of the d-dimensional hypercubic 
lattice (that is, κ_2 for the square lattice, and κ_3 for the
cubic lattice), or κ_H for the hexagonal (honeycomb) lattice.  We use similar
subscripts for μ.

For most lattices the values of κ or μ are not known exactly.  Best
estimates for μ have been determined by extrapolating from exact counts
of self-avoiding walks.  For example

    μ =
    
    2.6381585321(4),   square lattice <cit.>; 
    
    4.150797226(26),   triangular lattice <cit.>; 
    2.560576765(10),   Kagomé lattice (attributed to I Jensen - <cit.>); 
    4.684039931(27),   simple cubic lattice <cit.> .

In the case of the hexagonal lattice μ is known exactly, first determined
in reference <cit.>, and then proven in <cit.>: 

    μ_H = √(2+√(2)) = 1.847759 …

This result will be useful in section <ref>.  For other lattices rigorous 
lower and upper bounds are available <cit.>,  
and we shall make use of some of these bounds.  The lower bounds useful to
us in this paper are

    μ >
    
    2.625622,   square lattice <cit.>; 
    
    4.118935,   triangular lattice <cit.>; 
    2.548497,   Kagomé lattice <cit.>; 
    4.225,   simple cubic lattice <cit.> .

In this paper we shall be concerned with the connective constant of lattice combs,
and in particular compare bounds on the connective constants of combs to the 
connective constants of self-avoiding walks in the same lattice.

In a lattice with coordination number γ one can obtain upper bounds 
on μ by noting that c_nm≤ (c_n)^m((γ-1)/γ)^m-1 
(concatenate m walks of length n and note that each time a walk is added, 
then there are γ-1 directions available for its first step).  Taking logarithms 
of this, and dividing by nm, and then m→∞ with n fixed,

    logμ≤ (log c_n + log((γ-1)/γ))/n .

In the square lattice and for n=1 one obtains μ≤ 3, and the bound
improves as n increases.  For n=79 <cit.> it gives μ < 2.6848.
In the cubic lattice, n=36 <cit.> gives the bound μ < 4.7587.  We
shall use generalizations of this approach to find numerical upper bounds on the limiting
free energy of combs and brushes. 

We write g(m_a,m_b,t) for the number of combs (ie 
the number of lattice embeddings) with t teeth, each of length m_a, with 
m_b edges between adjacent pairs of degree 3 vertices along the backbone 
and with m_b edges between the first and last degree 3 vertices and the 
end vertices of degree 1 of the backbone.  

In <cit.> it was shown that the limit

    ζ(m_a,m_b) = lim_t →∞1/(m_a+m_b) tlog g(m_a,m_b,t)

exists.  The proof was for the simple cubic lattice but the same kind of concatenation 
argument works for other lattices.  The function ζ(m_a,m_b) is the free energy 
(or connective constant) of an infinite comb with backbone segments all 
of length m_b, and teeth all of length m_a. If m_a=0 and m_b>0 then 
clearly, ζ(0,m_b) = logμ=κ.  Generally the relations between 
the ζ(m_a,m_b) for various finite values of m_a and m_b are not known.  
For example,  one may expect that ζ(1,1) ≤ζ(2,1), but a proof of 
this is not known.  However, there are cases when ζ(m_a,m_b) < κ, 
and these cases are our concern.   In this paper we show this for various values 
of m_a and m_b.

The outline of the paper is as follows.  In section <ref> 
we review some results about self-avoiding walks and lattice stars that 
we shall need later.  The remainder of the paper is concerned with 
the exponential growth rate of the number of combs as t →∞.  
We begin in section <ref> where 
we make use of a pattern theorem for self-avoiding walks <cit.> 
to show that the growth constant of combs is strictly less than that of 
self-avoiding walks when the backbone segments are long enough, but finite, 
and the teeth are very short. 
In sections <ref> and <ref> we derive upper 
bounds on the growth rate of the number of combs when the teeth 
are short, and show that these establish in many cases that the growth 
rate is strictly less than that of self-avoiding walks.  In section 
<ref> we extend some of these approaches to brushes, 
where there is more than one tooth at each branch point, while in 
section <ref> we consider the situation where m_a or m_b 
goes to infinity, after the limit t →∞ is taken.



§ SOME BACKGROUND RESULTS AND NOTATION





 §.§ Self-avoiding walks and lattice combs


By equation (<ref>) c_n ≥μ^n generally so that the 
sequence (c_n^1/n) approaches μ from above (that is, 
1/nlog c_n ≥κ).  While not much is known about 
the rate of approach to the limit in (<ref>),  Hammersley and 
Welsh <cit.> showed for the d-dimensional hypercubic 
lattice that for any γ_0 > π√(2/3) there exists n_0(γ_0) such that 

    μ^n ≤ c_n ≤ A μ^n e^γ_0 √(n),
        for all n>n_0(γ_0).

The constant A is lattice dependent.

An important result about self-avoiding walks that we shall use in 
section <ref> is Kesten's pattern theorem <cit.>. 
A pattern is any fixed self-avoiding walk.  If there exists a self-avoiding 
walk on which the pattern occurs three times <cit.> 
then the pattern is a Kesten pattern.  Suppose that P is a Kesten 
pattern and suppose that c_n(P) is the number of n-edge 
self-avoiding walks on which the Kesten pattern P never occurs.  Then <cit.>

    lim_n→∞1/nlog c_n(P) 
      = κ(P) < κ.

That is, a Kesten pattern appears at least once on all except exponentially 
few sufficiently long self-avoiding walks. Additional results about self-avoiding 
walks can be found in references <cit.>.  

A lattice f-star or f-star is a model of a star polymer with 
f arms, and it is an embedding of a tree with one vertex of degree f (called 
the central node of the star) and f vertices of degree 1, in a lattice 
with coordination number at least f.
We denote the length of the i-th arm of the star by n_i, 
so that n=n_1+n_2+⋯+n_f is the total length or size of the star.  If n_i=n_j 
for all 1≤ i,j,≤ f, then the star is uniform.  In this case the total length 
of the star is n=mf, for some m∈.  If n is not a multiple of f, 
then a star can be almost uniform if ⌊ n/f ⌋≤ n_i ≤⌈ n/f ⌉.  
We shall write s_n(f) for the number of embeddings of a uniform or almost
uniform f-star with total length n, and we note the important fact that the
star's central node is rooted at the origin.  Then <cit.>

    lim_n→∞1/nlog s_n(f) = κ.

More generally, stars may be non-uniform and we denote the number of f-stars 
with arms of lengths (n_1,n_2,..,n_f) by s(n_1,n_2,…,n_f).  In the case that
n_i=n_j=m then s(m,m,…,m)=s_mf(f) is the number of uniform f-stars 
of size n=mf.

In the case of simple cubic lattice combs it can be shown that 

    lim_m_a,m_b →∞m_b=o(m_a)1/tm_a+ (t+1)m_blog g(m_a,m_b,t)=   κ_3  
    lim_m_a,m_b→∞m_a=o(m_b)1/tm_a+(t+1)m_blog g(m_a,m_b,t)  =   κ_3.

Lower bounds follow from theorems 2 and 3 in <cit.>
setting a=b=y=1 and using the fact that combs confined to a half-space and 
terminally attached to the dividing plane are a subset of all combs.  Upper 
bounds follow by regarding the teeth as independent of one another and 
independent of the backbone, giving the inequality

    g(m_a,m_b,t) ≤ c_m_b(t+1)  c_m_a^t.

Taking logarithms, dividing by tm_a+(t+1)m_b, and taking the 
appropriate limits with t fixed, gives the required upper bounds.    A similar 
argument, using theorem 1 in <cit.> with 
a=b=y=1 shows that

    lim_m→∞1/(2t+1)mlog g(m,m,t) = κ_3.

The expressions in equations (<ref>),  (<ref>) and
(<ref>) are independent of t and the limit t →∞ 
can be taken  after the limits m_a,m_b →∞ have been 
taken, with the same result. 

In reference <cit.> equations (<ref>) 
and (<ref>) correspond to the star and 
self-avoiding walk limits of lattice combs.  The limit as t→∞ 
shown in equation (<ref>), defining ζ(m_a,m_b),  is 
the limit as the number of teeth, t, increases with m_a and m_b fixed. 
In the next sections we consider the situation when the limit t→∞ is 
taken first, leaving m_a or m_b finite.  We shall see that in these cases 
the limits are in general functions of m_a and m_b, and in some cases 
are strictly less that κ_2 in the square lattice, and strictly less than 
κ_3 in the cubic lattice (this may also be the case in other lattices).





 §.§ Notation and branched structures


In figure <ref> we show the connectivity of some clusters in the
lattice, from the left, a self-avoiding walk, a 3-star, and then two 
branched structures or trees which may be created by concatenating
two stars using the endpoints of arms.  We shall call these 
double stars.  Note that the clusters are rooted at the origin
in a particular node; the self-avoiding walk has first vertex at the
origin, while the star has its central node fixed at the origin.
The double stars are rooted in the central node of one particular
component star.  Noting the location of the root node is important
when counting these, or when joining them, when one has to consider
symmetries in order to exhaust all the possible ways of putting two
or more together.



We proceed by introducing notation to describe the clusters in figure <ref>.
A self-avoiding walk of length n will be denoted by using square brackets,
so that the walk on the left in figure <ref> is denoted by [n].  The number 
of such walks is c_n.  For example, in the square lattice c_1=4, 
c_2=12, c_3=36 and c_4=100. 

The 3-star in figure <ref> with arms of lengths x, y and z will 
be denoted by [x,y,z], and the number of such stars, rooted in the central node, 
will be denoted by s([x,y,z]).  For example, one can check that, in the 
square lattice, s([1,1,1])=4, s([2,1,1])=36 and s([3,2,1])=520.  

There are symmetry factors associated with lattice stars.  If a star has
two arms of the same length, then there are two ways of labelling
those arms.  We denote this by σ([x,x,y])=2
if x≠y. Similarly, σ([x,x,x])=6 if all three arms have the same length,
but σ([x,y,z])=1 if x≠y≠z≠x.

The double stars in figure <ref> have clearly defined “middle” branches
joining the nodes, and arms attached at the two nodes.  We denote the
middle branch by an extra set of square brackets, so that [x,y,[z],a,b]
denotes the double star second from the right in figure <ref>, 
and [x,y,[z],a,b,c] is the right-most double star.  Notice that [x,y,0,[z],a,b,0]
= [x,y,[z],a,b], etc.

The numbers of branch structures and their symmetry factors are denoted 
similarly to stars.  For example, the number of double stars [x,y,[z],a,b,c]
is denoted by s([x,y,[z],a,b,c]) and its symmetry factor is
denoted σ([x,y,[z],a,b,c]).  In all these examples the cluster is rooted
at the fixed large node as shown in figure <ref> and arms of length zero 
are omitted.  For example, one may check that s([1,1,[1],1,1])=36 and 
σ([1,1,[1],1,1])=4 in the square lattice.  In the cubic lattice,
s([1,1,[1],1,1])=600 and σ([1,1,[1],1,1])=4.

This notation generalizes to combs in a natural way.  First, the backbone
of the comb is composed of segments of (say) lengths b_1,b_2,…,b_t+1
with t teeth of lengths u_1,u_2,…,u_t substituted along the backbone.
This is denoted by [[b_1],u_1,[b_2],u_2,…,[b_t],u_t,[b_t+1]], and notice
that the progression …,[b_j],u_j,[b_j+1],… means that a tooth of
length u_j is joined to the node joining backbone segments j and j+1
of lengths b_j and b_j+1.  For example, a comb with 3 teeth of length 
1 each, along a backbone with segments all of length 2, is denoted by 
[[2],1,[2],1,[2],1,[2]], and the 3-star in figure <ref> would be [x,z,y],
since it has no backbone. Depending on the choice of a backbone, the third 
and last cases in figure <ref> are [x,y,[z],a,b] or [[x],y,[z],a,[b]],
and [x,y,[z],a,b,c] or [x,y,[z],a,b,[c]], amongst other possibilities.

This can be generalized to other cases.






§ A PATTERN THEOREM ARGUMENT




In this section we show that ζ(m_a,m_b) < κ if  
m_b  large but finite and m_a is sufficiently small.

Consider the square lattice and write U, D, L and R for up, down, left and
right steps in the four lattice directions.  Fix a (large) natural number N=3m+5 
and consider the pattern P_N given by R^m+1DL^m+1DR^m+1.  

Suppose that m_b<N.  Since teeth occur on a comb every m_b 
steps of the backbone, the occurrence of P_N in the backbone prevents 
the existence of teeth in a section of the backbone of length N+1.  
Thus, P_N cannot occur in the backbone of a comb if m_b<N.

Denote the number of conformations of a self-avoiding walk not containing 
the pattern P_N by c_n(P_N) (see equation eqn:pattern).
Then the number of conformations of the backbone is less than or equal to
c_n(P_N).  Along the backbone each tooth can be appended 
at a node on the backbone with first step in two out of four possible directions.  
This show that, for any fixed m_b<N and m_a>0,

    g(m_a,m_b,t) ≤ c_(t+1)m_b(P_N)  ( 2 c_m_a/ 4 ) ^t,

since there are at most 2 c_m_a/4 conformations for each tooth. 
Taking logarithms, dividing by m_b(t+1) + m_at and letting t →∞ gives

    ζ(m_a,m_b) 
    ≤m_b κ_2(P_N)/m_a+m_b 
     + log c_m_a -  log 2/m_a+m_b,
       for any fixed m_b<N.

Since P_N is a Kesten pattern we know that κ_2(P_N) < κ_2
<cit.>.  Putting m_a=1, noting that c_1=4 and recalling that 
m_b<N for any finite fixed N, and that κ_2 > log 2, shows that
for any finite m_b, ζ(1,m_b) < κ_2.

Generalizations of the pattern P to walks in the cubic lattice can be
made, and here one similarly has κ_3(P_N) < κ_3.
In this case equation PLI becomes

    ζ(m_a,m_b) 
    ≤m_b κ_2(P_N)/m_a+m_b 
     + log c_m_a -  log (3/2)/m_a+m_b,
       for any fixed m_b<N.

since the number of ways a tooth can be attached at a node to the comb is
less than or equal to (4/6) c_m_a.  Putting m_a=1 and noting that
c_1=6 so that log c_1 - log (3/2) = log 4 < logμ_3 (see equation
lower:mu), one obtains that, in the cubic lattice, ζ(1,m_b) < κ_3.

These results give the following theorem.


In the square or cubic lattices, for any finite m_b>0, 
ζ(1,m_b) < κ. 

Since c_2=12 and log c_2 - log 2 = log 6 > κ_2 the above does not
give a similar result for m_a=2 in the square lattice.  More generally, 
since c_n is submultiplicative, log c_n > n κ_2 in general, so one 
cannot prove, using the above bounds, that ζ(m_a,m_b) < κ_2 
generally, for finite values of m_a and m_b in the square lattice.
Similar observations apply in the hypercubic lattices.  From these considerations
there may be values of (m_a,m_b) where ζ(m_a,m_b) exceeds κ.





In the square lattice more progress can be made using the (exact and rigorous) 
bounds log 2.625622 < κ_2 < log 2.679193 <cit.>.
Bounding κ_2(P) < log 2.679193 and 
κ_2 > log 2.625622 it follows from equation PLI
that ζ(m_a,m_b) < κ_2 when

    m_b < m_a log 2.625622  - log (c_m_a/2)/log 2.679193 -  log 2.625622.

Putting m_a=1 and c_m_a=4 then shows that ζ(1,m_b) < κ_2
when m_b≤ 13.  If m_a=2 then c_m_a=12 and using the above 
shows that ζ(2,m_b)<κ_2 when m_b≤ 6.  For m_a≥ 3
these approximations fail to give useful bounds.  Thus, the result is the following
theorem.


In the square lattice, for 1≤ m_b≤ 6, ζ(2,m_b) < κ_2.


Explicit upper bounds can be obtained using equation 
PLI and the rigorous numerical bounds 
and exact counts for c_m_a for small values of m_a <cit.>.  These are 
shown in table <ref>.  Square lattice upper bounds marked by * are 
below the lower bound κ_2> log 2.625622, and so are proofs that 
ζ(m_a,m_b) < κ_2.  The bounds for m_b=1 and m_b=2
are consistent with the statements of theorems <ref> and <ref>.

In the cubic lattice an upper bound on μ_3 can be obtained using 
exact enumeration data.  For n=36 <cit.> this gives the 
rigorous bound μ_3 ≤ 4.7827969, rounded up to 7 decimal places. 
Modifying equation eqn:patterninequality to the cubic lattice then gives

    ζ(m_a,m_b) 
    ≤m_b κ_3(P_N)/m_a+m_b 
     + log c_m_a -  log (3/2)/m_a+m_b
    < m_b log 4.7827969 + log c_m_a - log (3/2) /m_a+m_b ,

using κ_3(P) ≤κ_3 ≤log 4.7827969.  This gives, 
for example, ζ(1,1) ≤log 4.3740 which is well below the best numerical 
estimate for logμ_3 in equation exact:mu, but still above the 
rigorous lower bound on logμ_3 in equation lower:mu.  More 
cubic lattice bounds are shown in table <ref>.







§ UPPER BOUNDS ON THE GROWTH RATE FROM EXACT COUNTS OF SELF-AVOIDING WALKS



In this section we investigate the situation when the teeth 
are short, and show that there are cases where the growth constant 
is strictly less than that of self-avoiding walks.



 §.§ Short teeth


The next approach that we develop to investigate the case of short teeth 
makes use of an idea due to Lipson <cit.>.  The essential feature
is the existence of vertices of degree 3 that limit the number of ways of 
embedding the short teeth.  To obtain an upper bound we regard the teeth and the 
backbone as independent, and the teeth as independent of one another.  This gives 
the inequality

    g(m_a,m_b,t) ≤ c_(t+1)m_b( (γ-2)c_m_a/ γ) ^t ,

where γ is the coordination number of the lattice.  The first factor reflects the 
fact that the backbone of the comb consists of t+1 segments, each of length 
m_b.  The factor of (γ-2)/γ in brackets is due to the fact that each 
tooth can start in γ-2 ways so the number of walks making up the tooth is 
reduced by the factor (γ-2)/γ. If we take logarithms, divide by the total number 
of edges, tm_a + (t+1)m_b, and let t →∞ we have the inequality

    ζ(m_a,m_b) ≤m_b logμ/m_a+m_b 
      + log ((γ-2)/γ) + log c_m_a/m_a+m_b.




  §.§.§ The square lattice



For the square lattice (<ref>) gives 

    ζ(m_a,m_b) ≤ m_b logμ_2 /m_a+m_b + log(c_m_a/2)/m_a+m_b,

and ζ(m_a,m_b) < logμ_2 if μ_2^m_a > c_m_a/2.  
From <cit.> we know that μ_2 ≥ 2.625622
and for m_a=1, c_1/2=2.  Thus ζ(1,m_b) < κ_2 for all
m_b≥ 1 as seen in theorem <ref>.  If m_a=2, then 
μ_2^2 ≥ 6.89 > c_2/2=6 and ζ(2,m_b)<κ_2 for all 
m_b≥ 1, improving theorem <ref>. Continuing to 
m_a=3 gives μ_2^3 ≥ 18.1 > c_3/2=18 and thus
ζ(3,m_b)<κ_2 for all m_b≥ 1.  However, if
m_a=4 then μ_2^4>47.53 while c_4/2=50, and so this 
breaks down.  This gives the following theorem. 


If 1≤ m_a≤ 3, then ζ(m_a,m_b) < κ_2 for all finite m_b≥ 1
in the square lattice. 


To obtain rigorous numerical bounds this again reduces
to the results in table <ref>, in particular since the best numerical
bound is κ_2 <log 2.679193 <cit.> in theorem <ref>.

In the cubic lattice, using the lower bounds in equation lower:mu
<cit.> in a similar way, we see that ζ(m_a,m_b) < logμ_3
if μ_3^m_a > 2 c_m_a/3, which is true if 4.225^m_a> 2 c_m_a/3.
This is the case for m_a=1 and any finite m_b≥ 1.  It follows
that ζ(1,m_b) < logμ_3 for all finite m_b≥ 1 (as already seen
in theorem <ref>).




























































  §.§.§ The hexagonal lattice



For the hexagonal lattice the coordination number is 3 and we know <cit.> 
the exact value of μ = μ_H = √(2+√(2)) rigorously, and we 
define κ_H = 
logμ_H.  Equation (<ref>) then gives

    ζ(m_a,m_b) ≤ m_b κ_H /m_a+m_b + log(c_m_a/3)/m_a+m_b.

Rearranging gives

    ζ(m_a,m_b) ≤κ_H - ( m_a κ_H - log (c_m_a/3)/m_a+m_b).

ζ(m_a,m_b) < κ_H if μ_H^m_a = (√(2+√(2)))^m_a 
> c_m_a/3.   Exact counts of c_n for the hexagonal lattice are available to
n=105[Jensen: https://oeis.org/A001668] <cit.>.
Examining these shows that ζ(m_a,m_b) < κ_H for m_a ≤ 15
(where c_n=29892). This gives the following theorem.


In the hexagonal lattice, ζ(m_a,m_b) < κ_H for all 1≤ m_a ≤ 15
and for all finite m_b≥ 1. 







  §.§.§ Some other lattices



In the triangular lattice the growth constant has bounds
4.118935 ≤μ_T ≤ 4.25152 <cit.>.  Defining
κ_T = logμ_T, equation (<ref>) gives 

    ζ(m_a,m_b) ≤ m_b κ_T /m_a+m_b 
     + log(2 c_m_a/3)/m_a+m_b,

and ζ(m_a,m_b) < κ_T if m_a κ_T > log( 2  c_m_a/3).
This shows that for m_a=1, ζ(1,m_b)<κ_T.


In the triangular lattice, ζ(1,m_b) < κ_T for all finite m_b≥ 1. 


Notice that the numerical bounds above can be used with equation eqn22
to find explicit bounds, but that none of these are better than the result in
theorem <ref>.

Bounds on the growth constant μ_K in the Kagome lattice are 
2.548497 < μ_K < 2.590301 <cit.>.  The 
coordination number is q=4.  Defining κ_K = logμ_K and 
using exact counts <cit.> show that if m_a κ_K > log c_m_a/2 
then ζ(m_a,m_b) < κ_K if m_a≤ 3 and for all finite m_b≥ 1.


In the Kagome lattice, ζ(m_a,m_b) < κ_K if 1≤ m_a≤ 3 
and for all finite m_b≥ 1. 








§ UPPER BOUNDS DERIVED USING LATTICE STARS AND OTHER LATTICE CLUSTERS



In the last section combs were constructed from several self-avoiding walks.  
Self-avoidance was relaxed between the backbone and teeth, as well as between
teeth, to obtain upper bounds on ζ(m_a,m_b).  An alternative approach
would be to decompose the comb into other sub-units (that are self-avoiding 
clusters in the lattice). While these are not mutually avoiding, they do 
give useful bounds in many cases. Examples of such clusters are shown 
in figure <ref>.  In order to use these clusters to determine bounds, 
exact counts will be needed.  For this we implemented a back-tracking 
algorithm coded to enumerate clusters of various fixed connectivities and 
branches of given sizes.  In tables <ref> and <ref> some counts 
for lattice 3-stars in the square and cubic lattices are shown.



 §.§ Creating combs by concatenating walks and lattice 3-stars


In the case of combs there are natural decompositions into sub-units which
are combinations of lattice stars and walks, or even larger sub-clusters.
By counting the number of conformations of the clusters, upper bounds on the
number of conformations of the combs are obtained (since the sub-units, while
self-avoiding, are not mutually avoiding).

In what follows we limit the discussion to the hypercubic lattice.  Similar
results can be obtained in other lattices.












As an example, in figure <ref> a comb is decomposed into a sequence of
3-stars with arms of lengths (x,y,z) (these are denoted by [x,z,y]) 
alternating with walks of length z.  Putting z=m_a and x=y=m_b shows 
that this is a decomposition of a comb into 3-stars with arms of lengths 
(m_a,m_b,m_b) and walks of length m_a. The number of such stars rooted 
at the origin in the central node, is denoted s([m_b,m_a,m_b]).  

In order to determine an upper bound using the decomposition in figure
<ref>, careful accounting of symmetry factors is needed. 

Concatenating lattice stars at the endpoints of arms as shown in figure <ref> 
in effect moves the root from the central node to the endpoints of arms.  If there 
are several arms which have the same length, then this introduces a symmetry factor 
in the concatenation.  In the particular case that 3-stars are concatenated as in
figure <ref>, proceed as follows.  If x≠y≠z≠x then
the number of 3-stars rooted at the endpoint of an arm of length x is
s([x,y,z]).  If x=y, or x=z, but y≠z, then there are two 
choices for the root at the endpoint of an arm of length x, so the number
is then 2 s([x,x,z]).  If x=y=z, then the number is 3 s([x,x,x]), by a 
similar argument. 

With the above in mind, put x=y=m_b and z=m_a in figure <ref>.
There are 2 s([m_a,m_b,m_b]) choices for the left-most star with the endpoint
of an arm of length m_b rooted at the origin.  There are, similarly,
2 s([m_a,m_b,m_b]) choices for the second star.  In addition, concatenating
the stars at the endpoint of arms of length m_b requires that the last steps
of these arms have different orientations.  This reduces the number of
choices for the second star by a factor (2d-1)/(2d)) to 
((2d-1)/(2d))× 2 s(m_a,m_b,m_b).  

To attach a tooth, we note that there are ((2d-2)/(2d))× c_m_a 
ways of selecting the tooth, since two directions are already taken by
the stars where they are joined together.  This remains true when concatenating 
the subsequent stars.  Thus, if the comb has t teeth and m_a≠m_b, 
then an upper bound on the number of combs is obtained (since the component 
walks and stars are not mutually avoiding):

    g(m_a,m_b,t)    ≤    (2 s([m_a,m_b,m_b]))×( (2d-1)/(2d) × 2 s([m_a,m_b,m_b]))^⌈ t/2⌉-1
            ×( (2d-2)/(2d) × c_m_a)^⌊ t/2 ⌋ .

Taking logarithms, dividing by tm_a+(t+1)m_b, and taking t→∞ gives

    ζ(m_a,m_b) ≤( log((2d-1)(d-1)/d^2) + log s([m_a,m_b,m_b]) 
            + log c_m_a)/(2(m_a+m_b)) .

In the square lattice, this gives

    ζ(m_a,m_b) ≤ ( log(3/4) + log s([m_a,m_b,m_b]) + log c_m_a )/(2(m_a+m_b)) .


In the case that m_a=m_b a more careful accounting of joining a star to the previous
star gives the inequality

    ζ(m_b,m_b) ≤( log(3(2d-1)(d-1)/(d^2)) + log s([m_b,m_b,m_b]) 
            + log c_m_b)/(2(m_b+m_b)) .

In the square lattice this gives

    ζ(m_b,m_b) ≤ (log(9/4) + log s([m_b,m_b,m_b]) + log c_m_b )/(4m_b) .

We state these results as theorems.


If m_a≠m_b then the following bounds hold for ζ(m_a,m_b) and ζ(m_b,m_b):

    ζ(m_a,m_b) ≤ ( log((2d-1)(d-1)/d^2) + log s([m_a,m_b,m_b]) + log c_m_a )/(2(m_a+m_b)) ,   ζ(m_b,m_b) ≤ (log(3(2d-1)(d-1)/d^2) + log s([m_b,m_b,m_b]) + log c_m_b )/(4m_b) .

These inequalities are true in the hypercubic lattice.  






Using theorem <ref> and noting that s([1,1,1])=c_1=4 in the square lattice, 
ζ(1,1) ≤log√(6) < log 2.4495 (where the bound is rounded up in the 
4-th decimal place).  This improves the bound in table <ref>.  In table <ref> 
additional bounds are shown on ζ(m_a,m_b) in the square and cubic lattices,
for 1≤ m_a,m_b ≤ 6, by using the counts in tables <ref> and <ref>
in theorem <ref>. 


We note that the bounds in table <ref> are due to self-avoiding walk bounds
on both the backbone and teeth of the comb.  In contrast the construction in
figure <ref> takes into account the local reduction in conformations when
the stars and walks are concatenated, reducing the conformational degrees
of freedom of the backbone.  However, intersections between the components may
occur, so that the backbone is not a self-avoiding walk.  Generally, the quality
of the bounds in theorem <ref> should improve with m_b, since larger stars
will result in longer segments along the backbone being self-avoiding.
Indeed,  equation PLI
(using the upper bound κ_2 < log 2.679193 <cit.>) gives
the (rigorous) bound ζ(10,10) < log 2.6989 while
fixing m_a=m_b=10 in theorem <ref> gives the better bound 
ζ(10,10) < log 2.6682 (solely relying on the exact counts 
s([10,10,10])=1126242913156 and c_10=44100).  This is smaller
than the bound determined on ζ(6,6) in table <ref>.





 §.§ Concatenating asymmetric lattice stars


Similar to the approach above, it is possible to use lattice stars in other ways
to determine upper bounds on  ζ(m_a,m_b).  In this section we focus 
on the constructions shown in figures <ref> and <ref>.



The general methods here are the same as in the last section, and so we
simply state the upper bound as a theorem.   The factors arising from
symmetry considerations are accounted for by defining a function
ς([x,y,z]) on stars [x,y,z] by putting ς([x,x,x])=6, 
ς([x,x,z])=ς([x,z,x])=ς([z,x,x])=2 if z≠x,
and ς([x,y,z])=1 otherwise, for x,y,z∈ℕ.

The construction in figure <ref> gives the following theorem.


ζ(z,x+y) ≤(
log ((2d-1)ς([x,y,z])/2d ) + log s([x,y,z])
) / (x+y+z). 


The result in theorem <ref> is not very useful in determining bounds 
on ζ(m_a,m_b).  Instead, a decomposition as shown in figure <ref>
consisting of larger 3-star components may be considered.  Denoting the
growth constant of combs with t teeth of alternating lenghts z and y
and backbone segments of length x by ξ(x,y,z), the following
lemma is obtained from figure <ref>.


ξ(x,y,z) ≤(
log ((d-1)ς([x,x+y,z])/d ) + log s([x,x+y,z])
) / (2x+y+z). 







By lemma <ref>:

  
Since ζ(m_a,m_b) = ξ(m_b,m_a,m_a), 

ζ(m_a,m_b)  
  ≤ (log ((d-1)ς([m_a,m_b,m_a+m_b])/d)
   + log s([m_a,m_b,m_a+m_b])) / 2(m_a+m_b). 
 

This shows, for the square lattice, that

    ζ(1,1) ≤ (log(ς([1,1,2])/2) + log s([1,1,2]) )/4 = (log 1+log 36)/4 = 
    log√(6) < log 2.4495 .

This is the same result as obtained in theorem <ref>.  In table <ref> we 
give square and cubic lattice bounds due to theorem <ref> on
ζ(m_a,m_b) for 1≤ m_a,m_b≤ 6 (using the data in table <ref>).  








Better bounds can be obtained by concatenating larger lattice clusters 
to create the comb.  If χ(x,y,z) is the connective constant of the comb constructed
in figure <ref>, then

    χ(x,y,z) ≤ (log ((d-1)ς([x,z,[x],x+y,z])/d) +
     log s([x,z,[x],x+y,z]) )/(3x+y+2z) .

This gives the following bound.


Since ζ(m_a,m_b) = χ(m_b,m_a,m_a), 

0.9ζ(m_a,m_b) ≤
(log( (d-1)ς([m_b,m_a,[m_b],m_b+m_a,m_a])/d)
  + log s([m_b,m_a,[m_b],m_b+m_a,m_a]))/3(m_a+m_b). 



Using theorem <ref>, and noting that in the square lattice
s([1,1,[1],2,1])=184 and ς([1,1,[1],2,1])=2,
ζ(1,1) ≤ (log(2/2)+log 184)/6 < log 2.3849, improving on the bound
derived in tables <ref> and <ref>.  In the cubic lattice s([1,1,[1],2,1])= 5616 
and ς([1,1,[1],2,1])=2 so that
ζ(1,1) ≤ (log(2× 2/3) + log 5616)/6 < log 4.4232.  This is smaller
than the bound in table <ref>, but does not improve on the bound in 
table <ref>.















































§ EXTENSION TO BRUSHES



In this section we look briefly at extensions of some of these ideas to brushes.  
A brush is a modification of a comb where there are at least 2 teeth
at each branch point, so the backbone has t vertices of degree f where 
4 ≤ f ≤ q.  All the teeth have m_a edges.  We can obtain an upper bound 
on brushes by adaptations of the argument in section <ref>.  
We write b_f(m_a,m_b,t) for the number of brushes with t vertices of degree f,
t(f-2) teeth, all with m_a edges, and t+1 backbone segments, 
all with m_b edges. 

The arguments in section <ref> can be adapted to work for brushes 
with only minor modifications.  By treating the teeth and backbone as independent 
self-avoiding walks, the upper bound in (<ref>) becomes

    b_f(m_a,m_b,t) ≤ c_(t+1)m_bγ-2f-2^t 
    (c_m_a/γ)^(f-2)t.

The corresponding bound on the free energy is given by

    ζ_f(m_a,m_b) ≤m_b logμ/(f-2)m_a+m_b 
    + logγ-2f-2 
    - (f-2) logγ + (f-2) log c_m_a/(f-2)m_a+m_b.


For the square lattice with f=4 the brush must have 2 teeth at each branch point.  
This gives 

    ζ_4(m_a,m_b) ≤m_b logμ_2/2 m_a+m_b + 2 log (c_m_a/4)/2 m_a+m_b.

Rearranging gives

    ζ_4(m_a,m_b) ≤logμ_2 + 2 log (c_m_a/4) - 2 m_a logμ_2/2m_a + m_b.

ζ_4(m_a,m_b) < logμ_2 if c_m_a/4 < 2.625622^m_a≤μ_2^m_a, 
using the bound on μ_2 in equation lower:mu 
<cit.>.   This is true for m_a ≤ 23, for any  m_b such that 1 ≤ m_b < ∞..


In the square lattice, for 1≤ m_a ≤ 23 and 
1≤ m_b < ∞, ζ_4(m_a,m_b) < κ_2. 


For the simple cubic lattice γ=6 and f=4,5 or 6. 
ζ_f(m_a,m_b) < logμ_3 if

    4f-2 (c_m_a/6)^f-2 < 4.225^(f-2)m_a≤μ_3^(f-2)m_a

where we have used the bound on μ_3 in <cit.>.  Using the
values of c_n from <cit.> we find that this condition is satisfied if 
m_a ≤ 4 when f=4, when m_a ≤ 7 when f=5 and when m_a ≤ 11 
when f=6, for all m_b such that 1 ≤ m_b < ∞.


In the cubic lattice, for 1≤ m_b < ∞, the free energy ζ_f (m_a,m_b) < κ_3 
when 1≤ m_a≤ 4 if f=4, 1≤ m_a≤ 7 if f=5 and 1≤ m_b≤ 11
if f=6. 






We can also construct upper bounds on brushes by concatenating stars.  
In figure <ref> we show two decompositions of brushes into lattice stars, amongst
many other ways.  The top sequence shows a 4-brush decomposed into 3-stars,
and the bottom a decomposition into 4-stars.  In the square lattice the top 
decomposition (with m_b=x and m_a=y=z) gives

    ζ_4(m_a,m_b) ≤ ( log (ς([m_b,m_a,m_a])/4) + log s([m_b,m_a,m_a]) )/(2 m_a+m_b)

where ς([m_b,m_a,m_a]) is again a symmetry factor which is equal
to 3 if m_a=m_b (and equal to 1 otherwise) and the factor (1/4) is the 
result of the fact that only one direction is available when concatenating a star 
to its predecessor.  

If the coordination number of the lattice is more than 4 then 
using f-stars instead of 3-stars in the top decomposition gives (f+1)-brushes.
In the d-dimensional cubic lattice, using f stars with 3≤ f≤ 2d-1 
and the top decomposition in figure <ref>, one similarly obtains the bound

    ζ_f+1(m_a,m_b) ≤ ( log ((2d-f) ς([m_b,m_a,…,m_a])/2d )
    + log s([m_b,m_a,…,m_a]) )/((f-1)m_a+m_b) .

Here, ς([m_b,m_a,…,m_a]) = f if m_a=m_b, and it is equal to 1
otherwise.  Other, similar inequalities, can be obtained for other lattices, using
the same kind of analysis.  Notice that this reduces to equation eqn36
if d=2. We state this as the following theorem.


If 3≤ f≤ 2d-1 then the connective constant ζ_f(m_a,m_b) of 
f-brushes in the d-dimensional hypercubic lattice is bounded by

    ζ_f+1(m_a,m_b) ≤ 
    ( log ((2d-f) ς([m_b,m_a,…,m_a])/2d)
    + log s([m_b,m_a,…,m_a]) )
    /((f-1) m_a+ m_b) ,

where ς([m_a+m_b,m_a+m_b,m_a,m_a])=f if
m_a=m_b, and equal to 1 otherwise. 


In the square lattice with m_a=m_b=1 and f=3 one obtains the bound
ζ_4(1,1) ≤ ( log 3 )/3 < log 1.4423.  In the cubic lattice, one similarly
obtains ζ_4(1,1) < log 3.1073, ζ_5(1,1) < log 2.1148 and
ζ_6(1,1) < log 1.3780.  More estimates are shown in table <ref>,
using the data in tables <ref>, <ref>, <ref> and
<ref>.




A similar analysis using the bottom decomposition in figure <ref> gives, 
with m_a=y=z and m_b=x, the upper bound

    ζ_4(m_a,m_b) ≤ 
    ( log (2 (d-1)/d)
    + log s([m_a+m_b,m_a+m_b,m_a,m_a]) )
    /(8 m_a+4 m_b) .

Here, the factor of (d-1)/d is due to the fact that (2d-2) directions,
out of 2d, are available for the edge on the part of the arm between the
node and the vertex where the star is concatenated. The symmetry factor 
ς([m_a+m_b,m_a+m_b,m_a,m_a]) = 2 by default, for all 
values of m_a and m_b.  Using this and the data in table <ref>
gives ζ_4(1,1) ≤log 1.3855 in the square lattice, and the data
in table <ref> gives ζ_4(1,1) ≤log 1.9435 in the
cubic lattice, improving on the bounds in table <ref>.  Similarly, in
the square lattice this gives ζ_4(2,2) ≤log 1.4790, and in the cubic
lattice, ζ_4(2,2) ≤log 2.0424, as compared to the results in 
table <ref>.

More decompositions may be done, but it remains unclear which would give
optimal bounds.  In general, the larger the basic component used to determine
the bound, and the extent to which intersections between the components
are avoided, the better the bound.






§ SOME LIMITING CASES AND LOWER BOUNDS



In this section we shall examine situations where m_a →∞ 
or m_b →∞ after the limit t →∞ has been taken.  



 §.§ Lower bounds


To obtain lower bounds on ζ we shall consider the subset of combs on the 
simple cubic lattice, confined to a half-space (z ≥ 0) and terminally attached 
to the dividing plane (z=0).  This subset of combs has been investigated in 
reference <cit.>.  

We first look at the case where m_b →∞ and m_a →∞ but with 
m_a = o(m_b) so that the teeth grow less rapidly than the backbone segments.
This was called the self-avoiding walk limit in <cit.>.
We can use the result of theorem 6 in <cit.> 
with a=b=y=1 to show that

    lim_m_a,m_b →∞ m_a=o(m_b)ζ(m_a,m_b) ≥logμ_3.

The pattern theorem arguments used in section <ref> do not apply 
when m_b →∞ since the pattern P must be finite.

If m_a →∞ and m_b →∞ but with m_b = o(m_a) so that 
the teeth grow more rapidly than the backbone segments,
this was called the star limit in <cit.>.
We can use the result of theorem 4 in <cit.> 
with a=b=y=1 to show that

    lim_m_a,m_b →∞ m_b=o(m_a)ζ(m_a,m_b) ≥logμ_3.


Next consider the case where m_b = ⌊α p ⌋ and 
m_a = ⌊ (1-α)p ⌋ for some α∈ (0,1).  If we let
p →∞ then both m_a,m_b go to infinity.  Using the result of 
theorem 5 in <cit.> with a=b=y=1
(and remembering that κ(1)=λ(1) = logμ_3) we obtain the bound

    lim_p→∞ζ(⌊ (1-α)p ⌋, ⌊α p ⌋) ≥logμ_3,

for all α∈ (0,1).



 §.§ Upper bounds


We can obtain a upper bound by regarding the backbone as independent 
fom the teeth, and the teeth as independent of one another.  This gives

    g(m_a,m_b,t) ≤ c_(t+1)m_b c_m_a^t

and therefore

    log g(m_a,m_b,t)/(m_a+m_b)t≤m_b(t+1)/(m_a+m_b)t 1/m_b(t+1)log c_(t+1)m_b 
    +m_a/(m_a+m_b)1/m_alog c_m_a.

Letting t →∞ gives

    ζ(m_a,m_b) 
    ≤m_b/m_a+m_blogμ_3 
      + m_a/m_a+m_b 1/m_alog c_m_a.

We now consider three cases.  First, let m_b →∞ with m_a=o(m_b).  
The first term goes to logμ_3 and the second term goes to zero.  Next, 
let m_a →∞ and m_b = o(m_a).  In this case the first term 
goes to zero and the second term goes to logμ_3.  Finally, let 
m_b = ⌊α p ⌋ and m_a= ⌊ (1-α)p ⌋ with 
α∈ (0,1).  Then

    ζ(⌊ (1-α) p ⌋,  ⌊α p ⌋ ) 
    ≤αlogμ_3 + (1-α)   1/⌊ (1-α)p ⌋log c_⌊ (1-α)p ⌋

Letting p →∞ gives

    lim_p→∞ζ(⌊ (1-α) p ⌋, ⌊α p ⌋) ≤logμ_3.


This shows that, for the three limiting cases that we have considered, 
ζ (m_a,m_b) = logμ_3.  Similar arguments can be constructed for other lattices.



§ DISCUSSION



We have investigated the exponential growth rate of the numbers of combs embeddable in a lattice.  A comb consists of a backbone self-avoiding walk, and a number of teeth attached to the backbone at vertices of degree 3.  The comb has t teeth, all with m_a
edges, with m_b edges between adjacent vertices of degree 3 and between the end vertices of degree 3 and the end vertices of degree 1 of the backbone.
We have been mainly interested in the case where m_a and m_b are fixed and t goes to infinity, and we have derived upper bounds on the exponential growth rate, using several different approaches, that show that the growth rate is strictly less than that of self-avoiding walks, when m_a is small.

To derive upper bounds on the exponential growth rate, we have used two main approaches.  The first is to allow the backbone and teeth to intersect, and to
allow the teeth to intersect one another.  This is an extension of an approach due to Lipson <cit.>.
The second approach that we have used is to break up the comb into small lattice clusters (such as stars or walks) which are not necessarily mutually avoiding.  The two approaches give different upper bounds and both are useful for various values of m_a and m_b. In section
<ref> we have extended these ideas to brushes, where more than one tooth is attached to the backbone at each branch point.

There are a number of contributing factors.  Our use of a pattern theorem <cit.> in section <ref> shows that the teeth reduce the contribution to the entropy from the conformational freedom of the backbone.  The teeth may themselves contribute more entropy than they would as components of an infinite walk, since c_n 
> μ^n for small values of n.  This is offset by two factors.  The first is that the teeth must be mutually avoiding, and must avoid the backbone, resulting in a lower entropy
contribution.  The second is that the number of possibilities for the first step of each tooth is reduced by the fact that the backbone uses two of the possible positions at each degree 3 vertex.  These competing factors lead to a reduction in entropy when m_a is small.  Whether this is true for all finite values of m_a is an open question.




§ ACKNOWLEDGEMENT

EJJvR acknowledges financial support from the Natural Sciences and Engineering 
Research Council of Canada (NSERC) [funding reference number: 
RGPIN-2020-06339]. 



 


§ REFERENCES


plain










































































 Paper Title: Quantum Thermodynamics of Open Quantum Systems: Nature of Thermal Fluctuations.







neha@ctp-jamia.res.in

devvrat.1@iitj.ac.in


subhashish@iitj.ac.in

Department of Physics, Indian Institute of Technology, Jodhpur, India



§ ABSTRACT 

We investigate the thermodynamic behavior of open quantum systems through the Hamiltonian of Mean Force, focusing on two models: a two-qubit system interacting with a thermal bath and a Jaynes-Cummings Model without the rotating wave approximation. By analyzing both weak and strong coupling regimes, we uncover the impact of environmental interactions on quantum thermodynamic quantities, including specific heat capacity, internal energy, and entropy. Further, the ergotropy and entropy production are computed. We also explore the thermodynamic uncertainty relation, which sets an upper bound on the signal-to-noise ratio. 


Quantum Thermodynamics of Open Quantum Systems: Nature of Thermal Fluctuations
    Subhashish Banerjee
    
1Fusion@ÖAW, Institute for Theoretical Physics - Computational Physics, Graz University of Technology, 8010 Graz, Austria

2 Max-Planck-Institut für Plasmaphysik, 17489 Greifswald, Germany

3Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and Technology”, 61108 Kharkov, Ukraine

===============================================================================================================================================================================================================================================================================================================================



§ INTRODUCTION

The study of thermodynamics within the realm of open quantum systems is a burgeoning area that seeks to understand how classical thermodynamic principles apply when quantum effects are significant  <cit.>. This field is particularly relevant for systems interacting with their surroundings, leading to complex behaviors and phenomena not observed in isolated systems. 
A fundamental development in the theory of open quantum systems is the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation <cit.>, which offers a framework for describing the time evolution of a system S under the influence of weak interactions with a large external environment. Advances in theoretical and experimental domains implore one to go beyond the scope of GKLS dynamics, specifically where strong-coupling non-Markovian effects are prominent <cit.>.

Open quantum systems that are non-Markovian demonstrate the intricate connection between a quantum system and its surroundings. Unlike their Markovian counterparts, non-Markovian systems retain a memory of their past, resulting in complex dynamical behavior  <cit.>. A prominent feature of non-Markovian evolution is the resurgence of quantum properties  <cit.>, the examination of which is crucial for comprehending the system's dynamics under varying levels of coupling with the environment. Studying the dynamics of thermodynamic quantities when the system has strong coupling with the environment in the non-Markovian regime presents a formidable challenge  <cit.>. Numerous techniques have been developed recently to address the dynamics of such systems. These encompass the Hamiltonian of mean force (HMF)  <cit.>, the method of hierarchical equations of motion, a numerical method utilizing the influence-functional formalism <cit.>, the reaction coordinate method, which explores strong coupling effects through mapping the Hamiltonian with a reaction coordinate  <cit.> and pseudomodes technique  <cit.>. 

Here, we make use of the Hamiltonian of Mean Force to investigate the strong-coupling non-Markovian effects. This concept of HMF has been previously studied in the Caldeira-Leggett Model <cit.>, Spin-Boson Model <cit.>, Quantum Brownian Motion <cit.> to ascertain the system's effective dynamics and thermodynamics and also in molecular systems <cit.> to study solvation dynamics and chemical reaction rates in complex environments. It has been observed that HMF provides consistent generalizations of the laws of thermodynamics and fluctuation relations <cit.>.

Recent advances in the study of open quantum systems have yielded significant insights into the behavior of quantum thermodynamic quantities <cit.>, and with technological advancements, this is being leveraged towards quantum thermal devices <cit.>. The specific heat capacity of the systems coupled to thermal reservoirs demonstrates non-trivial temperature dependencies influenced by quantum effects <cit.>. Along with ubiquitous entropy, another quantity prominent in quantum thermodynamics is entropy production, which, along with system dynamics, also depends on global state evolution <cit.>. The Fluctuation Dissipation Relation (FDR), which connects response functions to correlation functions, is fundamental in understanding dissipation and fluctuations  <cit.>. It turns out that in the strong coupling regime of thermodynamics, FDR has modifications coming from the specific heat capacity and another term that is an essential ingredient in computing the system's internal energy <cit.>. Additionally, the concept of ergotropy, representing the maximum extractable work from a quantum system via unitary transformations, was introduced  <cit.> and is a crucial ingredient in characterizing the thermodynamic behavior of a system. These studies collectively advance our understanding of the complex behavior and properties inherent to open quantum systems.

Another development in this field is a bound on signal-to-noise ratio for estimation of temperature  <cit.>. This involves tools of quantum metrology  <cit.> in the form of the quantum Cramer-Rao inequality for an unbiased estimation of the temperature of the system. As is well known, the quantum Cramer-Rao inequality is bounded by the Fisher information  <cit.>. The other ingredient in this bound is the modified fluctuation-dissipation relation involving the specific heat capacity of the system as well as the Wigner-Yanase-Dyson skew information and another term that is pertinent to computing the system's internal energy  <cit.>.

Dynamics of two-qubit systems interacting with a squeezed thermal bath has revealed significant insights into entanglement dynamics, coherence, and quantum correlations <cit.>. The Jaynes-Cummings model  <cit.>—which describes a single two-level system interacting with a single-mode resonator—has seen significant progress in improving quantum measurement techniques specifically in quantum sensing and metrology and also in understanding the interplay between coherence and quantum thermal states <cit.>.
In this article, we discuss the quantum thermodynamics of the following open quantum systems models: (i) a two-qubit model interacting with a thermal bath and (ii) the Jaynes-Cummings Model without the rotating wave approximation, using the Hamiltonian of Mean force across all coupling regimes. 


The structure of the paper is as follows. The preliminary section (Sec. II) includes an introduction to the Hamiltonian of Mean Force, certain thermodynamical potentials, thermodynamic quantifiers like ergotropy and entropy production, and a brief explanation of quantum information theoretic quantities such as WYD skew-information and Quantum Fisher information. In Sec. III, we have briefly introduced the models under study and discussed their quantum thermodynamic properties using the Hamiltonian of Mean Force. This is followed by the conclusion in Sec. IV.


§ PRELIMINARIES

This section is divided into two categories. The first category focuses on Quantum Information-theoretic quantities, where we define quantum uncertainty, skew information, and Fisher information. The second category pertains to quantum statistical mechanics, introducing the Hamiltonian of Mean Force and thermodynamic potentials such as entropy and specific heat capacity. Additionally, we provide a brief introduction to entropy production and ergotropy.



 §.§ Quantum Information-Theoretic Quantities



  §.§.§ Quantum Uncertainty and Wigner-Yanase-Dyson Skew Information

Skew information <cit.>, quantifies the quantum uncertainty in a state relative to an observable defined by I(ρ, Y) = -1/2 Tr ([√(ρ), Y]^2). It measures the non-commutativity between the state and the observable and has been extensively studied and generalized. The concept of skew information was extended through the definition I_η (ρ, Y)= -1/2([ρ^η,Y][ρ^1-η,Y]), commonly referred to as WYD entropy <cit.>, where 0<η<1. When η = 1/2, it simplifies to the well-known skew information. This generalized form can be expressed as:

    I_η (ρ, Y)= (ρ Y^2)- (ρ^η Y ρ^1-η Y).

In comparison, the usual variance of an observable Y for the quantum state ρ is given by <cit.>,

    Var(ρ, Y) = (ρ Y^2) - ((ρ Y))^2= Q[ρ̂, Ŷ]+ K[ρ̂, Ŷ].

The variance of the observable Y is influenced by both quantum and classical uncertainties inherent in the system. The measures of quantum and classical uncertainties, Q[ρ̂, Ŷ] and K[ρ̂, Ŷ], respectively, are defined as follows <cit.>

    Q[ρ̂, Ŷ] = ∫_0^1 Q_a[ρ̂, Ŷ]   da,

where Q_a[ρ̂, Ŷ] = - 1/2( [Ŷ, ρ̂^a] [Ŷ, ρ̂^1-a] ) with a ∈ (0,1). This measure Q[ρ̂, Ŷ] quantifies the quantum uncertainty associated with the observable Ŷ in the state ρ̂. The negative sign reflects the non-commutativity of ρ̂ and Ŷ, capturing the essence of quantum fluctuations.
Similarly,

    K[ρ̂, Ŷ] := ∫_0^1 K_a[ρ̂, Ŷ]   da,

where K_a[ρ̂, Ŷ] := ( ρ̂^aδŶρ̂^1-aδŶ) with a ∈ (0,1). Here, δŶ = Ŷ - ⟨Ŷ⟩ represents the deviation of Ŷ from its expectation value. The measure K[ρ̂, Ŷ] captures the classical uncertainty, reflecting the statistical spread of outcomes due to classical probabilities. The parameter a indicates that there is no unique way of decomposing the variance into quantum and classical components. These quantum and classical uncertainty parameters play a very important role in the thermodynamics of quantum systems which is explored in the later sections.



  §.§.§ Quantum Fisher Information

Quantum Fisher Information (QFI) is a crucial concept in quantum metrology and information theory, representing the sensitivity of a quantum state to changes in a parameter. It is a fundamental tool for determining the ultimate precision limits in parameter estimation tasks, such as those encountered in quantum thermometry. Mathematically, QFI is defined through the symmetric logarithmic derivative (SLD), Λ_θ

    F(θ) = Tr[ρ(θ) Λ_θ^2],
 where ρ(θ) is the density matrix of the quantum state dependent on the parameter θ.
This quantity is pivotal as it establishes the quantum Cramér-Rao bound (QCRB), expressed as

    Δθ≥1/√(N F(θ)),

where N is the number of measurements, dictating that the variance of any unbiased estimator θ is inversely proportional to the QFI. It has been recently proved  <cit.> that for an exponential state ρ̂_θ= exp(-Φ̂_θ)/𝒵_θ the QFI concerning the parameter θ is bounded by,

    F(θ) ≤ K[ρ̂_θ, Ŷ_θ],
 where, Ŷ_θ is the thermodynamic observable defined as Ŷ_θ=∂_θΦ̂_θ ,and K[ρ̂_θ, Ŷ_θ] is the classical uncertainty defined in Eq. (<ref>).
The spectral decomposition of the state ρ̂_θ= exp(-Φ̂_θ)/𝒵_θ is given by ρ̂_θ= ∑_n p_n|ψ_n⟩⟨ψ_n| where the eigenstates satisfy 
Φ̂_θ| ψ_n⟩  = λ_n| ψ_n⟩. From this spectral decomposition, the QFI can be expressed as  <cit.>,

    F(θ) = 2 ∑_n,m| ⟨ψ_n| ∂_θρ̂_θ| ψ_m⟩|^2/p_n + p_m.
 

In this work, we have considered this unbiased estimator parameter θ as Temperature. For a quantum thermal state, the Fisher Information for the temperature is proportional to the specific heat capacity of the system as F(T)= C_S/T^2  <cit.>.



 §.§ Quantum Statistical Mechanics Related Quantities



  §.§.§ Hamiltonian of Mean Force

A quantum system 𝒮 interacting with a thermal bath ℬ is described by the Hamiltonian,

    Ĥ_𝒯 = Ĥ_S ⊗𝕀̂_B  + 𝕀̂_S ⊗Ĥ_B + 𝒱̂_SB,

where Ĥ_S and Ĥ_B are the hamiltonian of system and bath respectively, while 𝒱̂_ SB is an interaction term of arbitrary strength. Here, we will consider situations where the environment is huge compared to the system, i.e. the operator norms fulfill ||Ĥ_B|| ≫ || Ĥ_S||, ||𝒱̂_ SB||. The Hamiltonian of mean force takes into account the average effects of the environment on the subsystem.
The global equilibrium state of the system and the reservoir at temperature T is denoted by ζ̂_SB, which is of Gibbs form,

    ζ̂_SB := exp(-βĤ_𝒯)/𝒵_SB,

where, 𝒵_SB = _SB[exp(-βĤ_𝒯)]
is the partition function for SB. Also, β = (k_bT)^-1, the Boltzmann constant k_b is set to unity throughout. The interaction term causes the reduced state of the system S, ζ̂_S(T) =  Tr_B[ζ̂_SB(T)] to deviate from the thermal state unless the coupling is very weak (the magnitude of interaction term 𝒱̂_SB is much lesser than the magnitude of the bare system hamiltonian H_S). Thus, for the purpose of calculating internal energy in the strong coupling regime, the system's bare Hamiltonian would not be useful. 

This issue can be resolved by rewriting the state of the system S as an effective Gibbs state ζ̂_S(T) = exp(-βĤ_𝒮^* (𝒯))/𝒵_S^*, where the partition function for the system S can be expressed as the ratio 

    𝒵_S^* = 𝒵_SB/𝒵_B,
 where 𝒵_SB= _SB[e^-βH_SB] and 𝒵_B= _B[e^-βĤ_B] and,

    Ĥ_𝒮^* (𝒯) := - 1/βln(  Tr_B[exp(-βĤ_̂𝒯̂) ]/ Tr_B[exp(-βĤ_̂ℬ̂)]),

is the Hamiltonian of Mean Force  <cit.>. 
This operator can be interpreted as an effective Hamiltonian describing S, and unlike the bare Hamiltonian Ĥ_S, it implicitly depends upon both the temperature T and the interaction 𝒱̂_SB. In the weak coupling regime, this reduces to the bare system Hamiltonian. HMF is essential for understanding thermodynamic properties such as free energy, entropy, and heat capacities in systems where direct interactions with an environment cannot be ignored. It allows for the formulation of generalized thermodynamic potentials and facilitates the study of equilibrium and non-equilibrium processes, providing insights into phenomena such as quantum dissipation and decoherence.



  §.§.§ Thermodynamic Potentials

The partition function in statistical mechanics contains information regarding the distribution of occupation probabilities among the various microstates of the system S. It enables the calculation of important thermodynamic potentials like free energy, internal energy, and entropy.
The internal energy of the system S can  be written as <cit.>,

    U_S(T) = U_SB(T) - U_B^'(T).

This U_S(T) is simply the difference between the total energy U_SB(T)= -∂_βln𝒵_SB and the energy of the reservoir, U_B^'(T)= -∂_βln𝒵_B.
U_S(T) can be expressed as an expectation value <cit.>, 

    U_S(T)    = ⟨Ê_S^*(T) ⟩ =  Tr[ Ê_S^*(T). ζ̂_S ] 
       =  Tr[∂_β ( βĤ_𝒮^* (𝒯) ). ζ̂_S ] ,

of the observable,

    Ê_S^*(T) = ∂_β [ βĤ_𝒮^* (𝒯) ] 
        = Ĥ_S+  ∂_β[β (Ĥ_𝒮^* (𝒯)- Ĥ_S)].

This Ê_S^*(T) can be interpreted as the effective energy operator describing the system whose eigenstates are referred to as the system's energy eigenstates.  
With the inclusion of this operator, the fluctuations in the internal energy are considered as:

    U_S= √( Var[ζ̂_S, Ê_S^*]).

It is significant to remember that Ê_S^*(T) depends on both the temperature T and the system-environment coupling 𝒱̂_ SB. This Ê_S^*(T) differs from both the bare system hamiltonian Ĥ_S and hamiltonian of mean force Ĥ_S^*(T) <cit.>. From Eq. (<ref>) we can observe that only in those cases where Ĥ_S^*(T) becomes temperature independent will Ĥ_S^*(T) be almost equivalent to Ê_S^*(T). One example is the bilinear coupling model for Quantum Brownian motion  <cit.>. 

The entropy of the system is calculated as

    𝒮_S = -  Tr[ (lnζ̂_S) . ζ̂_S] + β^2  Tr[ (∂_βĤ_S^*).ζ̂_S  ].

Here, we recognize the first contribution as the von Neumann entropy for the system, while the second term stems from the temperature dependence of the HMF. This implies that the thermodynamic entropy is generally not equivalent to the information in the equilibrium state when interactions are present.

In the weak-coupling limit, it is well established that the specific heat capacity C_S is proportional to the variance in internal energy, expressed as  <cit.>

    C_S = β^2 Δ U_S^2.

This is a standard result of the fluctuation-dissipation relation, which connects the rate of change of the system's energy with temperature to the energy fluctuations at equilibrium. 
Recent research has indicated that for open quantum systems of the form given in Eq. (<ref>), the heat capacity can become negative at low temperatures, suggesting that it is not generally proportional to a positive variance. Further, in the presence of strong coupling, the fluctuation-dissipation relation (FDR) includes two additional contributions <cit.>,

    C_S(β) = β^2 Δ U_S^2 - β^2 Q[ζ̂_S, Ê_S^*] - 1/β^2⟨∂_βÊ_S^* ⟩,

where all the terms are as defined above.
This equation implies that C_S(β) can be less than β^2 Δ U_S^2 and even negative.
The first correction term arises from quantum fluctuations in energy, represented by the average WYD information related to the observable Ê_S^*. The second correction term ⟨∂_βÊ_S^* ⟩ accounts for the temperature dependence of Ê_S^* and remains in the classical limit. In the weak coupling limit, where Ê_S^* = Ĥ_S^*, these correction terms disappear, making the heat capacity proportional to the energy variance.
When considering temperature as the unbiased parameter θ, the Cramer-Rao bound inequality in Eq. (<ref>) can be written as Δβ≥1/√(F(β)) for a single measurement N=1. Thus, Eq. (<ref>) can be reformulated as F(β) ≤ K[ζ̂_S, Ê_S^*]. From this, a modified thermodynamic uncertainty relation was obtained as <cit.>,

    Δβ≥1/√(Δ U_S^2 - Q[ζ̂_S, Ê_S^*]) .

Comparing Eqs. (<ref>) and (<ref>), the uncertainty in the temperature in the strong coupling regime can be expressed as

    Δ T ≥T/√(C_S(T) - ⟨∂_TÊ_S^* ⟩) .

This expression sets a lower bound on the thermal fluctuations Δ T. In the scenario of strong coupling, the best possible signal-to-noise ratio T/Δ T for determining the temperature of the system S is constrained by both the specific heat capacity and an additional dissipation term ⟨∂_T Ê^*_S ⟩, which can be negative or positive. In the case of weak coupling, this additional dissipation term is zero. Therefore, in that case, the optimal signal-to-noise ratio T/Δ T depends solely on the specific heat capacity, indicating that accurate temperature measurement requires a large heat capacity.
 


 §.§ Quantum thermodynamic quantifiers

Here, we briefly discuss quantum thermodynamic quantifiers, particularly ergotropy and entropy production. 


  §.§.§ Ergotropy

Ergotropy is the maximum work that can be extracted from a quantum system <cit.>. Consider a quantum state ρ, whose spectral decomposition is given by 

    ρ = ∑_i r_i|r_i⟩⟨r_i|,

where r_i's (in the order r_1>r_2>r_3>...) are the eigenvalues of ρ, and the spectral decomposition of the system's Hamiltonian H_S is given by

    H_S = ∑_j e_j|e_j⟩⟨e_j|,

where e_j (in the order e_1<e_2<e_3...) are eigenvalues of the system's Hamiltonian. The ergotropy for this system is given by 

    𝒲[ρ(t)] = ∑_j, i r_ie_j(|⟨r_i||e_j⟩|^2 - δ_ij).




  §.§.§ Entropy production

The transfers undergone by the system during its evolution are alluded to as entropy production. It accounts for the irreversibility that arises due to discarding the bath degrees of freedom. 
Consider a system bath (S-B) evolution of an arbitrary initial state ρ_SB(0) dictated by the unitary operator U, such that ρ_SB(t) = Uρ_SB(t)U^†. Entropy production <cit.> is given by 

    Σ = ℐ(S:B) + S[ρ_B(t)||ρ_B(0)],

where ℐ(S:B) is the mutual information given by the von Neumann entropies (S(ρ) = - Tr(ρlnρ)) of the subsystems S and B and the joint system SB, that is, 

    I(S:B) = S(ρ_S) + S(ρ_B) - S(ρ_SB).

S(ρ||σ) =  Tr(ρlnρ - ρlnσ) is the quantum relative entropy. The above relation can further be simplified as

    Σ = S[ρ_SB(t)||ρ_S(t)⊗ρ_B(0)],

which is the Kullback-Leibler divergence between the joint system-bath evolved state and the state ρ_S(t)⊗ρ_B(0), where ρ_S(t) =  Tr_B(Uρ_SB(t)U^†). 



§ ILLUSTRATIVE OPEN QUANTUM SYSTEM MODELS

Here, we briefly discuss the models under consideration in this work. This is followed by a study of their corresponding quantum thermodynamic properties.




 §.§ Two-qubit system interacting with a thermal bath

We start with a two-qubit system interacting with a thermal bath via a dissipative interaction. 
The dissipative interaction between 2-qubits (a two-level atomic system) and the bath (represented as a three-dimensional electromagnetic field (EMF)) via the dipole interaction  <cit.> is described by the Hamiltonian,

    H    = H_S + H_B + H_SB
       = ∑_j=1^2 ħω_j S_j^z 
                     + ∑_k⃗sħω_k ( α_k⃗s^†α_k⃗s + 1/2) 
          - iħ∑_k⃗s⃗∑_j=1^2 [ μ⃗_j ·g⃗_k⃗s⃗( r⃗_j )(S_j^+ + S_j^-) α_k⃗s - h.c. ]

The two-qubits are modeled as two-level systems with excited state |e_j⟩, ground state |g_j⟩, transition frequency ω_j, and transition dipole moments μ_j. We assume that the qubits are located at different atomic positions r_j. The transition dipole moments are dependent on r_j.
S_j^+ = |e_j⟩⟨ g_j| and S_j^- = |g_j⟩⟨ e_j| are the dipole raising and lowering operators, respectively, satisfying the commutation relation and 
S_j^z = 12 (|e_j⟩⟨ e_j| - |g_j⟩⟨ g_j|)
is the energy operator of the jth-atom. 
α_k⃗s^†, α_k⃗s are the creation and annihilation operators of the field mode k⃗s with wave vector k⃗, frequency ω_k and the polarization index s. The system-reservoir coupling constant is

    g_k⃗s (r⃗_j) = (ω_j/2ϵħ V)^1/2e_k⃗s e^i k⃗.r⃗_j.

V is the normalization volume, and e_k⃗s is the unit polarization of the field. This equation implies that the system-reservoir coupling constant depends on the atomic position r_j. This leads to two regimes: one where the inter-qubit spacing is less than the length scale set by the bath, that is, the collective regime, and the other where the inter-qubit spacing is comparable with the length scale set by the bath, the independent regime <cit.>. In this work, the collective regime is considered for further analysis. 

For simplification, we assume that both the qubits have the same transition frequency and the first qubit is at the origin while the distance between them is ξ. Further, k⃗ is taken to be along r⃗_12. The dipole moments μ⃗_j's are taken to be equal for both qubits ( μ_1 = μ_2 = μ) and along the direction of a polarization vector, such that μ⃗_j.e⃗_k⃗s = μ. Incorporating these quantities, the Hamiltonian in Eq. (<ref>) (for ħ = c = 1) can be rewritten as

    H    = ω_0(S_1^z + S_2^z) + ∑_k⃗ω_k ( α_k^†α_k + 1/2) 
       -i ∑_kλ√(ω_k)[(S_1^+ + S_1^-)(α_k - α_k^†).
       . +(S_2^+ + S_2^-)(α_ke^iω_kξ - α_k^† e^-iω_kξ) ],

where λ  = (μ/√(2ϵ_0V)). 
The reduced density matrix of the two-qubit system in the interaction picture and the standard Born-Markov rotating wave approximation (RWA) was obtained in <cit.> by assuming separable initial system-bath correlation and taking a trace over the bath. 
When λ√(ω_k)∼ω_0, the interaction strength between the system and the bath is comparable to the system's natural frequency ω_0. This characterizes a strong coupling regime, wherein the system and the bath can efficiently exchange energy, resulting in significant mutual influence on each other's dynamics. 
Conversely, when λ√(ω_k)≪ω_0, the interaction strength is considerably smaller than the system's natural frequency. This corresponds to a weak coupling regime, where the interaction between the system and the bath is minimal. Consequently, the system's dynamics are less perturbed by the bath, and energy exchange between the two is limited.

HMF for this model is calculated using Eqs. (<ref>) and (<ref>), where ℋ_𝒯 is the total Hamiltonian of the system and bath H. The numerical form of the HMF for this model is given by 

    H^*_S = [ a 0 0 e; 0 b f 0; 0 g c 0; h 0 0 d ].

It is observed that in the weak coupling limit, the off-diagonal elements e, f, g, h tend to become zero, and the above Hamiltonian becomes equal to the bare Hamiltonian of the system H_S.   
The specific heat capacity and the corresponding variables in its expression are plotted in Fig. <ref>.

As can be observed, the quantum uncertainty and the quantity pertaining to E^*_S are zero in the weak coupling regime but non-zero in the strong coupling regime. This is consistent with the generalized fluctuation-dissipation relation, Eq. (<ref>).

An upper bound on the signal-to-noise ratio, obtained from the modified thermodynamic uncertainty relation, Eq. (<ref>), is depicted in Fig. <ref>. 

The bounds are well satisfied for both the weak and strong coupling regimes, with the weak coupling regime depicting the tighter bound, in fact, saturating the bound.

The entropy of the reduced dynamics of the system is plotted in Fig. <ref>. 

The weak coupling behavior of the entropy is along the expected line, whereas for strong coupling, the entropy is negative at lower temperatures. However, it approaches the entropy in the weak-coupling regime as the temperature is raised. 

The ergotropy is calculated using the state ζ_S(T) = e^-β H^*_S/Z^*, and is plotted in Fig. <ref>.

From this figure, one can infer that in the weak coupling regime, the HMF coincides with the bare Hamiltonian of the system, and the ergotropy becomes zero, as the state ζ_S(T) becomes the Gibbs state for the bare Hamiltonian, which is not the case in the moderate and strong coupling regimes. Further, ergotropy is composed of coherent and incoherent parts. It is observed that for the state ζ_S(T) of the two-qubit model, the incoherent contribution is zero. The non-zero ergotropy comes from the coherent part. This is due to the fact that the state ζ_S(T) has coherence terms present in it at stronger system-bath couplings, which are absent in the weak coupling regime.

Considering the single-mode resonator approximation of the above mode (for single k in Eq. (<ref>)), we calculate the entropy production, Eq. (<ref>), using the Bell state 1/√(2)(|00⟩ + |11⟩) as the initial state of the two-qubit system, and ρ_B(0) = e^-β H_B/ Tr(e^-β H_B) as the initial state of the bath. The dynamics of the total and the reduced systems are obtained using U {ρ_S(0) ⊗ρ_B(0)} U^† and Tr_B [U {ρ_S(0) ⊗ρ_B(0)} U^†], respectively, where U = e^-iHt. 

The variation of entropy production Σ as a function of temperature, at time t = 1, is plotted in Fig. <ref>. As can be observed from Fig. <ref>, in this case, the hierarchy of the entropy production with respect to the coupling strength is similar to that of the ergotropy. 



 §.§ A single two-level system interacting with a single-mode resonator

Here, we take an example of a two-level system interacting with a single-mode field, that is, the Jaynes-Cummings (JC) model without rotating wave approximation <cit.>. The Hamiltonian (for ħ=1) of the total system is given by 

    H    = H_S + H_B + H_SB
       = ω_0/2σ^z + ω_c â^†â + λσ^x(â + â^†).

We find out the HMF numerically for this model using Eq. (<ref>). It is found to be diagonal in form, though not equal to H_S in the strong coupling regime. It is, however, the same as the bare Hamiltonian in the weak coupling regime. The specific heat capacity C_S(T) and related quantities as a function of temperature are plotted in Fig. <ref>. 

It is observed that the quantum uncertainty is zero in both weak and strong coupling limits. The term ⟨∂_T Ê^*_S⟩ is non-zero for strong coupling but vanishes in the weak coupling limit. The generalized fluctuation-dissipation relation, Eq. (<ref>), is satisfied.
Further, C_S(T) is negative in the strong-coupling regime, which was also observed in <cit.>.


The upper bound on the signal-to-noise ratio is depicted in Fig. <ref> and is observed to be tight both for the weak and strong coupling regimes. From this, one could infer the local temperature attained by the system S. The entropy of the system using HMF is plotted in Fig. <ref> and is along expected lines. 


The ergotropy is calculated using the state ζ̂_S(T) = e^-β H^*_S/Z^*_S, which is zero due to the absence of coherence, and the diagonal terms having Tr[σ^zζ̂_S(T)]<0, that is, the population of the excited state is always lower than the ground state of the system in the state ζ_S(T) <cit.>. 

The entropy production for the JC model is calculated using the initial states |ψ⟩_S = 1/√(2)(|0⟩ + |1⟩) and ρ_B(0) = e^-β H_B/ Tr[e^-β H_B] of the system and the bath, using Eq. (<ref>), where the unitary evolution is given by U = e^-iHt. The variation of entropy production as a function of temperature is shown in Fig. <ref> and exhibits the same pattern as in the earlier model.



§ CONCLUSIONS

In this work, we studied the quantum thermodynamic behavior of open quantum system models. To understand the effect of weak and strong coupling, use was made of the Hamiltonian of mean force, which could be interpreted as an effective Hamiltonian of the system instead of the system's bare Hamiltonian in the strong coupling regime. The thermodynamic potentials, particularly specific heat capacity, internal energy, and entropy, were studied using this Hamiltonian of mean force.  Further, we also discussed the thermodynamic uncertainty relation that provides an upper-bound to signal-to-noise ratio. This upper bound was computed using the specific heat capacity of the system of interest and a quantity used for computing the internal energy of the system. The behavior of the above thermodynamic quantities was illustrated in two models. First, we considered a two-qubit model interacting with an electromagnetic field, and next the Jaynes-Cumming model without the rotating wave approximation. The generalized fluctuation-dissipation relation was satisfied for both models for all coupling limits, ranging from weak to strong coupling. The upper bound on the signal-to-noise ratio was tight for the two-qubit model in the strong coupling regime. In the weak coupling regime of the two-qubit model and for the single-qubit model for all coupling strengths, the inequality was saturated. Further, in the case of the two-qubit model, the Gibbs state found using the Hamiltonian of mean force had a non-zero capacity for doing work, as its coherent ergotropy was non-zero in strong and moderate coupling limits due to the presence of coherence in this state. The entropy production for both systems was greater in the strong coupling regime, reducing with decrease in the coupling strength. Our results provide insights into the complex behavior of open quantum systems in the context of quantum thermodynamics.




apsrev



 Paper Title: Generalised BBGKY hierarchy for near-integrable dynamics.





Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise Cedex, France


Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland


Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland


 Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise Cedex, France

§ ABSTRACT 
  We consider quantum or classical many-body Hamiltonian systems, whose dynamics is given by an integrable, contact interactions, plus another, possibly long-range, generic two-body potential. We show how the dynamics of local observables is given in terms of a generalised version of Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, which we denote as gBBGKY, which is built for the densities, and their correlations, of the quasiparticles of the underlying integrable model. Unlike the usual cases of perturbation theory from free gases, the presence of local interactions in the integrable model “lifts" the so-called kinetic blocking, and the second layer of the hierarchy reproduces the dynamics at all time-scales. The latter consists of a fast pre-equilibration to a non-thermal steady state, and its subsequent thermalisation to a Gibbs ensemble. We show how the final relaxation is encoded into a Boltzmann scattering integral involving three or higher body-scatterings, and which, remarkably, is entirely determined by the diffusion constants of the underlying integrable model. We check our results with exact molecular dynamics simulations, finding perfect agreement. Our results show how gBBGKY can be successfully employed in quantum systems to compute scattering integrals and Fermi's golden rule transition rates.


Generalised BBGKY hierarchy for near-integrable dynamics
    Jacopo De Nardis
    Submitted August 5, 2024
========================================================





  
Introduction –


The field of non-equilibrium many-body physics continually astounds us with its profound complexity, capable of driving the emergence of entirely new behaviors. While classical theories of chaos, turbulence, and active dynamics remain vibrant areas of research, recent years have seen quantum many-body systems enter the non-equilibrium domain <cit.>. 
In low spatial dimensions, a significant class of dynamical systems is represented by integrable systems, which are characterized by stable quasiparticles. These quasiparticles play essential roles in transport, entanglement spreading <cit.>, anomalous scrambling <cit.>, long coherence times <cit.>, as well as exotic hydrodynamics <cit.>. Typically, interacting integrable systems involve elementary constituents, bosons, spins, or fermions, with two-body contact (or very short-range) interactions. Remarkable examples include the Lieb-Liniger <cit.> gas and the Fermi-Hubbard chain <cit.>. This prompts the question: what happens to quasiparticles and the overall dynamics when additional, possibly longer-range, interactions are incorporated into the Hamiltonian? This problem of integrability breaking has garnered significant attention over recent years <cit.>, largely driven by cold atomic experimental settings <cit.> where anomalously long thermalization times have been observed for nearly two decades. While progress has been made in systems where the underlying integrable theory is free, much less has been accomplished for interacting integrable systems. In free theories, the matrix elements of local operators are typically known, enabling the use of Boltzmann scattering via the Fermi's golden rule. In contrast, interacting integrable models feature matrix elements that are complex functions of both the state and the model. Despite some recent advances in specific settings <cit.>, there remains a need for a general and practical framework to compute the dynamics of any integrable model with generic additional weak two- (or more) body potentials.




In this paper, we introduce a novel approach to the problem: we generalise the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy <cit.>, to the cases where the unperturbed theory  (classical or quantum) involves strong integrable contact interactions. This can be envisioned as a hard-core repulsion V^ int_a(r) with radius a on top of a long-range field φ(r), giving the two-body potential,  see also Fig. <ref>,

    U(r) =  V^ int_a(r) + V(r),      V(r) = (V_0/ξ)  φ(r / ξ  )  ,

with V(r) (repulsive V_0 >0 or  attractive V_0 < 0) acting on the characteristic distance ξ and it is well-known for example to appear in atomic dipolar interactions <cit.>. We demonstrate two main distinctions from the known free case: (a) the n-point correlation functions must be divided into spatially local contributions, namely δ-like terms from the contact integrable potential, and regular contributions induced by the extra potential; (b) the free propagator for all correlations is dressed by integrable scatterings, incorporating diffusive and higher-order derivative terms. This feature is crucial for the thermalization dynamics, resulting in a thermalization time (starting from any stationary state of the unperturbed theory) of order E_ Kin^3/2/(V_0  a)^2, with E_ Kin the mean energy of the unperturbed Hamiltonian. This contrasts with the the standard one-dimensional BBGKY for perturbed free gases, i.e. with a = 0, where the thermalisation time scales,  due to the so-called kinetic blocking, as  E_ Kin^7/2/V_0^4,  <cit.>.

 Our approach gives a fully predictive theory for generic dynamics close to integrability, encompassing classical systems but also quantum bosonic, fermionic, and spin chains.   








  
The gBBGKY hierarchy –
We consider the following Hamiltonian with the two-body potential of eq. (<ref>) with strength V_0

    Ĥ = Ĥ^ int_a + V̂,   V̂ = 1/2∫xx'V(x-x')q̂_0(x)q̂_0(x'),

where q̂_0(x) denotes the local particle density operator (which is a functional in the classical case) and Ĥ^ int_a is given by the sum of standard kinetic terms and an integrable interaction V̂^ int_a,  typically short ranged, and with interaction strength denoted by a, (for example in the Lieb-Liniger gas V^ int_a(x-x') ∼ a δ(x-x')) with a=0 corresponding to the free case. More generally, q̂_i(x) is the density of conserved charge Q̂_i, which commutes with the integrable Hamiltonian [Ĥ^ int_a,Q̂_i]=0. The first standard conserved quantities are particle number Q̂_0 = N̂, momentum Q̂_1 = P̂ and kinetic (unperturbed) energy Q̂_2 = Ĥ^ int_a.  Using relations for commutators between local charges,  the quantum generalisation of the classical Liouville's equation reads as  <cit.>, 

    ∂_t q̂_i(x)  +∂_x ( ĵ_i(x)  + ĵ_0,i(x) ∫ d x'  V(x-x') q̂_0(x') )=
       -1/2∫ dx' V'(x-x') [ q̂_0(x')ĵ_i,0(x)+ĵ_i,0(x)q̂_0(x') ] ,

where the (generalized) currents ĵ_k,i satisfy i[Q̂_k,q̂_i]+∂_x ĵ_k,i=0. For a system on the continuum and a potential coupling to the density as in eq. (<ref>) we have ĵ_0,i=0 due to Galilean invariance (the case of coupling to generic density will be considered in <cit.>). The equation (<ref>) is written for operators, and it is the action of taking expectation values that generates the hierarchy of dynamical equations for multipoint operators involving arbitrary products of charge densities. For all such equations, we take the average in a generic (Generalised) Gibbs ensemble, (G)GE, ⟨·⟩_β =  Tr(e^- ∫ dx β^i(x) q̂_i(x))/Z and isolate only their connected parts. The two-point correlations we denote S_o_1,o_2(x,y)= ⟨ô_1(x) ô_2(y) ⟩ _β - ⟨ô_1(x) ⟩_β⟨ô_2(y) ⟩_β and analogously for higher point correlations.  










In the same spirit as standard BBGKY theory, we move from the dynamics of the charges to the dynamics of the (quasi-)particles of the Hamiltonian Ĥ^ int_a. We introduce the particle density  ρ̂(θ, x) = ∑_j δ(θ - θ_j) δ(x-x_j), with average ⟨ρ̂(θ,x) ⟩ = ρ(θ,x),  (we use the generic labelling θ for each quasi-particle, denoted as momenta). Local charges (as well as other generic local operators) and their currents can be easily expressed in terms of the densities,  ⟨q̂_i ⟩ = ∫θh_i(θ)ρ(θ), with h_n(θ) = θ^n/n! the single-particle eigenvalue of Q̂_n. We can proceed analogously to introduce generic multi-particle correlations, obtaining this way the hierarchy involving ρ(θ,x) and its multi-point correlations as S_θ_1,θ_2(x_1,x_2), S_θ_1,θ_2,θ_3(x_1,x_2,x_3), and so on. 


It is crucial to stress the structure of the correlators such as S_θ_1,θ_2(x_1,x_2). In the unperturbed theory V_0=0 these objects carry non-trivial correlations induced by the local, integrable interactions. In particular, in any equilibrium state given by a generic density of quasiparticles ρ(θ), they read as 
    S_θ_1,θ_2(x_1,x_2)|_V_0=0≡δ_x_1,x_2(δ_θ_1,θ_2 g^(1,2)_(1,2)   +g^(1,2)_(1|2)),

 where we denoted the x dependence as upscript and the θ as subscript (see also eq. (<ref>)). 
Each correlator therefore consists of two main types of terms: one singular in space and one singular both in space and momenta. When interaction is added, V_0 ≠ 0,  the correlations also acquire an additional, smooth long-range component. Such a structure generalizes to higher point-functions and in order to have a well-defined and self-consistent hierarchy, the singular parts have to be extracted and the latter must be written only for the regular factors of the correlations.  Namely, we impose the following generic structure for the correlations S_θ_1,…,θ_s: 

    S_θ_1,…,θ_s(x_1,…,x_s) = ∑_P ∈𝒫_sδ^P(𝐱) ∑_R ∈ℛ_Pδ^R(θ) g^P_R.

Here 𝒫_s is the set of all partitions of a set of s elements. 
R is a refinement of P, that is a further fragmentation of the partition and by ℛ_P we denote a set of all refinements of P. With δ^P(𝐱) we denote a product of Dirac δ-functions that enforces the pairing of the coordinates according to P. For example, δ^(1,2|3,4)(𝐱) = δ(x_1 - x_2) δ(x_3 - x_4) with a standard notation for partitions in which vertical bar separates different subsets. In the momenta space we use analogous notation. Finally, g^P_R denotes contribution to the correlation functions of singular structure specified by the two partitions, see eq. (<ref>). We note that (<ref>) contains also regular terms corresponding to the full partition with cardinality |P|=s. Moreover, we stress that the correlators S_θ_1,…,θ_s(x_1,…,x_s) are symmetric under simultaneous exchange of position and momenta (we shall use the notation [· ]_(1,2) for symmetrization with respect to both), reducing therefore the complexity of (<ref>). The resulting hierarchy is written for the density of particles ρ and the functions g_R^P. We find the following first two lines of the gBBGKY hierarchy 


    𝒟^(1)_(1)ρ_(1)^(1)= ∫x_2θ_2𝒜^(1|2)_(1|2)( ρ_(1)^(1)ρ_(2)^(2) + g^(1|2)_(1|2)),





    𝒟^(1|2)_(1|2) g^(1|2)_(1|2)   = [𝒜^(1|2)_(1|2)ρ_(1)^(1)g^(2,3)_(2,3)+𝒜^(1|2)_(1|2) g^(1|2,3)_(1|2,3)+𝒜^(1|2)_(1|2)∫θ_3ρ_(1)^(1)g^(2,3)_(2|3)+𝒜^(1|2)_(1|2)∫θ_3g^(1|2,3)_(1|2|3)+
       +∫x_3θ_3𝒜^(1|3)_(1|3)ρ_(1)^(1)g^(2|3)_(2|3)+∫x_3θ_3𝒜^(1|3)_(1|3)g^(1|2|3)_(1|2|3)]_(1,2).


Here, the local 1-point correlations are denoted as densities of particles g_(1)^(1)≡ρ_(1)^(1) = ρ(θ_1, x_1), and we have introduced respectively the generalised acceleration induced by the perturbing potential 𝒜^(i|j)_(i|j)= V'(x_i-x_j)(∂_θ_i-∂_θ_j). 

The gBBGKY eqs. (<ref>) and (<ref>) read very similar to the known BBGKY equations for perturbed free particles, but with some major, important differences. First,  the left-hand side kinetic term 𝒟^(i|j)_(i|j), which describes the evolution of the multipoint correlations under the Hamiltonian Ĥ^ int_a depends on the interactions a: for free particles a=0 it reduces to the constant velocities spreading of the multipoint correlations 𝒟^(1|2)_(1|2) = ∂_t + ∑_j=1^2  θ_j ∂_x_j.  For finite interactions it is instead given (in a first order in V_0) in terms of the flux Jacobian <cit.> A_i^k = ∂⟨ĵ_i ⟩ /∂⟨q̂_k ⟩ (which is not diagonal in the interacting case) and by the diffusion matrix 𝔇_i^k <cit.>). For the case of two particles for example this reads 

    D^(1|2)_(1|2)= ∂_t + ∑_j=1^2∂_x_j A_θ_j^·  +  1/2∑_j=1^2∂_x_j 𝔇_θ_j^·∂_x_j  ,

where the operators A_θ_j^· and 𝔇_θ_j^·  acts on the momenta on their right. The flux Jacobian gives an effective ballistic spreading similarly to the free case, but where the velocities depends on the local density ρ(θ,x). Moreover, there is also a diffusive spreading of their trajectories, due to random scatterings with single particles. As at least two-body scattering are necessary, diffusion typically scales as 𝔇∼ (a ρ̅)^2, with ρ̅ characteristic density. Finally, we remark that higher order derivative terms in the hydrodynamic expansion of eq. (<ref>) can also be included, giving higher order corrections in a/ξ. 

The second major difference is that gBBGKY contain also correlations between particles at the same points, and different momenta, see eq. (<ref>). These correlations are not present in free gases, and they are a feature of Ĥ^ int_a being a short-range interacting Hamiltonian, which therefore can correlate particles at some microscopic scales, much smaller than the scale ξ at which the two-body perturbation acts.  

To truncate the hierarchy, one needs to estimate the order of the different contributions in (<ref>). Given that the system is characterized by the two interaction parameters a and V_0, the truncation of the hierarchy follows from scaling of correlation functions with these parameters, which reads g^P_R ∼ a^|R| - |P| V_0^|P|-1. We notice that the three-point function is of order g^(1|2|3)_(1|2|3) = O((V_0)^2) and enters the dynamics of ρ_(1)^(1) with order (V_0)^4. When interested in the dynamics at order (V_0)^2 we can therefore neglect it, together with the time evolution of g_(1,2)^(1,2), g_(1|2)^(1,2)  and of  g_(1|2|3)^(1|2,3), also of order (V_0)^2. 


We can recognize some of the terms in the RHS of Eqs. (<ref>) and  (<ref>). The O(V_0) term of (<ref>) is the  generalised Vlasov equation, which neglects two-particle correlations. It corresponds to a fully mean-field coupling of two space-dependent densities, and it gives no dynamics when the system is homogenous, namely when the density ρ(θ,x) does not depend on x. Similarly, the O(V_0) terms in (<ref>), the first on the right-hand side, are the driving terms of the generalised Landau equation, when the hierarchy is integrated into a single, effective kinetic equation. In one-dimensional free gas, setting a=0, the diffusive kinetic term in eq. (<ref>) vanishes and the  Landau equation gives only a short-time dynamics, and it cannot describe the thermalisation of the system at later times. This phenomenon is known as kinetic blocking in one-dimensional systems, and as we shall demonstrate later, the picture changes drastically when local integrable interactions are added.




  
The example of hard rods  –



We apply gBBGKY hierarchy truncated to its second layer,  eq. (<ref>), (<ref>) truncated to (V_0)^2 order in (<ref>), to a gas of classical impenetrable hard spheres with diameter a <cit.>, interacting with the long-range potential of eq. (<ref>). Hard spheres (or rods) gas in one dimension is integrable in the absence of extra interactions. The expressions for the  flux Jacobian and diffusion matrix are  well known <cit.> and they share the same hydrodynamics as quantum integrable systems, modulus changing the statistical factor <cit.>. We focus  on a homogenous system with constant density and we compare gBBGKY prediction with exact microscopic simulations.  We initialize the time-evolution starting from two boosted finite temperature states ρ_±; β; θ_0≡∑_σ = ± e^-β (θ- σθ_0)^2 /Z, and with g^(1|2)_(1|2)(t=0)=0. The state is stationary under the unperturbed Hamiltonian Ĥ^ int_a but it evolves non-trivially as long as V_0 is finite. As a probe of thermalisation dynamics, we compute the kurtosis of the (normalised) quasiparticle distribution and the two-point correlations. The kurtosis is expected to be zero in the thermal state, where quasiparticle distribution is Gaussian. In Fig. <ref> we show how gBBGKY reproduces the full molecular dynamics even at large values of V_0, modulo finite-time effects that involves higher order in the hierarchy. We can moreover observe two regimes in the evolution of the quasiparticle kinetic energy E_ Kin=v̅^2/2 =⟨Ĥ^ int_a⟩, Fig. <ref> (a): a fast evolution at short times, instated by the change in the interaction energy due to the build-up of the density-density correlations in the system (see. Fig. <ref> (b)), and the kinetic regime where it is mostly constant and where the dynamics is well described by the Boltzmann equation, which we describe in the coming section.
 

 







  
The interacting Landau equation and Boltzmann scattering kernel –
A common approach to handling the perturbed dynamics described by eq. (<ref>) is to use the Boltzmann scattering integral, which is a functional of the quasiparticle distribution at time t, ℐ_θ[ρ(·,t)]. This integral is derived by summing over all intermediate excitations induced by the operator V̂, as described by the celebrated Fermi's golden rule <cit.>. The matrix elements between the state |ρ⟩ at time t and its excitations, commonly referred to as form factors, define a typical timescale for each intermediate state | ex⟩, proportional to the square of its matrix element τ_ ex^-1∼ |⟨ρ | V̂ |  ex⟩ |^2 (in classical physics, a similar approach can be used, where cross-sections replace matrix elements). The Boltzmann scattering integral defines the Boltzmann equation as follows <cit.>:

    ∂_t ρ(θ,t) = ℐ_θ[ρ(· ,t)],

which, given that ℐ_θ∼ (V_0)^2, describes the late-time thermalization dynamics on timescales of order (V_0)^-2. Although efforts have been made to derive ℐ_θ for interacting integrable models, the general expression has remained elusive due to the complexity of the matrix elements involved.

Using algebraic manipulation (see <cit.>), we demonstrate that by truncating (<ref>) to terms of order V_0, and performing a time integration, we can precisely derive the Boltzmann scattering integral. By introducing the super-operators acting on two-point correlations 𝐀≡ (A ⊗1 - 1⊗ A), the diffusion matrix 𝐃 = (𝔇⊗1 + 1⊗𝔇), and 𝐂'_θ,θ'≡∫γ ( ∂_θρ_θC_γ,θ' - ∂_θ'ρ_θ'C_θ,γ ) (with 1 being the identity matrix in momentum space), we obtain the compact expression:

    ℐ_θ = ∂_θ∫_-∞^+∞k/2π k (V(k) )^2 [(k 𝐃 - i 𝐀)^-1]_θ,θ'^γ,γ' 1_θ'𝐂'_γ,γ'

where the repeated momentum indices are integrated over and V(k) denotes the Fourier transform of the potential (<ref>).  In <cit.>, we show that it possesses all the properties required for a scattering integral: it leaves thermal states steady, conserves the three canonical equations of motion, energy density, and momentum.

First, we observe that ℐ_θ∼ (V_0)^2, as expected. The first non-trivial terms in the expansion of the operator (k 𝐃 - i 𝐀)^-1 are of order k 𝐃∼ a^2, as the zeroth order in k integrates to zero. By expanding the inverse of the operator in (<ref>), we can see that ℐ_θ includes three or higher particle scatterings, where two particles scatter due to V̂ and all the others via V̂_a, see Fig. <ref> (b). Eq. (<ref>) provides a compact general form of the timescale for thermalization, even in a generic system given by Ĥ^ int_a with local interactions and a perturbing long-range potential V̂. It remarkably shows the role of the diffusion constants of Ĥ^ int_a in quantifying thermalization timescales: denoting D̅ the typical diffusion constant, we can extract the timescale τ^-1∼ (D̅/v̅^3)  ∫ dx (V'(x))^2 ∼ (a ρ̅ V_0)^2 /(ξ^3 E_ Kin^3/2).

The Boltzmann equation with the scattering integral given by eq. (<ref>) can also be understood as a generalization of the Landau equation for interacting integrable models. The latter clearly vanishes in the limit a → 0 and this can be either understood by the kinetic blocking or equivalently by the fact that density-density matrix elements of free systems cannot generate more than two particle scatterings <cit.>. Therefore, in this case 3-point correlations needs to be added to eq. (<ref>), which, after time integration, yield a higher-order scattering integral of order (V_0)^4 ρ̅^2/(ξ^3 E_ Kin^7/2)  <cit.>, which is also characterised by a dramatically different scaling with the kinetic energy.

 



 

  
Conclusions   –
 
In this work, we have generalized the BBGKY hierarchy (gBBGKY) for systems with integrable contact interactions by introducing ultra-local correlations and integrating kinetic theory with the underlying hydrodynamics of integrable models. We have demonstrated that the resulting kinetic equation overcomes the kinetic blocking, and the combination of short- and long-range interactions results in thermalization rates that are quadratic in the strength of both interactions. Kinetic blocking physics however still manifests itself from the fact that the thermalisation rate is O( (V_0 a)^2) which remains quartic in interaction strength. 

Our method is broadly applicable, with its main ingredient being generalized hydrodynamics, which offers a universal framework for many classical and quantum integrable models. Specifically, unlike previous approaches based on the Fermi's golden rule and matrix elements, the gBBGKY allows to directly access the Boltzmann scattering integrals and the time evolution of multi-point correlations  in generic perturbed integrable models.

While the partition separation of interactions between an integrable and a non-integrable part can be approached in various ways, we focused here on an experimentally relevant scenario of a short-range integrable model perturbed by long-range, dipole-like interactions. Nonetheless, many natural extensions, including dynamics from inhomogeneous initial states, perturbing operator coupling to other conserved charges, or systems on a lattice such as long-range spin chains or fermionic systems <cit.>, are now directly conceivable.




  
Acknowledgments–
MŁ and MP acknowledge support by the National Science Centre (NCN), Poland via project 2022/47/B/ST2/03334, J.D.N. and L.B. are founded by the ERC Starting Grant 101042293 (HEPIQ) and the ANR-22-CPJ1-0021-01. 




 























Supplementary Material 
 

Generalised BBGKY hierarchy for near-integrable dynamics






§ GBBGKY HIERARCHY IN THE HOMOGENOUS SETTING
 

 

We here specialize equations (<ref>) and (<ref>) from the main text to a homogeneous setup. In this case, the 1-point correlation does not depend on the spatial coordinate, while the 2-point correlation function depends only on the relative distance z = x_1 - x_2. This allows us to simplify the notation and use ρ(θ) and g^(2)_θ, θ'(z) for the two correlators respectively. We shall also make use of the filling function n(θ). The latter are the normal modes of the hydrodynamics,  defined as the modes that diagonalise the flux Jacobian of the model Ĥ^ int_a via the rotation matrix R

    A_θ,θ' = [R    diag(v^ eff) R^-1 ]_θ,θ',   δ n(θ)/δρ(θ') = R_θ,θ',    R_θ,θ'≡δ_θ,θ'-T_θ,θ'n_θ' ,

with T_θ,θ' the scattering shift of the integrable model. We also define v^ eff=(ε')^ dr/(p')^ dr with ε(θ) and p(θ) energy and momentum as function of momenta and with the dressing operator f^ dr≡ R^-1f.

The dynamics of the 1-point correlator is given by

    ∂_t ρ_θ=∫zθ'V'(z) ∂_θg^(2)_θ,θ'(z),

with the Vlasov term disappearing for a homogeneous system. In a similar way, the dynamics for the 2-point correlator is














    (D_(1|2)^(1|2) g^(2))_θ, θ'(z) =
       ∫ dz' γ V'(z-z') (∂_θρ( θ)g^(2)_γ,θ'(z')-∂_θ'ρ( θ')g^(2)_θ,γ(z'))
        +V'(z) (∂_θρ( θ)f(θ')ρ(θ')-∂_θ'ρ( θ')f(θ)ρ(θ))+ 
        
       
        + V'(z) (∂_θ' f(θ)g^(2)_θ,θ'(z)- ∂_θ f(θ')g^(2)_θ,θ'(z))
        +V'(z) ∫γ(∂_θρ( θ)g^(2)_{γ,θ'}-∂_θ'ρ( θ')g^(2)_{θ,γ})+ 
        
       
        + V'(z) ∫γ(∂_θ' g^(3)_{θ,γ},θ'(z)- ∂_θ g^(3)_θ,{γ,θ'}(z)) .

In writing this expression we introduced g^(2)_{γ,θ'} which corresponds to g_(1|2)^(1,2) of the main text and in a homogeneous system is space independent. Similarly, we write g^(3)_θ,{γ,θ'}(z) for g_(1|2|3)^(1|2,3) which depends only on the relative distance between particle at x_1 and group of particles (2,3) at x_2. The action of the kinetic term is 

    (D_(1|2)^(1|2) g^(2))_θ, θ'(z) = ∂_t g^(2)_θ,θ'(z) + ∂_z( A_θ^γ g^(2)_γ,θ'(z)- A_θ'^γ  g^(2)_θ,γ(z)) - ∂_z (𝔇_θ^γ∂_z  g^(2)_γ,θ'(z)+𝔇_θ'^γ∂_z  g^(2)_θ,γ(z) ).

Finally, f is the statistical factor depending on the statistics of the quasi-particles (fermions f= 1- n, bosons f=n, classical f=1).

The equation for g^(2)_θ,θ' depends on two more correlators namely the reduced 2-point  correlator g^(2)_{θ,θ'} and the reduced 3-point correlation g^(3)_{θ_1, θ_2 }, θ_3(z). They obey dynamic equations that are obtained following the procedure described in the main text. The kinetic part follows from the integrable dynamics, while the right hand side follows from the BBGKY hierarchy refined by the singularity structure of the correlations. In this way, the reduced 2-point correlator obeys

    ∂_t g^(2)_{θ,θ'} =  ∫zγ    V'(z)  (∂_θg^(3)_{θ,θ'},γ(z)-∂_θ'g^(3)_γ,{θ,θ'}(z)),

while the equation for the 3-point correlator, here written only in the leading term in V_0, is









    (D_(1,2|3)^(1|2|3) g^(3))_{θ_1, θ_2}, θ_3(z) = V'_1,3[ ∂_θ_1,θ_3g^(2)_{θ_1,θ_2}ρ_θ_3+  ∂_θ_1,θ_3∫θ_4 g^(3)_{θ_1,θ_2,θ_4}ρ_θ_3]_(1,2)
      +𝒪((V_0)^2),

where 

    (D_(1,2|3)^(1|2|3) g^(3))_{θ_1, θ_2}, θ_3(z) = ∂_t g^(3)_{θ_1,θ_2},θ_3 + 
       + 1/2∂_x_1( A_θ_1^γ g^(3)_{γ, θ_2},θ_3+H_θ_1^α,βg^(2)_{α ,θ_2}g^(2)_β,θ_3 +A_θ_2^γ g^(3)_{γ, θ_1},θ_3+H_θ_2^α,βg^(2)_{α ,θ_1}g^(2)_β,θ_3) + ∂_x_3( A_θ_3^γ g^(3)_{θ_1, θ_2},γ) +𝒪((V_0)^2).

In particular, here we defined

    H_θ^α,β≡δ  A_θ^α/δρ(β)= ∫γσ(R^t)^-1_θ ,γn_γ/ρ_γT^ dr_γ,σΔ v^ eff_σ,γ(R^t_σ, kR^t_γ,α+R^t_jkR^t_σ,β)  ,  Δ v^ eff_σ,γ≡ v^ eff_σ - v^ eff_γ .


Note that the kinetic term depends also on the lower order correlation functions. 
There are also higher orders contributions to the 3-point function that ultimately lead to higher order corrections to the dynamics of ρ_θ. However, the orders that we keep are enough to capture both short time dynamics and long-time thermalization. 

The dynamic equations has to be supplemented with the initial data. For the reduced 2-point and 3-point functions these are (see section <ref> for an explicit derivation)

    g^(2)_{θ,θ'}|_t =0 
       = [C]^ reg_θ,θ'   g^(3)_{θ,θ'}, θ” (z)
      |_t=0  = [(R^t)^-1_θ,ξ(R^t)^-1_θ',ξ
       f_ξg_ξ,θ”(z)]^ reg .


Results of the numerical solution to the set of equations (<ref>), (<ref>), (<ref>) and (<ref>) are presented in the main text.  

However, to capture the leading non-trivial dynamics the hierarchy can be further truncated to an equation for the density and

    (D_(1|2)^(1|2) g^(2))_θ, θ'(z)
        = V'(z) ∫γ(∂_θρ( θ)C_γ,θ'-C_θ,γ∂_θ'ρ( θ')) + 𝒪(V_0^2),

where in the equation for g^(2) we kept only the leading term of order V_0 and neglected all the terms that are of higher order. Whereas this set of equations is a further approximation to the dynamics, it has a benefit of being tractable analytically. In that regard we present two results. First, we will compute the correlations in the stationary state. Second, in the next Section, we will use it to derive a kinetic equation, the interacting Landau equation. 

To this end it is useful to introduce compact notation to express operations in the momenta space. The 2-point function depends on 2 momenta and the dynamic equations involve various contractions (integrations) of these momenta as actions of linear operators. The two momenta can be then thought as belonging to two different vector spaces. The operators acting on g^(2) naturally act in their tensor space. With this point of view we write 

    (𝐀g^(2)(z))_θ,θ' = 𝐀_θ,θ'^γ,γ'g^(2)_γ,γ'(z) = A_θ^γ g^(2)_γ,γ'(z) - A_θ'^γ g^(2)_γ,γ'(z),

where 𝐀 = A ⊗1  - 1⊗ A, with 1 being the identity operator which in the momenta space is represented by the δ-function. In a similar way, for the diffusion operator we have 𝐃 = 𝔇⊗1  + 1⊗𝔇. With this notation the kinetic term (<ref>) is given by

    (D_(1|2)^(1|2) g^(2)(z) )_θ, θ'= ((∂_t + 𝐀∂_z - 𝐃∂_z^2 ) g^(2)(z) )_θ,θ'.

Introducing as in the main text the matrix 

    𝐂_θ,θ' = ρ'_θ∫γC_γ,θ'-∫γC_θ,γρ'_θ',

the equation for g^(2) reads

    ((∂_t + 𝐀∂_z - 𝐃∂_z^2 ) g^(2)(z))_θ,θ' = V'(z) 𝐂_θ,θ'.

In the following we will also need that C_θ,θ'=(R^t)^-1fρ R^-1.

We conclude the discussion by computing the 2-point correlation in the stationary state. 












In order to find a stationary solution to this equation we suppose the following expression for correlations g^(2), Eq_θ,θ'(z)=V(x)F[ρ]_θ,θ' . We also assume that 𝔇_θ^γF[ρ]_γ,θ'=0.
The stationarity condition leads to the equation 𝐀 F[ρ] = 𝐂. Inverting this relation we find

    F[ρ]_θ, θ' = (𝐀^-1𝐂)_θ, θ' .

















We note that 𝐀^-1 operator has a simple form

    𝐀^-1= ((R^t)^-1⊗(R^t)^-1)1/v^ eff⊗1 -1⊗ v^ eff((R^t)⊗(R^t)) .

The expression for the stationary 2-point correlation is supplemented with an equation for the stationary 1-point function ρ. To simplify the discussion, we assume that the stationary state of the system is thermal. In that case we can evaluate F[ρ] explicitly as follows.

For a thermal state ρ_θ^ th we have the following expression

    ∂_θρ^ th_θ=δρ^ th_θ/δ n^ th_γ∂_γn^ th_γ=-β(R^t)^-1_θ,γρ_γf_γv^ eff_γ 1^ dr_γ ,

which follows from the general structure of thermal states in integrable models as described by the Thermodynamic Bethe Ansatz. With this expression it is a matter of simple manipulation to evaluate F[ρ^ th]. In the process we find the following formula

    𝐂^ th = -β((R^t)^-1⊗ (R^t)^-1)( v ⊗ 1 - 1 ⊗ v ) (ρ_ tot f ρ^ th⊗ρ_ tot f ρ^ th),

which implies that 

    (𝐀^-1𝐂^ th)_θ, θ' = [(R^t)^-1ρ_ tot fρ^ th]_θ  [(R^t)^-1ρ_ tot fρ^ th  ]_θ'.

The 2-point correlator in the stationary and thermal state is then

    g^(2),λ, th_θ,θ'(z)= - β V(z)[(R^t)^-1ρ_ tot fρ^ th]_θ  [(R^t)^-1ρ_ tot fρ^ th  ]_θ' + 𝒪(V_0^2)  .

We observe that the spatial profile follows the shape of the long-range potential, while correlations in the momenta simply factorize.

We can also verify that the diffusion operator acting on the stationary correlations is zero. This follows from the fact that 𝐃𝐀^-1𝐂^ th =0. To prove this relation we observe that diffusion operator 𝐃 acts independently on the θ and θ' variables and then use 𝔇=(R^t)^-11^ dr𝔇̃(1^ dr)^-1R^t together with 𝔇̃ρ f=0. This confirms that indeed g^(2),λ, th_θ,θ'(z) is the stationary correlation also in the presence of the diffusion.
 


 §.§ Derivation of three point correlations at equilibrium, eq. (SA.9)


We derive here an analitic expression for g^(3)_{θ,θ'}, θ”(z) in a GGE. In particular, reintroducing x and y, where z=x-y, we write

    g^(3)_{θ,θ'}, θ”(x,y) = ⟨δρ(ξ,x)δρ(ξ',x)δρ(θ”,y) ⟩ = δρ_θ/δ n_ξδρ_θ'/δ n_ξ'⟨δ n(θ,x)δ n(θ',x)δρ(θ”,y) ⟩
         =
       (R^t)^-1_θ,ξ(R^t)^-1_θ',ξ
       (ρ^ tot_ξ(x))^2
       ⟨δ[n(ξ,x)/ρ^ tot_ξ(x)] δρ(θ”,y)⟩=
       
       
       = (R^t)^-1_θ,ξ(R^t)^-1_θ',ξ(ρ^ tot_ξ(x))^2δ/δρ_α(x)[n(ξ,x)/ρ^ tot_ξ(x)] 
       ⟨δρ(α,x)δρ(θ”,y)⟩
       
       = (R^t)^-1_θ,ξ(R^t)^-1_θ',ξ(ρ^ tot_ξ(x))^21/(ρ^ tot_ξ(x))^2[R^t_ξ,α+n_ξT_ξ,α] 
       ⟨δρ(α,x)δρ(θ”,y)⟩
       
       = (R^t)^-1_θ,ξ(R^t)^-1_θ',ξ⟨δρ(ξ,x)δρ(θ”,y)⟩ .

where the repeated indices are integrated over. We also used the following relations

    δ n(x,θ) δ n(x,θ') =   δ_θ,θ'δ[ n(x,θ)/ ρ^ tot(x) ]  ,  δρ^ tot_α/δρ_β = T_α,β  ,  δρ_α =(R^t)^-1 _α,βρ^ tot_βδ n_β .
 
 
 


§ DERIVATION OF THE BOLTZMANN SCATTERING INTEGRAL


As discussed on the previous section, the BBGKY hierarchy for near-integrable dynamics, in the leading order in the perturbing potential V_0 yields the following set of equations 

    ∂_t ρ_θ =∫zθ'V'(z) ∂_θg^(2)_θ,θ'(z), 
       ((∂_t + 𝐀∂_z - 𝐃∂_z^2 )
        g^(2)(z))_θ,θ' = V'(z) 𝐂_θ,θ',








In the following we will derive the kinetic equation for ρ_θ which supersedes the Landau equation for interacting models.

The equation for the correlations is a linear equation and has the shape of a forced convective-diffusive equation. It can be solved by the Green's function technique. Its Green's function obeys

    ((∂_t + 𝐀∂_z - 𝐃∂_z^2 ) 𝐆(z, t) ) = δ(τ) δ(z) ( 1 ⊗ 1 ).

This equation can be solved by usual techniques and 

    𝐆(z,τ)=θ_H(τ)/2π∫ke^ikz[e^ik𝐀τ-k^2𝐃τ].

Note that the operators 𝐀 and 𝐃 depends also on τ through ρ_θ and C_θ, θ'.
At time t the correlations read

    g^(2)_θ,θ'(z,t)=∫_0^tτ∫z' V'(z') (𝐆(z-z', t - τ) 𝐂)_θ, θ'.




Inserting this result into the equation for the density we obtain

    ∂_t ρ_θ = ∫_0^tτ∫zz' V'(z) V'(z') ∫θ'∂_θ(𝐆(z-z', t - τ) 𝐂)_θ, θ'.

At this stage it is customary to perform two approximations. First, we assume that all operators 𝐀, 𝐂 and 𝐃 depend on time t and not on t - τ as expected from the formulas. The difference between the two evaluations is of order V_0 and hence does not contribute to the leading term. Second, we assume that the lower integration limit over τ can be extended from 0 to the minus infinity. This is motivated by assuming that the particles' distributions and correlations evolve at different timescale. The former being much shorter than the latter implies that the correlations relax to the temporary stationary state before the distribution itself changes. This permits to extend the temporal integral. With these two assumptions we obtain

    ∂_t ρ_θ = ∫_0^∞τ∫zz' V'(z) V'(z') ∫θ'∂_θ(𝐆(z-z',τ) 𝐂)_θ, θ',








where τ appears only explicitly in the expression for the Green's function and 𝐀 and 𝐃 are computed at time t. Hence, upon inserting the expression for 𝐆, the integral over τ can be performed. At the same time we express V'(x) through the Fourier transform of the potential,

    V(x) = ∫k/2π e^i k x Ṽ(k),

to obtain

    ∂_t ρ_θ= ∂_θ∫k/2π∫θ'
        k Ṽ^2(k)[(k𝐃-i𝐀)^-1𝐂]_θ,θ'.

This is the kinetic equation generalizing the Landau equation. To show that the RHS of Eq.(<ref>) is real we expand the kernel

    (k𝐃-i𝐀)^-1 = -i ∑_m≥ 0 (ik)^m (𝐀^-1𝐃) ^m 𝐀^-1 .

finding 

    ∂_t ρ_θ= -∂_θ∫θ'∑_m≥ 0∫k/2π (ik)^m+1Ṽ^2(k)[ (𝐀^-1𝐃) ^m 𝐀^-1𝐂]_θ,θ'.

Finally, we observe that each complex term of the sum with even m is vanishing because of the symmetry of the potential Ṽ(k)=Ṽ(-k).



























 §.§ Stability of the thermal state


We will show, that the thermal state is a stationary state of the interacting Landau equation. This guarantees the stability of the thermal equilibrium. The prove is simple. From the expansion (<ref>) we see that each term contains factor 𝐃𝐀^-1𝐂. However as derived in Appendix <ref>, this expression, when specialized to a thermal state, vanishes, 𝐃𝐀^-1𝐂^ th = 0. Therefore, every thermal state is a stationary state. 




 §.§ Conservation of density, energy and momentum


The interacting Landau equation respects the usual 3 conservation laws, that is conservation of particle number, momentum and the energy. In the standard Landau equation, the conserved form of the energy is the kinetic energy. Here, the integrable dynamics, which governs the left hand sides of the dynamic equations, takes into account also short-range interactions. Hence, the resulting kinetic equation conserves the sum of the kinetic energy and the potential energy of the short-range interactions. Equivalently, we interpret it as the kinetic energy of the quasiparticles. 

To prove the conservation laws it is convenient to rewrite the equation in a form of the diffusion equation

    ∂_t ρ_θ= 
        ∂_θ(𝔇^ LR_θ,γ∂_γρ_γ),

where we introduced

    𝔇^ LR_θ,γ=∫θ'∫k/2π  kṼ^2(k)(𝐌_θ,θ'^γ,γ'-𝐌_θ,θ'^γ',γ)∫αC_γ',α  ,  𝐌_θ,θ'^γ,γ'= Re[ (k 𝐃 - i 𝐀)^-1]_θ,θ'^γ,γ',

and used eq. (<ref>) for 𝐂. We observe that the second operator 𝐌 is partially transposed (only in γ momenta) compared with the first one. Such partial transposition is not a symmetry of the operator, unlike the full transposition, 𝐌^T = 𝐌. 

The particle number Q_0 is given by an integral over ρ_θ. Its conservation is immediate, (where L the volume of the system)

    L^-1∂_t Q_0=  ∂_t ∫θρ_θ= ∂_t ∫θ∂_θ(𝔇^ LR_θ,γ∂_γρ_γ) = 0 .

The total momentum Q_1 is given by

    L^-1 Q_1 =  ∫θ p_θρ_θ.

In a Galilean invariant theory p_θ = θ and 

    L^-1∂_t Q_1 = ∫θp_θ∂_tρ_θ   =- ∫θ(
        𝔇^ LR_θ,γ∂_γρ_γ).

The integral over 𝔇^ LR_θ,γ is 

    ∫θ𝔇^ LR_θ,γ   =∫θθ'∫k/2π kṼ^2(k)(𝐌_θ,θ'^γ,γ'-𝐌_θ,θ'^γ',γ)∫αC_γ',α=0  ,

upon exchanging in the second term the dummy integration variables θ and θ' and using the symmetry of 𝐌.

Finally we show that kinetic energy is also conserved 

    L^-1∂ Q_2 = ∫θϵ_θ∂ρ_θ.

 For this proof it is convenient to go back to the original formulation (<ref>) of the kinetic equation,

    L^-1∂_t Q_2    = ∫θε_θ∂_tρ_θ = -∫k/2π k Ṽ^2(k)∫θθ'ε'_θ[(k𝐃-i𝐀)^-1𝐂]_θ,θ'
       
        = -1/2∫k/2π k Ṽ^2(k)∫θθ'(ε'⊗1-1⊗ε')
        [(k𝐃-i𝐀)^-1𝐂]_θ,θ'
       
        = -1/2∫k/2π k Ṽ^2(k)∫θθ'𝐀[(k𝐃-i𝐀)^-1𝐂]_θ,θ' =0

In going to the last line we used that for any function f_θ of the momenta, it holds

    ∫θε'_θf(θ)=∫θγξε'_θ(R^t)^-1_θ,γ1^ dr_γ/1^ dr_γR^t_γ,ξf_ξ=∫θγξ(R^t)^-1_θ,γv^ eff_γR^t_γ,ξf_ξ=∫θγA_θ,γf(γ) .

In the last we used that upon expanding the inverse operator we get ∫θ𝔇_θ,γ which is indeed zero. This concludes our proof. 












§ NUMERICAL SIMULATIONS


In this section we explain in details how we performed the numerical simulations used to obtain the reults presented in the main text.

Firstly, we detail the method used to integrate the set of gBBGKY equations (<ref>), (<ref>), (<ref>) and (<ref>). This is a set of coupled non-linear integro-differential equations and its solution represents a highly expensive numerical task in terms of computational time and memory consumption. 
In order to overcome this complexity we perform the following crucial approximation

    g^(2)_{θ,θ'} (t)
       ≃ [C]^ reg_θ,θ'(t)+𝒪(V_0)    g^(3)_{θ,θ'}, θ” (z,t)  ≃ [(R^t)^-1_θ,ξ(R^t)^-1_θ',ξ
       f_ξg_ξ,θ”(z)]^ reg(t)+𝒪(V_0) .

that permits to reduce the initial set of equations to the set of equations (<ref>) and (<ref>). This simplification neglects only subleading corrections at order ∼ aV_0^2 to the evolution of g^2_θ,θ'(z,t).

Then, we discretize space and momentum and we solve the two coupled equations (<ref>) and (<ref>) using the 4^ th-order Runge Kutta method with time interval Δ t=0.02. More precisely we used the following grid of points: θ∈[-θ_ max,θ_ max] with θ_ max=7 and Δθ=14/100; z∈[-L,L] with L=200 and Δ z=2. 
Additionally, in order to avoid spurious space reflections at the borders during the space propagation of g^(2)_θ,θ'(z), we forced it to be vanishing at z=± L.
The choice of Δ t,Δθ, Δ z, L and θ_ max was made such that it guarantees the convergence of the numerical solution.



The numerical simulations for the evolution of the hard rods gas with long range interaction was made using the same algorithm presented in  Appendix A of <cit.>. In particular we used the symplectic integrator 
implemented in the  function of the  library, i.e. an explicit Runge-Kutta method of order 8 <cit.>.
We used a time step Δ t_ HR=0.0005 that guarantees an error in the total energy of the order of 10^-10.

More precisely, we simulated a system on a ring of total length L_ HR=500 with particle density ρ̅=0.2, namely with ∼ 100 particles per realisation. We performed the ensemble average over a total of 2000 independent initial states for each set of parameters. 















































































































































































 Paper Title: Comprehensive Study of Dynamical Phase Transitions in One-Dimensional XY Model.








School of Physics and Optoelectronic Engineering, Ludong University, YanTai 264025, China

xbmv@bit.edu.cn
Shandong Key Laboratory of Biophysics, Institute of Biophysics, Dezhou University, Dezhou 253023, China






§ ABSTRACT 
Short-time transient dynamical phase transition (SDQPT), serving as the theoretical framework on far-from-equilibrium physics of quantum many-body systems, has been experimentally observed recently. However, the existence of different types of SDQPTs and the associated conditions are yet to be elusive. In this paper, we comprehensively classify SDQPTs by considering the one-dimensional transverse field quantum XY model initialized in the ground state. We find that there are only six types of SDQPTs, and three of which are completely new that had not yet been identified previously. The associated conditions for different SDQPTs are discussed in detail, and based on this, the dynamical phase diagrams are given. Our analytical method used to classify SDQPTs in one-dimensional transverse field quantum XY model can be applied to any other two-band models in one- and two-spatial dimensions.


Comprehensive Study of Dynamical Phase Transitions in One-Dimensional XY Model
    Bao-Ming Xu
    Submitted August 5, 2024
==============================================================================





§ INTRODUCTION

The past decades have witnessed the flourishing of dynamical quantum phase transitions (DQPTs), both in experimental and theoretical research, among condensed matter physicists, because quantum simulators have nowadays achieved experimental access to the real-time dynamics of closed quantum many-body systems; see Refs. <cit.> for the recent reviews. Such quantum simulators have been realized on various experimental platforms, such as ultra-cold atoms <cit.> or trapped ions <cit.>. DQPTs are the sudden or non-analytic changes in the behaviour of a large quantum system during its time evolution, and generally refer to two largely independent concepts: long-time relaxation DQPT (LDQPT) and short-time transient DQPT (SDQPT). LDQPT resembles the equilibrium Landau theory <cit.>: A system undergoes a DQPT at the critical value of the control parameter if the long-time limit of physically relevant observables, namely the order parameters, are finite for one phase, whereas they vanish for another. These asymptotic values are somehow connected with pre-thermalization <cit.>. Furthermore, LDQPT also entails criticality in the transient dynamics of the order parameter and two-point correlators before reaching the steady state, giving rise to effects such as dynamic scaling and aging, which have been extensively explored both theoretically <cit.> and experimentally <cit.>. SDQPT generalizes the nonanalytic behavior of the free energy at a phase transition in the thermodynamic limit to the out-of-equilibrium case <cit.>. To this end, the Loschmidt echo, measuring the return probability of the time-evolving system onto its initial state, has been introduced. SDQPT is defined by the zeros in the Loschmidt echo or the nonanalyticities in its rate function obtained by large deviation principle. It is analogous to the zeros in the partition function and the nonanalyticities in the free energy in equilibrium phase transitions, and therefore the rate function of Loschmidt echo is also called the dynamical free energy per particle. Some SDQPTs obey the dynamical scaling defined by a corresponding out-of-equilibrium analog of the universality class <cit.>. Both kinds of DQPTs have aroused great interests, and this paper focuses exclusively on SDQPT.


SDQPTs were originally introduced in the nearest-neighbor transverse-field Ising model <cit.> and was generalized to mixed state <cit.>, finite temperature <cit.>, Floquet systems <cit.>, slow quench process <cit.> and Kitaev honeycomb model <cit.>. Efforts have also been made to generalize this concept to non-Hermitian systems <cit.>, nonintegrable models <cit.> and disorder systems <cit.>. Experimentally, SDQPTs have been observed in various systems, ranging from trapped ions <cit.>, ultracold atoms <cit.> to superconducting qubits <cit.>, and some other platforms <cit.>. The creation of different phases is not immediately apparent from the nature of the effect, but some dynamical order parameters, such as longitudinal magnetization <cit.>, string order parameter <cit.>, the winding number <cit.>, vortex loops <cit.> and some others <cit.> have been identified so far. Entanglement, as a key concept for the characterization of equilibrium quantum phases and criticality <cit.>, is naturally expected to be generalized to DQPTs. In pursuit of this goal, the connections between entanglement dynamics and SDQPTs were explored <cit.>, although the principles underlying these connections are unclear on a general level. In addition, quantum coherence has been shown to be a valuable resource for SDQPTs <cit.>.

Although the links between the SDQPTs and many physical observables have been established, a thorough understanding of this transition still calls for more studies. To the best of our knowledge, the general relation between SDQPTs and the underlying equilibrium quantum critical point is unclear. The seminal paper <cit.> suggests that SDQPT occurs when the quenching process is ramped through an equilibrium critical point, seemingly implying that SDQPT is closely related to equilibrium phase transitions <cit.>. However, it was later demonstrated that this connection lacks rigor, meaning that SDQPT can occur for the quench within the same phases, and it may not occur for the quench crossing the equilibrium critical point <cit.>. Although SDQPT has been disentangled from equilibrium phase transitions, the conditions for it remain hazy. Beyond that, some SDQPTs lead to topological numbers with integer jumps <cit.>, while others cause topological numbers with half-integer jumps <cit.>. The existence of other types of SDQPTs and the associated conditions require further investigation. This paper conducts a comprehensive study of SDQPTs by taking into account the one-dimensional two-band noninteracting translationally invariant Hamiltonian, identifying six distinct types of SDQPTs and delineating their respective conditions. Two-band model is simple and has analytical results, but captures some of the essential physics, such as  symmetry protected topological phases <cit.>. Two-band models are the effective model of spin chains, including the transverse field Ising and XY chains. In this study, we examine the one-dimensional transverse field XY model, chosen as a prototypical representation of the broader class of two-band models. Such two-band systems are widely studied in the literature, and several of them have been realized experimentally, in condensed matter or with analog quantum simulators using ultracold atoms in optical lattices <cit.>.


This paper is organized as follows: In the next section, we introduce the quenched one-dimensional XY model, a prototypical two-band model, and briefly review the crucial method for the analytical solution, which will be used in the following. Sec. III is devoted to study the central objects of SDQPT, including Loschmidt echo, the rate function, Fisher zeros and the topological order parameter of the winding number. In Sec. IV, a relatively comprehensive classification of SDQPTs is provided, along with the associated criteria. According to the conditions under which different SDQPTs occur, the dynamical phase diagrams are given in Sec. V. Finally, Sec. VI closes the paper with some concluding remarks.





§ QUENCHED ONE-DIMENSIONAL XY MODEL

The Hamiltonian of the one-dimensional XY model is

    Ĥ(λ,γ)=J/2∑_j=1^N[1+γ/2σ̂_j^xσ̂_j+1^x
      +1-γ/2σ̂_j^yσ̂_j+1^y+λσ̂_j^z],

where J is longitudinal spin-spin coupling, γ governs the anisotropic coupling between spins along the x and y directions, λ is a dimensionless parameter measuring the strength of the transverse field with respect to the longitudinal spin-spin coupling. In this work, we set J=1 as the overall energy scale without loss of generality. σ̂^α_j (α=x,y,z) is the spin-1/2 Pauli operator at lattice site j and the periodic boundary conditions are imposed as σ̂^α_N+1=σ̂^α_1. Here we only consider that N is even. The model exhibits competitions between anisotropic and magnetic couplings, which results in the existence of multiple phases. The quantum phase transition from the ferromagnetic phase (FM) to the paramagnetic phase driven by the transverse field λ is called the Ising transition with the quantum critical point λ_c=1. On the other hand, the quantum phase transition between two FMs, with magnetic ordering in the x-direction and the y-direction, respectively, driven by the anisotropy parameter γ, is called the anisotropic transition with the critical point γ_c=0. In fact, in the absence of the transverse field, the ground state of the system is in the Luttinger liquid phase at γ_c=0.

The Hamiltonian is integrable and can be mapped to a system of free fermions and therefore be solved exactly. By applying the Jordan-Wigner transformation and the Fourier transformation, the Hamiltonain converts from spin operators into spinless fermionic operators as

    Ĥ(λ,γ)=∑_k
    [ ĉ^†_-k    ĉ_k ][  λ+cos k -iγsin k;  iγsin k -λ-cos k ][  ĉ_-k; ĉ^†_k ],

where ĉ_k and ĉ^†_k are respectively fermion annihilation and creation operators for mode k=(2n-1)π/N with n=1⋯ N/2, corresponding to antiperiodic boundary conditions for N is even. Each Ĥ_k(λ,γ) acts on a two-dimensional Hilbert space generated by {ĉ^†_-kĉ^†_k|0⟩, |0⟩}, where |0⟩ is the vacuum of the Jordan-Wigner fermions ĉ_k and ĉ_-k, and can be represented in that basis by a 2×2 matrix

    Ĥ_k(λ,γ)=𝐝_k·σ̂

with

    𝐝_k=(0,γsin k,λ+cos k).

The instantaneous eigenvalues are d^±_k=± d_k with

    d_k=√((λ+cos k)^2+γ^2sin^2k).

The corresponding eigenvectors are

    |d^+_k⟩=
    [1+d_k^z/d_k/√(2(1+d_k^z/d_k))ĉ^†_-kĉ^†_k+id_k^y/d_k/√(2(1+d_k^z/d_k))]|0⟩

and

    |d^-_k⟩=
    [id_k^y/d_k/√(2(1+d_k^z/d_k))ĉ^†_-kĉ^†_k+1+d_k^z/d_k/√(2(1+d_k^z/d_k))]|0⟩,

respectively.

At time t≤0, the system is prepared in the ground state

    |G⟩=⊗_k|d^-_k⟩.

Then at time t=0, the transverse field is suddenly changed from λ to λ' or the anisotropic parameter is suddenly changed from γ to γ'. We assume that this process is so sudden that the system state has no time to change. After quenching, the dynamics of the system is governed by the evolution operator (ħ=1)

    Û(t)=e^-iĤ(λ',γ')t=⊗_ke^-iĤ_k(λ',γ')t,

so that the state |ψ(t)⟩ at a time t after the quench is given by

    |ψ(t)⟩=⊗_ke^-iĤ_k(λ',γ')t|d^-_k⟩.





§ THE FUNDAMENTAL THEORY OF SDQPT


The central object within the theory of SDQPTs is the Loschmidt overlap amplitude quantifying the deviation of the time-evolved state from the initial condition. Here, the Loschmidt overlap amplitude is generalized to general density matrices defined as

    𝒢(t)=⟨ G|Û(t)|G⟩.

Because different modes are independent with each other, the Loschmidt overlap amplitude can be written as

    𝒢(t)=∏_k𝒢_k(t)
      =∏_k⟨ d^-_k|e^-iĤ_k(λ',γ')t|d^-_k⟩.

The behavior of the momentum-wise Loschmidt overlap amplitude can be visualized by the vector introduced by Ding <cit.>:

    R⃗_k = (x_k, y_k)= |𝒢_k(t)|e^iϕ_k^G(t),

where

    ϕ_k^G(t)=ϕ_k(t)-ϕ_k^D(t)

is the geometric phase of the Loschmidt overlap amplitude with

    ϕ_k(t)=𝒢_k(t)

being the total phase of 𝒢_k(t) and

    ϕ_k^D(t)=-⟨ d^-_k|Ĥ_k(γ',λ')|d^-_k⟩ t.

being the dynamical phase. At a given time, the winding number of the geometric phase in the first Brillouin zone is defined as <cit.>

    ν(t)=1/2π∫_0^π∂ϕ_k^G(t)/∂ kdk.

In general, it takes a quantized jump whenever the evolution of the system passes through a critical time of SDQPT; therefore, it is generally regarded as the dynamical topological order parameter of SDQPT. It is obvious that ν(t) can be understood as the winding number of the trajectory R⃗_k around the origin.









The rate function of the Loschmidt echo, measuring the return probability |𝒢(t)|^2, is defined as

    r(t)=-lim_N→∞1/Nln|𝒢(t)|^2.

It is the dynamical analogues of the equilibrium free energy, namely the dynamical free energy. In the thermodynamic limit one can derive an exact result for r(t):

    r(t)=-1/πRe[∫_0^π dk ln𝒢_k(t)].

A SDQPT will occur when r(t) exhibits a nonanalyticity at some times t^*, namely critical times.
A powerful method to analyse the nonanalyticity of r(t) is Fisher zeros <cit.>. In order to apply this concept, we should expend time t into the complex plane and focus on the boundary partition function Z(z)=⟨ G|e^-zĤ(λ',γ')|G⟩. The boundary partition function can be expressed as

    Z(z)=∏_k⟨ d^-_k|e^-zĤ_k(λ',γ')|d^-_k⟩

where z∈ℂ. For imaginary z=it this just describes the overlap amplitude of Eq. (<ref>). The Fisher zeros are the values of z that make Z(z)=0. Due to the multiplicative properties of Eq. (<ref>), a zero in Z(z) is equivalent to finding at least one mode k and one z making ⟨ d^-_k|e^-zĤ_k(λ',γ')|d^-_k⟩ = 0. In the thermodynamic limit the zeros of the boundary partition function in the complex plane coalesce to a family of lines labeled by a number n∈ℤ

    z_n(k)=1/2d_k'[ln1-𝐝_k/d_k·𝐝_k'/d_k'/1+𝐝_k/d_k·𝐝_k'/d_k'+i(2n+1)π].

SDQPTs, manifested as the non-analytic behaviors of the dynamical free energy, are therefore contingent upon the existence of purely imaginary zeros of the boundary partition function, corresponding to real-time zeros of the Loschmidt overlap amplitude. From Eq. (<ref>), it can be seen that the condition of SDQPT is 𝐝_k·𝐝_k'=0, i.e.,

    (λ+cos k)(λ'+cos k)+γγ'sin^2 k=0.

In other words, given a set of parameters (λ,γ,λ',γ'), a SDQPT can only occur if at least one mode satisfy Eq. (<ref>) (or 𝐝_k·𝐝_k'=0), this mode is called critical mode k^*. Conversely, solving Eq. (<ref>) yields the critical mode k^*, and discussing the existence of the solution yields the concrete parameters (λ,γ,λ',γ') for SDQPT. Then, there is a series of critical times t^*_n

    t^*_n=1/2d_k^*'(2n+1)π

depending on the corresponding critical mode k^*. Fisher zeros are the key basis for determining whether and what kind of SDQPT occurs, which will be used to achieve the main goal of this paper: classifying SDQPTs and discussing the classification criteria. SDQPTs in the two-band mode have been classified by the ground state topology of both the initial and final Hamiltonians <cit.>. In contrast, our classification is based on the attribution and number of critical modes, and each resulting type of SDQPTs has its own topological properties.

Physically, the condition of SDQPT can be understood with the help of the Bloch sphere. According to the rate function, SDQPT occurring is equivalent to making ⟨ d^-_k^*|e^-iĤ_k^*(λ',γ')t|d^-_k^*⟩ = 0, which means a flip of the vector of |d^-_k^*⟩ in the Bloch sphere (see Fig. <ref>). It is conceivable that the axis of rotation must be perpendicular to a vector in order to flip it (see the green axis in Fig. <ref>). Therefore, SDQPT is only possible if the axis corresponding to the evolution operator Û_k^*(t)=e^-iĤ_k^*(λ',γ')t is perpendicular to the initial state |d^-_k^*⟩, otherwise it is impossible (see the blue axis in Fig. <ref>). The axis corresponding to the evolution operator can be understood as 𝐝_k^*' and the vector of initial state can be understood as 𝐝_k^*, therefore the condition of SDQPT can be demonstrated as  𝐝_k^*·𝐝_k^*'=0.

However, it must be noted that 𝐝_k^*·𝐝_k^*'=0 is not always effective in causing SDQPT. Specifically, the discussions above have acquiesced d_k^*≠0 and d_k^*'≠0, i.e., the energy levels of the pre- and post-quench systems are both non-degenerated. Breaking this acquiescence, i.e., d_k^*=0 or d_k^*'=0, 𝐝_k^*·𝐝_k^*'=0 does not always imply the perpendicularity, because the orientation of zero vector is uncertain and can no longer be used to represent the orientation of the initial state or the axis of evolution. In this subtle case, we use whether the initial state can be flipped as a criterion to judge whether SDQPT occurs, which is a more fundamental and thoroughgoing approach to determine SDQPT. If d_k^*'=0, the initial state |d^-_k^*⟩ would not evolve at all, let along undergo SDQPT, because Û_k^*(t)=e^-iĤ_k^*(λ',γ')t=I (I is an identity matrix). If d_k^*=0 but d_k^*'≠0, the orientation of the evolution operator would still be represented by 𝐝_k^*', and as long as it is perpendicular to the orientation of the initial state (not 𝐝_k^*), the initial state would be flipped and SDQPT would occur. The analysis above will be concretized in the following sections.



§ THE CLASSIFICATION OF DYNAMICAL PHASE TRANSITIONS

In this section, we mainly classify SDQPTs and discuss the parameter spaces of different SDQPTs based on Eq. (<ref>). First, we give the solution to Eq. (<ref>). If γγ'=1, Eq. (<ref>) tells us that

    cos k=-1+λλ'/λ+λ'

with the condition λ+λ'≠0.
If γγ'≠1, the solutions of Eq. (<ref>) are

    cos k_±=1/2(1-γγ')[-(λ+λ')±√(Δ)]

with the condition

    Δ=(λ+λ')^2-4(1-γγ')(γγ'+λλ')≥0.

Specially, if γγ'=0, these two solutions would be simplified as

    cos k_1=-λ, cos k_2=-λ'.

And if λ=±1 or λ'=±1, the solutions (<ref>) would be reduced to

    cos k_1=∓1, cos k_2=∓λλ'+γγ'/1-γγ'.

These solutions allow for single, two, or even infinite critical modes to cause SDQPT. In this paper, the modes k≠0 and k≠π is named as the body modes, and the mode k=0 or k=π is named as the edge mode. The critical mode can be either a body or an edge mode. Based on the number and attribution of the critical mode, as well as the properties of Fisher zeros, SDQPTs can be classified into six types.



 §.§ SDQPT-1

SDQPT-1 is caused by a single critical body mode, wherein the line of Fisher zeros cuts Im(z) axis. First, we discuss the parameter space of SDQPT-1.

For γγ'=1, a critical body mode k^*=arccos[-1+λλ'/λ+λ'] can be found if |1+λλ'/λ+λ'|<1. And we find that (𝐝_k<k^*·𝐝_k<k^*')(𝐝_k>k^*·𝐝_k>k^*')<0, which means that the line of Fisher zeros cuts Im(z) axis at this critical body mode, thereby implying the occurrence of SDQPT-1. The condition |1+λλ'/λ+λ'|<1 means that

    (|λ|-1)(|λ'|-1)<0.

In other words, if the transverse field is quenched across the quantum critical point |λ_c|=1, i.e., |λ|<1→|λ'|>1 or |λ|>1→|λ'|<1, SDQPT-1 would occur.







If γγ'=0, two critical body modes k^*_1=arccos(-λ) and k^*_2=arccos(-λ') could be found for |λ|<1 and |λ'|<1. However, whether SDQPT-1 occurs depends on the quench protocol of anisotropic parameter. If γ=0→γ'≠0, the initial state would be |d_k^-⟩=|0⟩, which could be flipped by the evolution operator Û_k^*_2(t)=e^-id_k^*_2'σ_yt but could not by the evolution operator Û_k^*_1(t)=e^-iĤ_k^*_1(λ',γ')t, therefore SDQPT only occurs at the critical body mode k^*_2=arccos(-λ'). This SDQPT belongs to SDQPT-1 because the line of Fisher zeros cuts Im(z) axis at this critical mode. If γ≠0→γ'=0, SDQPT-1 would occur at the critical mode k^*_1=arccos(-λ), but would not at the critical mode k^*_2=arccos(-λ') because d_k^*_2'=0, which prevents the initial state from evolving at all.

Given γγ'(γγ'-1)≠0, one critical mode k^*=arccos(cos k_±) (|cos k_±|<|cos k_∓|) could be found if (|cos k_+|-1)(|cos k_-|-1)<0, implying that SDQPT-1 occurs. The condition (|cos k_+|-1)(|cos k_-|-1)<0 also means (|λ|-1)(|λ'|-1)<0 that the transverse field is quenched crossing the quantum critical point |λ_c|=1. It can be proven that (|λ|-1)(|λ'|-1)<0 implies Δ>0, therefore only (|λ|-1)(|λ'|-1)<0 deserves consideration in the parameter space of SDQPT-1. In addition, for |λ'|=1, two critical modes k^*_1=π and k^*_2=arccos-λλ'+γγ'/1-γγ' (λ'=1) or k^*_1=0 and k^*_2=arccosλλ'+γγ'/1-γγ' (λ'=-1) could be found if

    |λλ'+γγ'/1-γγ'|<1.

The critical edge mode k^*_1=π or 0 does not cause any SDQPT because d_k^*_1'=0, but the critical body mode k^*_2=arccos-λλ'+γγ'/1-γγ' (λ'=1) or k^*_2=arccosλλ'+γγ'/1-γγ' (λ'=-1) can cause SDQPT-1.

All the elements consisting the parameter space of SDQPT-1 are investigated above. To sum up, they include (see Tab. <ref>) 

(i) γγ'≠0, (|λ|-1)(|λ'|-1)<0; 

(ii) γ=0, γ'≠0, |λ'|<1; 

(iii) γ≠0, γ'=0, |λ|<1, λ'≠λ; 

(iv) γγ'(γγ'-1)≠0, λ≠λ', |λ'|=1, and |λλ'+γγ'/1-γγ'|<1.

Now, we discuss the properties of SDQPT-1. Consider γ=0.5→γ'=2 and λ=0.5→λ'=±1.5 as an example, we plot the lines of Fisher zeros, the time evolution of rate function r(t), the winding number ν(t) and the trajectory of vector in Fig. <ref>. It can be seen that the line of Fisher zeros cuts Im(z) axis at the critical mode [see Figs. <ref>(a1) and (a2)], giving rise to nonanalytic behavior (cusp singularity) of the rate function of the Loschmidt echo [see the blue curves in Figs. <ref>(b1) and (b2)]. The trajectory of vector R⃗_k forms a closed loop anticounterclockwise [see Fig. <ref>(c1)] or counterclockwise for λ'<0 [see Fig. <ref>(c2)], as k varies from 0 to π. The trajectory does not encircle the origin until it firstly crosses it at the first critical time [see Figs. <ref>(c1) and (c2)], and then the winding number jumps up into 1 [see the red curves in Fig. <ref>(b1)] or jumps down into -1 [see the red curves in Fig. <ref>(b2)]. After the first critical time, the trajectory will encircle the origin until the second critical time, when the trajectory secondly crosses the origin and the winding number jumps up into 2 or jumps down into -2, and so on. It should be noted that SDQPT-1 is just the result discovered by the seminal paper <cit.>.



 §.§ SDQPT-2

SDQPT-2 is caused by two critical body modes. These two critical body modes can only be k_+^*=arccos(cos k_+) and k_-^*=arccos(cos k_-) derived from Eq. (<ref>). The presence of them requires (|cos k_+|-1)(|cos k_-|-1)>0 and |cos k_+cos k_-|<1. (|cos k_+|-1)(|cos k_-|-1)>0 implies that 
    (|λ|-1)(|λ'|-1)>0,

i.e., the transverse field is quenched within the same phases. |cos k_+cos k_-|<1 is equivalent to |γγ'+λλ'/1-γγ'|<1 [Eq. (<ref>)]. In summary, the parameter space of SDQPT-2 is given by the conditions γγ'(γγ'-1)≠0, (|λ|-1)(|λ'|-1)>0, Δ>0 and |γγ'+λλ'/1-γγ'|<1 (see Tab. <ref>). Considering γ=1→γ'=-0.5 and λ=1.1→λ'=1.3 as an example, we can find that the line of Fisher zeros cuts Im(z) axis twice at two critical modes [see Figs. <ref>(a)], giving rise to nonanalytic behavior (cusp singularity) of the rate function which implies SDQPT occurring [see the blue curves in Figs. <ref>(b)]. The trajectory of vector R⃗_k forms a closed loop anticounterclockwise as k varies from 0 to π. The trajectory does not encircle the origin until it firstly crosses it at the first critical time t_1^*≈2.07 [see Figs. <ref>(c)], and then the winding number jumps up into 1 [see the red curves in Fig. <ref>(b)]. After the first critical time, the trajectory will encircle the origin until the second critical time t_2^*≈3.95, when it puts the origin out the loop, resulting in the winding number jumping down to 0. SDQPT-2 has been discussed in Ref. <cit.>, and has been shown to occur with the aid of quantum coherence <cit.>.



 §.§ SDQPT-3

Notably, two Fisher zeros cutting Im(z) axis [see Figs. <ref>(a)] can approach each other until they merge, and then leave Im(z) axis, by adjusting the external field and anisotropic parameter. The merged Fisher zero means that there is only one solution to Eq. (<ref>), i.e., cos k_+=cos k_-=-λ+λ'/2(1-γγ'), which requires Δ=0. One critical body mode k^*=arccos[-λ+λ'/2(1-γγ')] can be obtained if |λ+λ'/2(1-γγ')|=|γγ'+λλ'/1-γγ'|<1. And it should be noted that (𝐝_k<k^*·𝐝_k<k^*')(𝐝_k>k^*·𝐝_k>k^*')>0, which implies that the line of Fisher zeros only touches but not cross Im(z) axis at this critical body mode. The SDQPT caused by this kind of critical body mode is named SDQPT-3, an entirely new SDQPT, and the parameter space of SDQPT-3 is determined by the conditions γγ'(γγ'-1)≠0, Δ=0 and |γγ'+λλ'/1-γγ'|<1 (see Tab. <ref>). Considering γ=1→γ'=-0.5 and λ=1.1→λ'=(1-2γγ')λ-√(4γγ'(1-γγ')(1-λ^2)) as an example, we plot Fisher zeros, the rate function, winding number and the trajectory in Fig. <ref>. The line of Fisher zeros touches Im(z) axis at the critical mode [see Figs. <ref>(a)], but the rate function does not show the nonanalytic behavior (cusp singularity) [see the blue curves in Figs. <ref>(b)]. How to describe SDQPT in this case? The trajectory of vector R⃗_k forms a closed loop anticounterclockwise for λ'>0 [see Fig. <ref>(c)], as k varies from 0 to π. It should be noted that the trajectory only crosses the origin at the critical times but never encircle it [see Figs. <ref>(c)]. In this case, the winding number shows singularities at the critical times.










 §.§ SDQPT-4

SDQPT-4 is caused by one edge mode. This edge mode requires that one of the solutions of Eq. (<ref>) should be cos k=±1, implying |λ|=1 or |λ'|=1. If the transverse field is quenched to the quantum critical point λ'=±1, the critical edge mode would not cause SDQPT because d_k^*'=0 and the initial state does not evolve at all. The transverse field quenched from the quantum critical point λ=±1 to others λ'≠λ is a prerequisite of SDQPT-4. In this case, the energy levels at critical edge mode k^* will be degenerate with d_k^*=0. However, whether the SDQPT occurs also depends on the anisotropy parameter γ. If γ=0, the initial state would be |d_k^-⟩=|0⟩ and the evolution operator at critical edge mode would be Û_k^*(t)=e^-id_k^*'σ_zt, so that the initial state can not be flipped not by the evolution operator. Therefore, γ≠0 is also a prerequisite of SDQPT-4. If γ≠0, the ground state (initial state) would continuously become the superposition of |d_k^*^-⟩=√(1/2)(i|1⟩+|0⟩), but the evolution operator would continuously become e^-i(λ'∓1)σ̂^zt, as k approaches into critical edge mode k^*. The initial state can be flipped by the evolution operator, so SDQPT-4 occurs. It should be noted that, in addition to this critical edge mode there could be another critical body mode, if γγ'(γγ'-1)≠0, Δ>0 and |γγ'+λλ'/1-γγ'|<1, which would cause SDQPT-5, discussed in the next section. To sum up, the parameter spaces of SDQPT-4 are (see Tab. <ref>) 

(i) γγ'=1, |λ|=1, λ'≠λ; 

(ii) γ≠0, γ'=0, |λ|=1, λ'≠λ; 

(iii) γγ'(γγ'-1)≠0, |λ|=1, λ≠λ', Δ>0, |γγ'+λλ'/1-γγ'|>1.

Consider γ=0.5→γ'=2 and λ=1→λ'=±1.5 as an example, we plot the lines of Fisher zeros, the time evolution of rate function r(t), the winding number ν(t) and the trajectory of vector in Fig. <ref>. It can be seen that the Fisher zeros coalesce into a continuous curve that touches Im(z) axis at a critical edge mode k^* [see Fig. <ref>(a1) and (a2)], giving rise to nonanalytic behavior (cusp singularity) of the rate function of the Loschmidt echo which implies SDQPT occurring [see the blue curves in Fig. <ref>(b1) and (b2)]. It should be noted that the winding number in this SDQPT jumps up with half-integer, differing from SDQPT-1 in Fig. <ref>, because the trajectory of R⃗_k only swipes half a circle [see Fig. <ref>(c1) and (c2)], which is consistent with the observation in Ref. <cit.>. The trajectory of vector R⃗_k swipes half a circle anticounterclockwise for λ'>0 [see Fig. <ref>(c1)] and counterclockwise for λ'<0 [see Fig. <ref>(c2)], as k varies from 0 to π. The trajectory does not half-encircle the origin until it firstly crosses it at the first critical time [see Figs. <ref>(c1) and (c2)], and then the winding number jumps up into 1/2 [see the red curves in Fig. <ref>(b1)] or jumps down into -1/2 [see the red curves in Fig. <ref>(b2)]. After the first critical time, the trajectory will half-encircle the origin until the second critical time, when the trajectory secondly crosses the origin and the winding number jumps up into 2/2 or jumps down into -2/2, and so on.





 §.§ SDQPT-5

SDQPT-5 is caused by one critical edge mode and one critical body mode. These two critical modes are derived from Eq. (<ref>) for |γγ'+λλ'/1-γγ'|<1. In summary, the parameter space satisfies the following conditions γγ'(γγ'-1)≠0, |λ|=1, λ≠λ', Δ>0 and |γγ'+λλ'/1-γγ'|<1 (see Tab. <ref>). Consider γ=1→γ'=0.3 and λ=1→λ'=0.1 as example, we plot SDQPT-5 in Fig. <ref>. From Fig. <ref> we can see that the line of Fisher zeros cuts Im(z) axis at the critical mode k_2^* and touches it at the critical mode k_1^* [see Figs. <ref>(a)], giving rise to nonanalytic behavior of the rate function of the Loschmidt echo [see the blue curves in Figs. <ref>(b)]. As k varies from 0 to π, the coordinate of R⃗_k moves first clockwise and then counterclockwise, forms an open trajectory [see Figs. <ref>(b)]. The end of trajectory crosses the origin at the first critical time, and then the winding number jumps down into -1/2 [see the red curves in Fig. <ref>(b)]. After the first critical time, the trajectory will half-encircle the origin until the second critical time, when the trajectory secondly cross the origin counterclockwise. Notably, it is not the end of trajectory that crosses the origin secondly. As a result, the winding number will jumps up from -1/2 to 1/2 [see the red curves in Fig. <ref>(b)]. SDQPT-5 is also an entirely new SDQPT that is not discovered before.




 §.§ SDQPT-6

It is worth noting that, if the transverse field is quenched from one critical point λ=±1 to another one λ'=∓1 and the anisotropic parameters satisfy γγ'=1, Eq. (<ref>) is an equality independent of k. In other words, all the Fisher zeros are located on Im(z) axis [see Fig. <ref>(a)], implying a very anomalous DQPT that the critical time is no longer periodic. The trajectory of R⃗_k is always along the imaginary axis [see Fig. <ref>(c)], in this case, the winding number is always 0 [see Fig. <ref>(b)]. This anomalous SDQPT is also an entirely new SDQPT, and is referred to in this paper as SDQPT-6.

In summary, we demonstrate that there are only six possible SDQPTs in the one-dimensional XY model, three of which were previously discovered, and the others are entirely new SDQPTs that have not been discovered before. Furthermore, the associated conditions for different SDQPTs are shown in the Table <ref>.




§ DYNAMICAL PHASE DIAGRAM

Now we give the dynamical phase diagram according to the discussions above. We should consider six cases: γγ'<0, γγ'=0 including γ=0→γ'≠0 and γ≠0→γ'=0, 0<γγ'<1, γγ'=1 and γγ'>1 to investigate the dynamical phase diagrams, and plot them in Fig. <ref>. It can be seen that for γγ'≠0, i.e., the anisotropic parameters (γ and γ') before and after quench are both not at the critical point γ_c=0, SDQPT-1 always happen if the transverse field is quenched across quantum critical point |λ_c|=1 [see the purple areas in Fig. <ref>(a)-(d)]. However, this does not mean SDQPT must depend on the quantum phase transition because SDQPT-2 (the yellow areas) and SDQPT-3 (the red curves) can happen when the transverse field is quenched within the same phases. To be specific, SDQPT-2 and SDQPT-3 can happen at the ferromagnetic phase if 0<γγ'<1, but can happen at the paramagnetic phase if γγ'>1 [see Fig. <ref>(b) and (c)]. Beyond that, SDQPT-2 and SDQPT-3 can happen at both ferromagnetic and paramagnetic phases, if the anisotropic parameter is quenched across the critical point γ_c=0 [see Fig. <ref>(d)]. It is worth noting that for a given suitable field λ=λ', only the quench of the anisotropic parameter across the critical point γ_c=0 can still cause SDQPT-2 and SDQPT-3, and even SDQPT-1.

At the critical boundaries, the situations are more complicated. If the transverse field is quenched from the quantum critical point |λ_c|=1, SDQPT can always occur, either SDQPT-4 or SDQPT-5 (see the blue and cyan lines in Fig. <ref>), depending on λ' and γγ', as shown in Table <ref>. On the other hand, if the transverse field is quenched to the quantum critical point |λ'|=1, only SDQPT-1 can occur, but not always, depending on λ' and γγ' (see the purple lines in Fig. <ref>). If the anisotropic parameter is quenched from the critical point γ_c=0, SDQPT-1 can always occur when the transverse field is quench to the ferromagnetic phase |λ'|<1, independent of the pre-quenched field λ [see the purple areas in Fig. <ref>(e)]; on the contrary, if the anisotropic parameter is quenched to the critical point γ_c=0, SDQPT-1 can always occur when the transverse field is quench from the ferromagnetically ordered phase |λ'|<1, independent of the post-quenched field λ'≠λ [see the purple areas in Fig. <ref>(f)]. These results furthermore demonstrate that SDQPT is independent of quantum phase transition.



§ CONCLUSIONS

In this paper, we have comprehensively investigated SDQPTs by focusing the Fisher zeros, the rate function of Loschmidt echo and the winding number after quenching the strength of transverse field and the anisotropic parameter in the one-dimensional XY model. Accordingly, SDQPTs have been classified into six types based on the attribution and number of critical modes, and each type has its own topological properties. Specifically, SDQPT-1 is caused by a single critical body mode at which the line of Fisher zeros cuts the imagery axis, and its topological parameter order jumps up or down with an integer unit at the critical times. SDQPT-2 happens when two critical body modes exist. In SDQPT-2, the winding number jumps up at one set of critical times determined by one critical mode, but jumps down at another set of critical times determined by another critical mode, the jump heights are all integer units. By adjusting the external field and anisotropic parameter, two critical body mode above can approach each other until they merge. In this case, SDQPT-3 occurs. SDQPT-3 can not be described by the rate function of the Loschmidt echo, but can be described by the singularities of the winding number. SDQPT-4 is caused by a single critical edge mode, wherein the winding number jumps up or down with a half-integer unit. SDQPT-5 is commonly caused by two critical modes: one critical body mode and one critical edge mode. The critical body mode makes the winding number jump with the integer unit, whereas the critical edge mode makes it jump with the half-integer unit. SDQPT-6 is very anomalous that all the modes locate at the imagery axis. Although SDQPT-1, SDQPT-2 and SDQPT-4 have been previously discovered in other papers, SDQPT-3, SDQPT-5 and SDQPT-6 are entirely new and had not been discussed before. Furthermore, the parameter spaces of these six SDQPTs have been determined and the phase diagrams have been given. The analytical method used in this paper can be applied to any other two-band models in one and two dimensions (e.g. SSH model, Kitaev-chain, Haldane model, p+ip superconductor, etc.).






§ ACKNOWLEDGEMENT

This work was supported by the National Natural Science Foundation of China (Grant No. 11705099) and the Talent Introduction Project of Dezhou University of China (Grant Nos. 2020xjpy03 and 2019xgrc38).




Heyl2019 M. Heyl, Dynamical quantum phase transitions: A brief survey, Europhys. Lett. 125, 26001 (2019).
Heyl2018 M. Heyl, Dynamical quantum phase transitions: a review, Rep. Prog. Phys. 81, 054001 (2018).
Marino2022 J. Marino, M. Eckstein, M. S. Foster and A. M. Rey, Dynamical phase transitions in the collisionless pre-thermal states of isolated quantum systems: theory and experiments, Rep. Prog. Phys. 85, 116001 (2022).

Levin2012 K. Levin, A. Fetter, and D. Stamper-Kurn, Ultracold Bosonic and Fermionic Gases, Contemporary Concepts of Condensed Matter Science (Elsevier Science, New York, 2012).
Yukalov2011 V. I. Yukalov, Equilibration and thermalization in finite quantum systems, Laser Phys. Lett. 8, 485 (2011).
Bloch2008 I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
Greiner2002 M. Greiner, O. Mandel, T. W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature (London) 419, 51 (2002).
Porras2004 D. Porras and J. I. Cirac, Effective quantum spin systems with trapped ions, Phys. Rev. Lett. 92, 207901 (2004).
Kim2009 K. Kim, M.-S. Chang, R. Islam, S. Korenblit, L.-M. Duan, and C. Monroe, Entanglement and tunable spin-spin couplings between trapped ions using multiple transverse modes, Phys. Rev. Lett. 103, 120502 (2009).
Jurcevic2014 P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Quasiparticle engineering and entanglement propagation in a quantum many-body system, Nature (London) 511, 202 (2014).
Eisert2015 J. Eisert, M. Friesdorf and C. Gogolin, Quantum many-body systems out of equilibrium, Nature Phys. 11, 124 (2015).

Hofferberth2007 S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Non-equilibrium coherence dynamics in one-dimensional Bose gases, Nature (London) 449, 324 (2007).
Gring2012 M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer, Relaxation and Prethermalization in an Isolated Quantum System, Science 337, 1318 (2012).
langen2013 T. Langen, R. Geiger, M. Kuhnert, B. Rauer, and J. Schmiedmayer, Local emergence of thermal correlations in an isolated quantum many-body system, Nat. Phys. 9, 640 (2013).
Langen2015 T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, Experimental observation of a generalized Gibbs ensemble, Science 348, 207 (2015).
Neyenhuis2017 B. Neyenhuis, J. Zhang, P. W. Hess, J. Smith, A. C. Lee, P. Richerme, Z.-X. Gong, A. V. Gorshkov, and C. Monroe, Observation of prethermalization in long-range interacting spin chains, Sci. Adv. 3, e1700672 (2017).
Le2023 Y. Le, Y. Zhang, S. Gopalakrishnan, M. Rigol, and D. S. Weiss, Observation of hydrodynamization and local prethermalization in 1D Bose gases. Nature 618, 494 (2023).
Bertini2015 B. Bertini, F. H. L. Essler, S. Groha, and N. J. Robinson, Prethermalization and Thermalization in Models with Weak Integrability Breaking, Phys. Rev. Lett. 115, 180601 (2015).
Mallayya2019 K. Mallayya, M. Rigol, and W. De Roeck, Prethermalization and Thermalization in Isolated Quantum Systems, Phys. Rev. X 9, 021027 (2019).
Mori2018 T. Mori, T. N. Ikeda, E. Kaminishi and M. Ueda, Thermalization and prethermalization in isolated quantum systems: a theoretical overview, J. Phys. B: At. Mol. Opt. Phys. 51 112001 (2018).



Chiocchetta2015 A. Chiocchetta, M. Tavora, A. Gambassi, and A. Mitra, Short-time universal scaling in an isolated quantum system after a quench, Phys. Rev. B 91, 220302(R) (2015).
Marcuzzi2016 M. Marcuzzi, J. Marino, A. Gambassi, and A. Silva, Prethermalization from a low-density Holstein-Primakoff expansion, Phys. Rev. B 94, 214304 (2016).
Chiocchetta2017 A. Chiocchetta, A. Gambassi, S. Diehl, and J. Marino, Dynamical Crossovers in Prethermal Critical States, Phys. Rev. Lett. 118, 135701 (2017).

Nicklas2015 E. Nicklas, M. Karl, M. Höfer, A. Johnson, W. Muessel, H. Strobel, J. Tomkovič, T. Gasenzer, and M. K. Oberthaler, Observation of Scaling in the Dynamics of a Strongly Quenched Quantum Gas, Phys. Rev. Lett. 115, 245301 (2015).
Heyl2013 M. Heyl, A. Polkovnikov, S. Kehrein, Dynamical Quantum Phase Transitions in the Transverse Field Ising Model, Phys. Rev. Lett. 110, 135704 (2013).

Heyl2015 M. Heyl, Scaling and universality at dynamical quantum phase transitions, Phys. Rev. Lett. 115, 140602 (2015).
Trapin2021 D. Trapin, J. C. Halimeh, and M. Heyl, Unconventional critical exponents at dynamical quantum phase transitions in a random ising chain, Phys. Rev. B 104, 115159 (2021).

Bhattacharya2017a U. Bhattacharya, S. Bandyopadhyay, and A. Dutta, Mixed state dynamical quantum phase transitions, Phys. Rev. B 96, 180303(R) (2017).
Heyl2017 M. Heyl and J. C. Budich, Dynamical topological quantum phase transitions for mixed states, Phys. Rev. B 96, 180304(R) (2017).
Lang2018 H. Lang, Y. Chen, Q. Hong, and H. Fan, Dynamical quantum phase transition for mixed states in open systems, Phys. Rev. B 98, 134310 (2018).
Mera2018 B. Mera, C. Vlachou, N. Paunković, V. R. Vieira, and O. Viyuela, Dynamical phase transitions at finite temperature from fidelity and interferometric Loschmidt echo induced metrics, Phys. Rev. B 97, 094110 (2018).


Sharma2014 S. Sharma, A. Russomanno, G. E. Santoro, and A. Dutta, Loschmidt echo and dynamical fidelity in periodically driven quantum systems, Europhys. Lett. 106, 67003 (2014).
Yang2019 K. Yang, L. Zhou, W. Ma, X. Kong, P. Wang, X. Qin, X. Rong, Y. Wang, F. Shi, J. Gong, and J. Du, Floquet dynamical quantum phase transitions, Phys. Rev. B 100, 085308 (2019).
Zamani2020 S. Zamani, R. Jafari, and A. Langari, Floquet dynamical quantum phase transition in the extended XY model: Nonadiabatic to adiabatic topological transition, Phys. Rev. B 102, 144306 (2020).
Zhou2021b L. Zhou and Q. Du, Floquet dynamical quantum phase transitions in periodically quenched systems, J. Phys.: Condens. Matter 33, 345403 (2021).
Jafari2021 R. Jafari and A. Akbari, Floquet dynamical phase transition and entanglement spectrum, Phys. Rev. A 103, 012204 (2021).
Hamazaki2021 R. Hamazaki, Exceptional dynamical quantum phase transitions in periodically driven systems, Nature Commun. 12, 5108 (2021).
Zamani2022 S. Zamani, R. Jafari, and A. Langari, Out-of-time-order correlations and Floquet dynamical quantum phase transition, Phys. Rev. B 105, 094304 (2022).
Luan2022 L.-N. Luan, M.-Y. Zhang, and L. Wang, Floquet dynamical quantum phase transitions of the XY spin-chain under periodic quenching, Physica A 604, 127866 (2022).
Jafari2022 R. Jafari, A. Akbari, U. Mishra, and H. Johannesson, Floquet dynamical quantum phase transitions under synchronized periodic driving, Phys. Rev. B 105, 094311 (2022).
Kosior2018a A. Kosior and K. Sacha, Dynamical quantum phase transitions in discrete time crystals, Phys. Rev. A 97, 053621 (2018).
Kosior2018b A. Kosior, A. Syrwid, and K. Sacha, Dynamical quantum phase transitions in systems with broken continuous time and space translation symmetries, Phys. Rev. A 98, 023612 (2018).
Naji2022 J. Naji, R. Jafari, L. Zhou, and A. Langari, Engineering Floquet dynamical quantum phase transitions, Phys. Rev. B 106, 094314 (2022).

Sharma2016 S. Sharma, U. Divakaran, A. Polkovnikov, and A. Dutta, Slow quenches in a quantum Ising chain: Dynamical phase transitions and topology, Phys. Rev. B 93, 144306 (2016).
Sedlmayr2018 N. Sedlmayr, P. Jaeger, M. Maiti, and J. Sirker, Bulk-boundary correspondence for dynamical phase transitions in one-dimensional topological insulators and superconductors, Phys. Rev. B 97, 064304 (2018).
Maslowski2020 T. Masłowski and N. Sedlmayr, Quasiperiodic dynamical quantum phase transitions in multiband topological insulators and connections with entanglement entropy and fidelity susceptibility, Phys. Rev. B 101, 014301 (2020).
Zamani2024 S. Zamani, J. Naji, R. Jafari and A. Langari, Scaling and universality at ramped quench dynamical quantum phase transitions, J. Phys.: Condens. Matter 36 355401 (2024).
Schmitt2015 M. Schmitt and S. Kehrein, Dynamical quantum phase transitions in the Kitaev honeycomb model, Phys. Rev. B 92, 075114 (2015).



Zhou2018 L. Zhou, Q.-h. Wang, H. Wang, and J. Gong, Dynamical quantum phase transitions in non-Hermitian lattices, Phys. Rev. A 98, 022129 (2018).
Zhou2021a L. Zhou and Q. Du, Non-Hermitian topological phases and dynamical quantum phase transitions: A generic connection, New J. Phys. 23, 063041 (2021).
Mondal2022 D. Mondal and T. Nag, Anomaly in the dynamical quantum phase transition in a non-Hermitian system with extended gapless phases, Phys. Rev. B 106, 054308 (2022)
Mondal2023 D. Mondal and T. Nag, Finite temperature dynamical quantum phase transition in a non-Hermitian system, Phys. Rev. B 107, 184311 (2023).
Mondal2024 D. Mondal and T. Nag, Persistent anomaly in dynamical quantum phase transition in long-range non-Hermitian p-wave Kitaev chain. Eur. Phys. J. B 97, 59 (2024).
Jing2024 Y. Jing, J.-J. Dong, Y.-Y. Zhang, and Z.-X. Hu, Biorthogonal Dynamical Quantum Phase Transitions in Non-Hermitian Systems, Phys. Rev. Lett. 132, 220402 (2024).

Karrasch2013 C. Karrasch and D. Schuricht, Dynamical phase transitions after quenches in nonintegrable models, Phys. Rev. B 87, 195104 (2013).
Sharma2015 S. Sharma, S. Suzuki, and A. Dutta, Quenches and dynamical phase transitions in a nonintegrable quantum Ising model, Phys. Rev. B 92, 104306 (2015).
Cheraghi2021 H. Cheraghi, S. Mahdavifar, Dynamical Quantum Phase Transitions in the 1D Nonintegrable Spin-1/2 Transverse Field XZZ Model, Ann. Phys. (Berlin) 533, 2000542 (2021).
Kennes2018 D. M. Kennes, D. Schuricht, and C. Karrasch, Controlling dynamical quantum phase transitions, Phys. Rev. B 97, 184302 (2018).
Andraschko2014 F. Andraschko and J. Sirker, Dynamical quantum phase transitions and the Loschmidt echo: A transfer matrix approach, Phys. Rev. B 89, 125120 (2014).
Halimeh2021a J. C. Halimeh, M. V. Damme, L. Guo, J. Lang, and P. Hauke, Dynamical phase transitions in quantum spin models with antiferromagnetic long-range interactions, Phys. Rev. B 104, 115133 (2021).
Hagymasi2019 I. Hagymási, C. Hubig, Ö. Legeza, and U. Schollwöck, Dynamical Topological Quantum Phase Transitions in Nonintegrable Models, Phys. Rev. Lett. 122, 250601 (2019).

Cao2020 K. Cao, W. Li, M. Zhong, and P. Tong, Influence of weak disorder on the dynamical quantum phase transitions in the anisotropic xy chain, Phys. Rev. B 102, 014207 (2020).
Kuliashov2023 O. N. Kuliashov, A. A. Markov, and A. N. Rubtsov, Dynamical quantum phase transition without an order parameter, Phys. Rev. B 107, 094304 (2023).
Jafari2024 R. Jafari, A. Langari, S. Eggert, and H. Johannesson, Dynamical quantum phase transitions following a noisy quench, Phys. Rev. B 109, L180303 (2024).


Jurcevic2017 P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos, Direct Observation of Dynamical Quantum Phase Transitions in an Interacting Many-Body System, Phys. Rev. Lett. 119, 080501 (2017).
Flaschner2018 N. Fläschner, D. Vogel, M. Tarnowski, B. S. Rem, D.-S. Lühmann, M. Heyl, J. C. Budich, L. Mathey, K. Sengstock, and C. Weitenberg, Observation of dynamical vortices after quenches in a system with topology, Nat. Phys. 14, 265 (2018).
Guo2019 X.-Y. Guo, C. Yang, Y. Zeng, Y. Peng, H.-K. Li, H. Deng, Y.-R. Jin, S. Chen, D. Zheng, and H. Fan, Observation of a Dynamical Quantum Phase Transition by a Superconducting Qubit Simulation, Phys. Rev. Applied 11, 044080 (2019).
Tian2019 T. Tian, Y. Ke, L. Zhang, S. Lin, Z. Shi, P. Huang, C. Lee, and J. Du, Observation of dynamical phase transitions in a topological nanomechanical system, Phys. Rev. B 100, 024310 (2019).
Wang2019 K. Wang, X. Qiu, L. Xiao, X. Zhan, Z. Bian, W. Yi, and P. Xue, Simulating Dynamic Quantum Phase Transitions in Photonic Quantum Walks, Phys. Rev. Lett. 122, 020501 (2019).
Xu2020 X.-Y. Xu, Q.-Q. Wang, M. Heyl, J. C. Budich, W.-W. Pan, Z. Chen, M. Jan, K. Sun, J.-S. Xu, Y.-J. Han, C.-F. Li, and G.-C. Guo, Measuring a dynamical topological order parameter in quantum walks, Light Sci. Appl. 9, 7 (2020).
Heyl2014 M. Heyl, Dynamical Quantum Phase Transitions in Systems with Broken-Symmetry Phases, Phys. Rev. Lett. 113, 205701 (2014).

Budich2016 J. C. Budich and M. Heyl, Dynamical topological order parameters far from equilibrium, Phys. Rev. B 93, 085416 (2016).
Bhattacharya2017b U. Bhattacharya and A. Dutta, Emergent topology and dynamical quantum phase transitions in two-dimensional closed quantum systems, Phys. Rev. B 96, 014302 (2017).
Kosior2024 A. Kosior and M. Heyl, Vortex loop dynamics and dynamical quantum phase transitions in three-dimensional fermion matter, Phys. Rev. B 109, L140303 (2024).

Yu2021 W. C. Yu, P. D. Sacramento, Y. C. Li, and H.-Q. Lin, Correlations and dynamical quantum phase transitions in an interacting topological insulator, Phys. Rev. B 104, 085104 (2021).
Halimeh2021b J. C. Halimeh, D. Trapin, M. Van Damme, and M. Heyl, Local measures of dynamical quantum phase transitions, Phys. Rev. B 104, 075130 (2021).
Li2023 C.-X. Li, S. Yang, J.-B. Xu, and H.-Q. Lin, Exploring dynamical quantum phase transitions in a spin model with deconfined critical point via the quantum steering ellipsoid, Phys. Rev. B 107, 085130 (2023).

Amico2008 L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008).
Laflorencie2016 N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rep. 646, 1 (2016).

Torlai2014 G. Torlai, L. Tagliacozzo and G. De Chiara, Dynamics of the entanglement spectrum in spin chains, J. Stat. Mech. (2014) P06001.
Canovi2014 E. Canovi, E. Ercolessi, P. Naldesi, L. Taddia, and D. Vodola, Dynamics of entanglement entropy and entanglement spectrum crossing a quantum phase transition, Phys. Rev. B 89, 104303 (2014).
Schmitt2018 M. Schmitt, M. Heyl, Quantum dynamics in transverse-field Ising models from classical networks, SciPost Phys. 4, 013 (2018).
Surace2020 J. Surace, L. Tagliacozzo, and E. Tonni, Operator content of entanglement spectra in the transverse field Ising chain after global quenches, Phys. Rev. B 101, 241107(R) (2020).
Nicola2021 S. De Nicola, A. A. Michailidis, and M. Serbyn, Entanglement View of Dynamical Quantum Phase Transitions, Phys. Rev. Lett. 126, 040602 (2021).
Modak2021 R. Modak and D. Rakshit, Many-body dynamical phase transition in a quasiperiodic potential, Phys. Rev. B 103, 224310 (2021).
Nicola2022 S. De Nicola, A. A. Michailidis, and M. Serbyn, Entanglement and precession in two-dimensional dynamical quantum phase transitions, Phys. Rev. B 105, 165149 (2022).
Xu2023 B.-M. Xu and C.-Q. Wang, The singularities of the rate function of quantum coherent work in one-dimensional transverse field Ising model, New J. Phys. 25 073020 (2023).
Xu2024 B.-M. Xu, Quantum coherence assisted dynamical phase transition, arXiv:2305.08400 (2023).
Vajna2014 S. Vajna and B. Dóra, Disentangling dynamical phase transitions from equilibrium phase transitions, Phys. Rev. B 89, 161105(R) (2014).
Bhattacharjee2018 S. Bhattacharjee and A. Dutta, Dynamical quantum phase transitions in extended transverse Ising models, Phys. Rev. B 97, 134306 (2018).
Qiu2018 X. Qiu, T.-S. Deng, G.-C. Guo, and W. Yi, Dynamical topological invariants and reduced rate functions for dynamical quantum phase transitions in two dimensions, Phys. Rev. A 98, 021601(R) (2018).

Wong2023 C. Y. Wong, H. Cheraghi, and W. C. Yu, Quantum spin fluctuations in dynamical quantum phase transitions, Phys. Rev. B 108, 064305 (2023).
Ding2020 C. Ding, Dynamical quantum phase transition from a critical quantum quench, Phys. Rev. B 102, 060409(R) (2020).
Chiu2016 C.-K. Chiu, J. C.-Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
Atala2013 M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, Direct measurement of the Zak phase in topological Bloch bands, Nature Phys. 9, 795 (2013).
Jotzu2014 G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Experimental realization of the topological Haldane model with ultracold fermions, Nature (London) 515, 237 (2014).
Mourik2012 V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices, Science 336, 1003 (2012).
Perge2014 S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor, Science 346, 602 (2014).
Fisher1965 M. E. Fisher, in Boulder Lectures in Theoretical Physics (University of Colorado, Boulder, 1965), Vol. 7.
Vajna2015 S. Vajna and B. Dóra, Topological classification of dynamical phase transitions, Phys. Rev. B 91, 155127 (2015).







 Paper Title: Force and kinetics of fast and slow muscle myosin determined with a synthetic sarcomere-like nanomachine.




headings	



	
Finding Defective Elements in Intelligent Reflecting Surface via Over-the-Air Measurements
    
   Ziyi Zhao1,
                     Zhaorui Wang2,
                     Shuowen Zhang1,
                     and Liang Liu1
   1
                    Department of Electrical and Electronic Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, China,

                     E-mail: ziyii.zhao@connect.polyu.hk, {shuowen.zhang,liang-eie.liu}@polyu.edu.hk
   2
                     School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Shenzhen, China 

                     E-mail: wangzhaorui@cuhk.edu.cn
 The work was supported in part by the National Key R&D Project of China under Grant No. 2022YFB2902800; by the National Natural Science Foundation of China with Grant No. 62101474; and by the Research Grants Council, Hong Kong, China, with Grant No. 15203222 and Grant No. 15230022.
    August 5, 2024
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

15cm0.5pt
    § ABSTRACT 
    Myosin II is the muscle molecular motor that works in two bipolar arrays in each thick filament of the striated (skeletal and cardiac) muscle, converting the chemical energy into steady force and shortening by cyclic ATP–driven interactions with the nearby actin filaments.
    Different isoforms of the myosin motor in the skeletal muscles account for the different functional requirements of the slow muscles (primarily responsible for the posture) and fast muscles (responsible for voluntary movements). 
    To clarify the molecular basis of the differences, here the isoform–dependent mechanokinetic parameters underpinning the force of slow and fast muscles are defined with a unidimensional synthetic nanomachine powered by pure myosin isoforms from either slow or fast rabbit skeletal muscle. 
    Data fitting with a stochastic model provides a self–consistent estimate of all the mechanokinetic properties of the motor ensemble including the motor force, the fraction of actin–attached motors and the rate of transition through the attachment–detachment cycle. 
    The achievements in this paper set the stage for any future study on the emergent mechanokinetic properties of an ensemble of myosin molecules either engineered or purified from mutant animal models or human biopsies.
    
	^*Corresponding authors 
	E-mail addresses: vincenzo.lombardi@unifi.it, duccio.fanelli@unifi.it

	

	

	Keywords: Myosin motor; muscle myosin isoforms; synthetic myosin–based nanomachine
	
15cm0.4pt




§ INTRODUCTION


Steady force and shortening in the half–sarcomere, the functional unit of the muscle cell, are due to ATP–driven cyclic interactions of the subfragment 1 (S1, the head) of the heavy meromyosin fragment (HMM) of the myosin II molecule extending from the thick filament with the actin monomers on the nearby overlapping thin filament (Figure <ref>). 


In each interaction the free energy of the splitting of MgATP to MgADP and inorganic phosphate (Pi) in the head is associated to a structural working stroke consisting in a tilting between the motor domain firmly attached to actin (red) and the light chain binding domain (or lever arm, violet) connected to the myosin filament backbone through the subfragment 2 (S2, the tail, green). 
In isometric contraction, lever arm tilt raises the force exerted by the half–sarcomere, increasing the strain of all the elastic elements represented for simplicity in Figure <ref>, state (b) by the bending of the lever arm. 

When the load is lower than the maximum steady force exerted under isometric conditions (T_0, that is conventionally expressed as force per cross–sectional area of the contractile material, □), lever arm tilting results in relative filament sliding with a reduced strain in the elastic components (Figure <ref>, state (c)). 
Cyclic asynchronous interactions of myosin motors with the actin filament account for the generation of steady force and shortening.
The shortening velocity V is inversely proportional to the force T (force–velocity relation, T-V hill1938heat). 
At physiological concentrations of ATP, ADP release is the rate–limiting step for motor detachment from actin (step (b)/ (c)→ (d)).
The rate of ADP release is conformation–dependent, increasing during steady shortening when motors at the end of the working stroke would become negatively strained.
This explains the increased rate of energy liberation E (and the underlying ATP hydrolysis rate, ϕ) when the load is reduced and the shortening speed is increased hill1938heat, huxley1957muscle, fenn1924relation, woledge1985energetic, smith1995strain, smith1995strainII, smith2008towards, linari2020straightening. 
Faster detachment of negatively strained (compressed) motors prevents the ones at the end of their working stroke to oppose positively strained motors, a requirement for the maximisation of the power and efficiency of an array of motors working in parallel.  
The performance of different types of skeletal muscles depends on the myosin II isoform expressed in the muscle. 
Slow muscles, which are primarily involved in maintenance of posture and are characterised by the dominant presence of the isoform 1 of Myosin Heavy Chain (MHC-1 isoform), exhibit lower shortening speed at any given load, thus develop lower power and consume ATP at a lower rate than fast muscles which are involved in movement and are characterised by the dominant presence of the isoforms MHC-2A, -2B or -2X isoforms schiaffino2011fiber. 
Strikingly, the functional difference between slow and fast isoforms is due to a difference of only 20% in the amino–acid composition.  

During an isometric contraction the power is zero, so that the rate of energy consumption accounting for the steady force T_0 (denoted E_0) corresponds to the rate of heat production (H_0) hill1938heat. 
E_0 has been found ∼ 5–fold larger in fast muscles than in slow muscles barclay1993energetics, barclay2010efficiency, wendt1973energy, wendt1974energy. 
The underlying rate of ATP hydrolysis at T_0 can be obtained from E_0 by dividing it by the energy liberated per molecule of ATP hydrolysed (Δ G_ATP = 60 in mammalian muscle according to barclay2010inferring). 
In this way the energetic cost of the isometric force in the intact muscle can be compared with that in the demembranated fibres, in which the rate of energy liberation is determined by measuring the rate of ATP hydrolysis.
A further normalisation for the concentration of myosin motors in the mammalian muscle (0.18, barclay2010efficiency) gives the rate of ATP hydrolysed per myosin motor (ϕ). 
In demembranated fibres of rat fast muscle bottinelli1994unloaded, reggiani1997chemo, rabbit potma1994effects, reggiani2000sarcomeric and human muscle he2000atp at 12, ϕ is 5–fold (or more) larger than in slow muscle, in agreement with muscle measurements (Supplementary Table 1).  
In both fast muscles barclay1993energetics, barclay2010efficiency, wendt1973energy, wendt1974energy and fast demembranated muscle fibres bottinelli1994unloaded, reggiani1997chemo, potma1994effects, reggiani2000sarcomeric T_0 is either similar or at max 1.5–fold larger than in slow muscles and muscle fibres.
Thus the tension cost of the isometric contraction E_0 / T_0 results to be systematically larger in fast muscles by on average 5–fold (with a minimum of 3–fold). 
The justification for the elevated tension cost of the fast muscle can be only partly found in the intrinsic larger actin–activated myosin ATPase in solution, which for the fast myosin is twice that of the slow myosin barany1965myosin.

The bulk of data characterising the energetics of slow and fast muscles at cell and tissue levels, first of all the ∼ 5–fold larger isometric tension cost, leaves open the question of the underlying molecular mechanism.  
Inferring the definition of the molecular mechanism from cell and tissue is complicated by the scaling factors related to the structural organisation of the molecular motors in the three–dimensional lattice, the co–presence of different isoforms in the same muscle and even in the same muscle fibre and the possible confounding contribution of the other sarcomeric (cytoskeleton and regulatory) proteins. 
Even assuming that the tension cost is solely related to intrinsic properties of the motor isoform, the question remains about the role played by the differences in the mechanokinetic properties of the motor, as the force developed in a single motor interaction or the fraction of the ATP-ase cycle time each motor spends attached (the duty ratio) while working in situ in the half–sarcomere of the striated muscle.
In vitro, the definition of the emergent properties of the half–sarcomere, which cannot be studied with single molecule mechanics finer1994single, ishijima1996multiple, became recently accessible by exploiting a unidimensional synthetic nanomachine powered by myosin motors purified from the skeletal muscle pertici2018myosin.
The nanomachine allows the performance of the half–sarcomere, the generation of steady force and shortening, to be reproduced by an ensemble of pure myosin isoforms interacting with the actin filament without the confounding effects of other sarcomeric proteins and higher hierarchical levels of organisation of the muscle. 
In the nanomachine 8 HMM fragments extending from the functionalised surface of a micropipette carried on a three–way nanopositioner acting as a length transducer interact with an actin filament attached with the correct polarity to a bead trapped by a Dual Laser Optical Tweezers (DLOT) acting as a force transducer (Figure <ref>).


In solution with physiological ATP concentration, in which the two motors of each dimer act independently pertici2018myosin, myosin motors, after entering in contact with the actin filament, establish continuous interactions underpinning force development to a steady maximum value (equivalent to the force generated by the muscle in isometric contraction).
In the original design pertici2018myosin the system was operated either in position clamp (achieved using as feedback signal the position of the nanopositioner carrying the motor array (x), red branch in Figure <ref>), to reproduce the isometric contraction, or in force clamp (achieved using as feedback signal the position of the bead in the laser trap (x_bead), green branch in Figure <ref>), to reproduce isotonic contraction. 

A major limit of the nanomachine working in position clamp was the large trap compliance in series with the motor array, two order of magnitude larger than the native compliance in series with the half–sarcomere.
This makes each addiction–subtraction of force by individual motor attachment–detachment to induce substantial sliding undermining the condition of independent force generators of the motors in the native half–sarcomere. 
Consequently, in position clamp, the kinetics of the attached motors is influenced by the push–pull experienced when actin slides away–toward the bead for the addition–subtraction of the force contribution by a single motor (pertici2018myosin, Supplementary Figure 7). 

In the present experiments the system has been implemented to operate in length clamp (achieved using a feedback signal the difference between the position of the bead and that of the nanopositioner x_bead-x=L), blue branch in Figure <ref>.
In length clamp the sliding between the actin filament and the motor array caused by force generating interactions is eliminated because any movement of the bead is counteracted by the movement of the nanopositioner.
In this way the condition of the motors as independent force generators in the array is recovered and the rate of development of the steady isometric force as well as the force fluctuations superimposed on the steady force are direct expression of attachment/force–generation and detachment of the myosin motors. 
The data collected from either myosin isoform are used to feed a stochastic model providing a self–consistent estimate of all the relevant mechanokinetic parameters of the isometric performance of the motor ensemble: the force of a motor, f_0, the fraction of actin–attached motors, r, and the rate of transition through the attachment–detachment cycle, ϕ, without assumptions from cell mechanics and solution kinetics as in previous studies pertici2018myosin, pertici2020myosin, pertici2021muscle. 
The combined experimental and theoretical achievements reported in this paper set the stage for any future study on the emergent mechanokinetic properties of an ensemble of myosin molecules, either engineered or purified from mutant animal models or human biopsies.



§ RESULTS
 



 §.§ Estimate of the number of HMM molecules on the support available for the interaction with the actin filament.


The number of motors on the micropipette surface able to interact with the actin filament (N) is initially determined by measuring the number of mechanical rupture events when the motor array is brought to interact with the actin filament in ATP–free solution pertici2018myosin. 
Following the formation of rigor bonds between the HMM–coated support and the actin filament (panel 1 in Figure <ref> a), the HMM support is moved away from the actin filament, first by 1–2 in the direction orthogonal to the plane of the support, in order to raise a force from the trapped bead to the first bound HMM at an angle greater than 30 with the plane of the support, and then in the direction parallel to the plane, at constant velocity (50), to pull the motors away from the actin filament diagonally.
This allows the first bonded HMM to undergo a pulling force that is higher than the axial component shared among the other motors. 
In this way the myosin–actin bonds brake one at a time and the attached motors cannot bind back once detached. 
Moreover, following each detachment the force drops because the length of actin filament segment between the bead and the next attached motor is transiently increased. 
Thus an additional pull is necessary to get to the next rupture event, the occurrence of which will vary in time according to the distance between the two neighbouring motors.


With HMMs purified from soleus muscle the number of rupture events per interaction (Figure <ref>b) attains a saturating value of 7.9 ± 1.1 (n=8), with [HMM] used to coat the pipette of 0.2. 
A similar saturating value of rupture events, 8.1 ± 1.4 (n=8), is obtained for the HMM purified from psoas muscle with a [HMM] of 0.1 (Figure <ref>c). 
Notably, similar saturating values of [HMM] and number of rupture events (8.1 ± 1.2) were found for the psoas motors in the previous study in which an optical fibre etched to the same diameter was used as support pertici2018myosin. 
In 2 ATP each head of an HMM dimer works independently and thus the number of available motors is twice the number of HMM ruptures: N= 16 ± 2 
and 16 ± 3 
for the soleus and psoas respectively.



 §.§ Isometric force development by the nanomachine powered by slow and fast myosin motors.
 

The experiment starts in position clamp, because, for the system to operate in length clamp, it is necessary that first the feedback loop is closed by the establishment of actin–myosin interactions.
When an array of motors from the slow soleus muscle is brought to interact with a bead–tailed actin filament in solution with 2M ATP (Figure <ref>a), the establishment of continuous ATP–driven actin–myosin interactions causes the force (F, blue trace) to rise pulling on the actin filament, which in position clamp (HMM support position x = 0, red trace), slides in the shortening direction (Δ L, black trace, negative for shortening) due to the trap compliance (phase 1). 


A steady maximum force (F_0) of ∼12 is attained with a shortening of ∼55. 
The control is switched to length clamp in correspondence of the vertical dashed line separating phase 1 and 2.
The switch time is marked by the increase in noise of the force trace as a consequence of the reduction of the compliance in series with the motor system. 
In fact in length clamp the force change generated in each individual attachment and detachment is no longer dissipated in filament sliding against the large in series trap compliance.
A shortening of ∼500 completed within ∼700 is superimposed on the steady isometric force in correspondence of the second vertical dashed line to drop and keep the force at zero (phase 3). 
When actin filament sliding stops (third vertical dashed line) force starts to redevelop towards F_0 (phase 4) with just a minimum delay, indicating that the motor array was able to cope with the imposed 500 shortening maintaining continuous interactions under zero load. 
The extent of shortening minus the amount taken by the trap compliance, (500 - 55 =)  445, divided by the time passed from the imposition of the shortening to the start of force redevelopment (0.88) gives an estimate of the velocity of unloaded shortening (V_0) of 0.5. 
Force redevelopment in length clamp is much faster than the original force rise in position clamp and occurs in truly isometric conditions, as the movement of the bead due to trap compliance is counteracted by a corresponding movement of the nanopositioner in the lengthening direction (red trace, ∼55), that keeps Δ L= 0 (black trace).
Notably, the force redevelopment following a 500 release attains the same F_0 value as that attained during the original rise in position clamp thanks to the architecture of the machine, in which the dimension of the motor array remains constant independent of the amount of reciprocal sliding pertici2018myosin. 
F_0 from 33 records shows a Gaussian distribution with centre 10.5 (Figure <ref> b).
The rate of force redevelopment in length clamp only depends on the attachment/detachment kinetics of myosin motors in isometric conditions. 
Force redevelopment is roughly exponential, and its time course is quantified by the rise time t_r (the time from 10 % to 90 % of F_0). 
t_r estimated on the record (Figure <ref>c, black) obtained by averaging the blue 6 traces from as many experiments (grey) is 238 ±13.  
The time constant of the underlying exponential force rise of the soleus powered nanomachine, τ is (t_r/2.2=) 108 ±5, and the rate of force development, a is (1/τ=) 9.3 ±0.5.  
The sequence of events accompanying the interaction of the array of motors purified from psoas muscle with the actin filament is the same as for the soleus motors (Figure <ref> d).  
The Force develops in position clamp (phase 1), while the actin filament slides in the shortening direction due to the trap compliance. 
A steady isometric force F_0 (15.9), is attained with a shortening of 70. 
In the 47 records of the psoas HMM F_0 shows a Gaussian distribution with centre 17 (Figure <ref>e).  
Following the switch to length clamp, a rapid shortening of ∼500 is imposed so that the force drops to zero. 
The shortening in this case is just sufficient to drop the force to zero, given the much faster shortening velocity afforded by the fast motor array, so that, as soon as the actin filament sliding stops (third vertical dashed line), V_0, calculated by the extent of shortening minus the amount taken by the trap compliance, (500 - 70 =)  430, divided by the time passed from the imposition of shortening to the start of force redevelopment (0.22), is 1.95 (3.9 times larger than that of slow muscle). 
It must be considered, however, that V_0 in this case is somewhat underestimated, as most of the shortening occurs with force greater than zero. 
Force redevelopment in length clamp (phase 4) occurs with a rate that is not influenced by the trap compliance and thus is the expression of the kinetics responsible for the transition to the steady force F_0 by the fast isoform array. 
A t_r of 77 ±4 
is estimated on the record (black in Figure <ref>f) obtained by averaging the traces from 7 experiments (grey). 
The corresponding τ (= t_r/2.2) and a (= 1/τ) are 35.0 ±1.8 and 28.6 ±1.4 respectively. 

The - 3 upper frequency characterising the force rise f_c = 0.35/t_r is 4.5 ±0.2.

Two main points emerge from these measurements on the synthetic machine operating in length clamp conditions. 
The first is that the rate of force redevelopment, which only depends on the attachment/detachment kinetics of myosin motors in isometric conditions, is three times slower in the soleus powered nanomachine than in the psoas powered nanomachine.  
The second point is that the force fluctuations around the average value displayed by the force record at the steady state are stemming from individual attachment/detachment events. 
Both pieces of information can be used to feed the stochastic model described in the next section. 




 §.§ Modelling the mechanical output of the nanomachine powered by the slow and the fast isoform ensembles.
 

In the stochastic model each motor exists in three possible states (or motor configurations): one detached state and two different force–generating attached states.
Fitting the experimental records allows a self–consistent estimate of all the relevant mechanokinetic parameters of the nanomachine including the force exerted by a single myosin motor and the average number of attached motors in the stationary state, without assumptions from cell and solution kinetic studies. 

As detailed in the Introduction, the implementation of the length clamp mode allows to recover the condition of the motors as independent force generators in the array. 
Therefore we consider an ensemble of N independent ATP–fuelled molecular motors interacting with an actin filament in isometric conditions. 
Each motor can be found in the detached state D, in the attached low force–generating state A_1, or in the attached high force–generating state A_2.
The corresponding  kinetic scheme, which exemplifies the possible transitions between distinct allowed motor states is:

    D <=>[ k_1 ][ k_-1 ] A_1 <=>C[ k_2  ][ k_-2 ]A_2 ->[ k_3  ][]D

The rate constants k_j, j∈{1,-1,2,-2,3} represent the probability per unit of time for the reaction j to occur, and are expressed in s^-1.
The state of the system at any time t is characterised by the vector n(t)=(n_D(t),n_1(t),n_2(t)) whose entries specify the number of molecular motors in each of the considered configurations. 
Specifically, n_D stands for the number of motors in the state D, n_1 is the number of motors in the state A_1 and n_2 denotes the number of motors in the state A_2. 
The system admits the obvious conservation law N= n_D+ n_1+ n_2 where N stands for the total number of motors in any of the considered states. 
Accounting for the above relation enables one to employ just two scalar (discrete) entries to photograph the state of the system, namely n(t)= (n_1(t),n_2(t)). 
Under the Markov hypothesis, the stochastic dynamics of the scrutinised system is ruled by a master equation which sets the evolution of the probability P(n,t) of finding the system in the state specified by the vector n at time t. 
The master equation accounts for the balance of opposing contributions: on the one side the transitions towards the reference state (the associated terms bearing a plus sign). 
On the other, the transitions from the reference state (terms with a minus). 
From the master equation one can readily derive the mean field equations that govern the deterministic dynamics for the continuous concentrations y and z of the molecular motors in states A_1 and A_2, as detailed in Methods. Further, a self-consistent elimination of the variable y can be performed to yield a simpler description of the examined process in terms of the variable z (the derivation is given in Methods, see also Supplementary Figure 1): 

    z(t)= b/a(1- e^-at)= k_1/k_1+ Gk_2/k_2+k_-2+k_3(1- e^-[k_-2+k_3+ k_2 (k_1- k_-2)/k_1+k_-1+k_2]t)

where a and b are positive quantities, self-consistently defined by the latter equality and 
G= (k_-1(k_-2+k_3)+k_2k_3)/(k_2+k_-2+k_3). 
We define the duty ratio r as the fraction of attached motors (or the fraction of the ATPase cycle time a motor spends attached). 
Assuming that for mammalian muscle myosin at temperature T ≃24 motors are prevalently found in the state A_2, a straightforward analysis outlined in the Methods yields r ≃ z^* = k_1/k_1+G.
Here, z^* stands for the equilibrium concentration of A_2 motors. 
Through parameter a, we have also access to a closed estimate for the characteristic time scale of the exponential evolution of z. 
Let us notice that a is the inverse of the time constant of the development of the steady force, τ as defined in the experiment, hence a=2.2/t_r. 
According to this simplified scheme, the effective rate of ATP consumption can be estimated as the flux of motors through the cycle per unit time. 
This equals to the rate of motors in A_2 detaching from the actin, in formula ϕ= z^* (a-b). 
 
Starting from these premises we can characterise the average force exerted by a small ensemble of molecular motors in isometric conditions. 
This is obtained by combining the contributions from each individual motor of the collection: motors in the configuration A_1, each exerting a force f_1 and motors in A_2, each exerting a force f_2.
In the nanomachine, motors have a random orientation with respect to the actin filament.
As a consequence, we assume that the intensity of the force exerted by a motor depends on the binding angle θ, as measured from the correct orientation. 
Depending on the specific orientation of the molecule, the force progressively decreases up to a minimum value that is 0.1 f_0 ishijima1996multiple.
In particular, the force of a single motor can change within a bounded interval: the largest value of the force f_0 is exerted when the motor orientation is correct (corresponding to the in situ orientation). 
Then, in accordance with pertici2020myosin (see Supplementary Figure 2) we postulate that the exerted force f_1 is a random variable, uniformly distributed within the interval ℐ_1=[-f_0, f_0]. 
Similarly, the force f_2 is randomly extracted from the interval ℐ_2=[f_0/10, f_0 ]. 
The mean field average force exerted by the ensemble of myosin motors, at any time t, is F(t) = n_1(t)f_1+n_2(t)f_2 = n_2(t)f_2, given that f_1=0 since the interval ℐ_1 is symmetric with respect to zero. 
In the stationary state,  F(t) converges to the asymptotic plateau F_0, and thus:

    r f_0  = 1/N20/11 F_0

where use has been made of the conditions f2 = (11/20)f_0 and z^* ≃ r. 

The experimental value of the stationary force exerted by a pool of N motors acting in the state A_2 solely constrains the product of f_0 and r. 
That is, on deterministic means, we cannot access a direct estimate of the  maximum force exerted by an individual motor (f_0) and the associated duty ratio (r), but just constraint this latter pair to fall on a hyperbole, set by  F_0. 
Accounting for the fluctuations superimposed on F_0, and thus by properly gauging the stochastic component of the dynamics, enables us to resolve the above degeneracy. 

To this end we consider the dynamics of the system at finite N, so as to account for the role played by finite size fluctuations. 
To quantify the contribution as stemming from the intimate graininess of the investigated system, we ought to solve the master equation, focusing in particular on the stationary state probability distribution P^st. 
As discussed in Methods, we are in a position to solve exactly the stochastic model in its stationary state, and thus get a closed expression for P^st, as function of the parameters of the model. This knowledge can be used to compute P(F) the distribution of the exerted force F (see Methods). Remark that P(F) is ultimately shaped by the kinetic constants of the model (namely, k_1,k_-1, k_2, k_-2, k_3) and also reflects the maximum force f_0, as applied by individual motors. Recall that N is directly determined (Figure <ref>). 
We can therefore construct an inverse procedure to recover information on the underlying parameters, by confronting the predicted distribution of the force P(F) to the homologous curve recorded experimentally. 
In particular we will prove that, by exploiting the information stemming from the fluctuations, it is eventually possible to unambiguously determine both f_0 and r (see Supplementary Table 2, Supplementary Figure 8 and Supplementary Figure 9).
The relevant steps that define the envisaged fitting strategy are summarised in the following and made explicit in the Methods:

	
  * The first step amounts to analyse the time evolution of the force in its mean field approximation: the asymptotic force F_0 and the time scale a, as defined above, are extracted via a direct - two parameters - fit that exploits expression (<ref>).
	
  * We turn to study the distribution of the fluctuation of the force around the equilibrium value. To this end we make use of P^st. 
	
  * From P^st we extract the N+1 marginal probabilities ρ_k, namely the probabilities to find k ≤ N motors in the force-generating configuration A_2. This is achieved by  summing over n_1= 0,… , N the stationary probability distribution P^st. 
	
  * We then make use of the marginal probabilities (ρ_0, ρ_1, ρ_2, … , ρ_N) to weight the probability distributions Π_k (f) of the force exerted by a set of k motors.  These latter are computed as generalised Irwing-Hall distributions for independent and identically distributed random variables f drawn from the considered interval I_2. The distribution of the force is hence estimated as P(F)= ∑_k=0^N ρ_k Π_k (f). 
	
  * For fixed size N (previously estimated by the counting of rupture events in ATP–free solution, see also Supplementary Figure 10 where the possibility to modulate N is accounted for) we adjust the kinetic rate constants k_1,k_-1, k_2, k_-2, k_3, so as to minimise the root mean square distance between the recorded distribution and its analytic estimate. The best fit values are used to compute the parameter b and thus determine the sought estimates for r and f_0, as well as the rate of motors detaching from the actin, i.e. ϕ=z*(a-b). 	

The above fitting strategy is successfully challenged against synthetic data as discussed in Methods. 
Then we proceed by applying the validated procedure to the experimental data collected with the nanomachine powered by either HMM isoforms. 
As mentioned, the number of available molecular motors (N = 16) estimated from number of ruptures in rigor for both isoforms (Figure <ref>), is assumed as the reference value in the following, unless otherwise specified. 
We interpolate the distribution of the fluctuations as recorded experimentally, given the analytical solution obtained above. 
A representative example of the fitting outcome for the soleus HMM ensemble is reported in the Supplementary Figure 11.
The estimated values for f_0, r and ϕ, for both the psoas and the soleus HMM, are listed in the Supplementary Note 1 (Supplementary Table 3 and Supplementary Table 4 respectively), and their respective average values are reported in Table <ref>. 


In Figure <ref> the results of the analysis are plotted in the parameters plane (f_0,r) (symbols and lines refer to different isoforms according to the colour: blue for psoas, red for soleus; different tones identify different experiments).
The solid lines highlight the ensemble of distinct – though equivalent – solutions ensuing from the average force profile.
By accounting for the fluctuations one breaks the degeneracy inherent to the system when analysed in its mean–field version, getting just one pair (f_0,r) (identified by the symbol) compatible with each individual experimental curve. 

One can relax the constraint N= 16 obtained from the rigor experiments (Figure <ref>) and scan the range of N that yields convergence of the optimisation algorithm, for the imposed level of accuracy.
The results of the analysis for the soleus isoform is reported in the Supplementary Note 1 (Supplementary Figure 12), where the estimated f_0 is plotted against N. 
The histogram computed from the collection of fitted parameters can be conceptualised as an indirect imprint of the degree of experimental variability as associated to f_0 and N.






§ DISCUSSION


We use the one–dimensional synthetic nanomachine described in pertici2018myosin 
to define the isometric mechanical output of an array of 16 myosin motors purified from either fast (psoas), or slow (soleus), muscle of the rabbit.
To eliminate the large trap compliance and recover the condition for the motors to operate as independent force generators as in the native half–sarcomere, once the interaction is established the system control is switched from position clamp to length clamp.
The array of 16 motors in physiological ATP concentration (2) at 24 exhibits a steady isometric force that in the fast isoform is 17, and in the slow isoform is 10.5.  
The finding that the force exerted by the same number of motors is 1.6–fold larger in the fast isoform disagrees with the most common finding in muscles and muscle fibres that the isometric force normalised for the cross sectional area of the fibre T_0, is either similar or at max 1.5–fold larger in the fast isoform barclay1993energetics, barclay2010efficiency, wendt1973energy, wendt1974energy, bottinelli1994unloaded, reggiani1997chemo, potma1994effects, reggiani2000sarcomeric, he2000atp.
Notably in skinned fibres from the same rabbit muscles from which the nanomachine motor proteins are purified, T_0 in psoas at 25 has been found 317 ±14, 1.9–fold larger than T_0 in soleus, 165 ±12 caremani2022force. 

Recording the development of the steady isometric force in length clamp eliminates the contamination of the large trap compliance, showing a roughly exponential time course characterised by the parameter t_r that is 238 for the slow isoform and 77 for the fast isoform. 
Thus the rise of the force to the maximum steady value takes a 3–fold longer time for the slow isoform than for the fast isoform.
How this emergent property of the motor ensemble relates to the corresponding event in situ and how it is affected by the different isoforms has been tested by comparing the nanomachine output with that of  Ca^2+ -activated skinned fibres, from the same rabbit muscles from which the motor proteins were purified. 
According to the sarcomere-level mechanics for skinned fibres  developed in our laboratory, the compliance of the attachments of the skinned fibre segment to the transducer levers is negligible (see the Supplementary Note 2, Supplementary Figure 13). 
Under these conditions, the force redevelopment following a fast shortening able to drop the isometric force to zero is characterised by a t_r = 265 ±15 in soleus fibres and 62 ±5 in psoas fibres. Thus, the time course of force development recorded by the nanomachine in length clamp and its modulation by the two isoforms, are in quite satisfactory agreement with those recorded at the cell level.
The corresponding rates of force development (a) by the nanomachine are 28.6 for the fast isoform and 9.3 for the slow isoform. 
Considering that in length clamp a is direct expression of the sum of the effective rate constant of attachment/force–generation and the effective rate constant of detachment of the myosin motors, we conclude that the interaction kinetics in isometric condition is 3–fold higher in the fast isoform than in the slow isoform. 
The attachment/detachment kinetics is expected to increase if the load on the motor ensemble is reduced, due to the strain-dependent increase in rate constant of detachment, which underpins the maximum velocity of shortening (V_0) attained under zero load.  
V_0 estimated by the time taken by the ensemble to redevelop force following a release able to drop the isometric force to zero (Figure <ref>a,d) is 0.5 and 1.95 in the slow and fast HMM respectively, showing a V_0 ∼ 4–fold larger in the fast isoform. 
Thus, the isoform-dependent increase of V_0 is 33 % larger than the increase in a and even larger if one considers that V_0 of the fast isoform is underestimated by the proportionally larger fraction of time spent for the force to drop to zero following the release (compare the records in Figure <ref>a,d). 
This suggests that the fast isoform exhibits a specifically larger strain dependence of the detachment rate constant. 
 
The rate of development of the isometric steady force and the force fluctuations superimposed on the steady force in length clamp have been exploited to implement a stochastic three-state model which is able to fit the experimental responses, allowing self-consistent estimates of all the relevant mechanokinetic parameters underlying the isometric performance of the motor ensemble: f_0, the force of a single correctly oriented motor, r, the fraction of attached motors, and ϕ, the rate of transition through the attachment/detachment cycle (Table <ref> in the Results). 
f_0 of the fast isoform (6.8 ±1.0) is 2.8–fold larger than f_0 of the slow isoform (2.4 ±0.4), while the ensemble force F_0 is only 1.6 times larger (Figure <ref>b,e). 
This is in a great part explained by the different fraction of attached motors r, which in the fast isoform (0.32 ± 0.02) is 0.64 that of the slow isoform (0.50 ± 0.03). 
The corresponding number of attached motors (Nr) is ∼ 5 and ∼ 8 for the fast and the slow isoform respectively. 
The average force of a single randomly oriented motor (0.55 f_0) is 3.7 for the fast isoform and 1.3 for the slow isoform, from which the predicted ensemble force is (3.7× 5 =)  18.5 and (1.3 × 8 =)  10.4 respectively. 
These values are in quite good agreement with the observed values: 17 ±3 (σ) for the fast isoform and 10.5 ±1.8 (σ) for the slow isoform.

The model predicts a rate of transition of a motor through the interaction cycle, and thus a frequency of ATP splitting per motor (ϕ) 2.6 times higher for the fast isoform (6.0) than for the slow isoform (2.3) (Table <ref>). 
The value of ϕ of the slow isoform array is in a remarkably good agreement with that estimated on the slow muscle of mouse and rat (2.3–2.9, Supplementary Table 1). On the other hand, ϕ for the fast isoform array is less than half of the one estimated in the fast muscle of the same animals (12.4–13.3, Supplementary Table 1). The same discrepancy for the isoform-dependent increase in ϕ is found between the model prediction and the skinned fibre experiments (Supplementary Table 1, bottinelli1994unloaded, reggiani1997chemo, potma1994effects, reggiani2000sarcomeric, he2000atp).
However, it must be noted that (i) the absolute values of ϕ in skinned fibres is 10–fold smaller than the one in the muscle for both slow and fast myosin isoforms bottinelli1994unloaded, reggiani1997chemo, potma1994effects, reggiani2000sarcomeric; (ii) the difference can only in minor part be explained by the different temperature of the experiments (21–27 for the muscle and 12 for the skinned fibres), taking into account that the Q_10 of ϕ is ≤ 2.5 in either preparation barclay2010efficiency, he2000atp, stienen1996myofibrillar. 
ϕ predicted by the model for the output of the  fast isoform nanomachine is 2.5–fold larger than that predicted for the slow isoform, but still 2–fold smaller than that indicated by the energy rate measured in the fast muscle. 
Thus the 5–fold larger ϕ of the fast isoform with respect to the slow isoform found in muscle is only partly explained by higher rate constants of transitions through the conventional attachment/force generation and detachment cycle operating in isometric conditions and recorded by the nanomachine force fluctuations.  
The actin–activated myosin ATPase activity in solution is 2.5 times larger in fast than in slow muscle barany1965myosin, which can be accounted for by a higher rate of ADP release (which is followed by a fast ATP binding and detachment, step (c)–(d) in Figure <ref>, iorga2007slow) and/or a higher rate of the hydrolysis step (d)–(e), and/or a higher rate of actin attachment (step (a)–(b)).
In isometric contraction at physiological ATP concentration, ADP release is the rate–limiting step for detachment and is 10–fold slower in slow myosin than in fast myosin iorga2007slow and this may per se explain the finding that during steady isometric force generation the duty ratio of the fast myosin nanomachine is lower than that of the slow myosin nanomachine. 
However, it must be taken into account that under isometric conditions (or high load) the transitions through the different force-generating states of the motor (step (a)–(b)/(c) in Figure <ref>, huxley1971proposed, piazzesi1995cross) slow down due to the strain dependence of the transition rate and thus the subsequent conformation-dependent release of ADP also gets slower (5, caremani2015force). 
As far as the difference in ϕ  between slow and fast myosin ensembles in isometric contraction, the finding that the force of fast myosin is 2.5–fold higher should suggests that the equilibrium distribution between different force–generating states is shifted toward the end of the working stroke in the fast myosin, in this way explaining a larger flux through the detachment step and thus the reduction in the duty ratio and the increase in ϕ with respect to the slow myosin (Table <ref>). 
However, it must be considered that the stiffness of the myosin motor, determined in situ with fast sarcomere–level mechanics applied to skinned fibres from rabbit muscle, is larger in the fast muscle in proportion to the motor force, so that the extent of the force–generating structural change is the same in either fast or slow myosin motor percario2018mechanical. 

In conclusion, the 2.5–fold larger isometric ϕ of the fast myosin isoform found with the analysis of force fluctuations is accounted for by an intrinsic faster rate of the relevant kinetic steps of the fast myosin isoform which underpins a 2.5–fold larger ATPase rate in solution barany1965myosin. 
Instead, the 5–fold larger isometric ϕ of the fast isoform reported in the literature Supplementary Table 1, exceeds by a factor of 2 the one recorded by the nanomachine force fluctuations at 25 and could be explained by a further kinetic adaptation of fast myosin isoform hypothesising that, also in isometric conditions, a futile faster actin-activated ATPase cycle is present.
In terms of the kinetic scheme in Ref. zeng2004dynamics, this cycle implies the working stroke transition to occur in the motor undergoing weak actin-binding interactions and does not imply strong/force-generating attachment unless the load is reduced and the muscle shortens.   

A comparison of the parameters estimated in this work with those obtained in previous nanomechanical approaches is possible for the fast isoform purified from rabbit psoas investigated by Yanagida’s group ishijima1996multiple through the microneedle manipulation technique. 
In close-to-isometric conditions, obtained through a stiff microneedle, both the force of the motor (5.9) and the fraction of actin-attached motors (0.36) estimated in that work are in exceptional good agreement with the values calculated here from the output of the nanomachine. 
A peculiar difference that makes our nanomachine unique is the possibility to define the performances emerging from the array arrangement of the motors in the half-sarcomere, as the force - velocity relation and the maximum power output. 
The novelty of the present nanomachine application in relation to the previous ones pertici2018myosin, pertici2020myosin, pertici2021muscle, is the interpretation of the output of the motor ensemble and of the isoform-dependent differences on the basis of the mechanokinetic molecular properties of either isoform is self-consistent way without any assumptions from cell mechanics and solution kinetics. 
The combined experimental and theoretical achievements in this paper set the stage for any future studies on the emergent mechanokinetic properties of the half-sarcomere like arrangement of any myosin motors, either engineered or purified from mutant animal models or human biopsies.






§ METHODS
 



 §.§ Preparation of Proteins


Adult male rabbits (New Zealand white strain), provided by Envigo, were housed at Centro di servizi per la Stabulazione Animali da Laboratorio (CeSAL, University of Florence), under controlled conditions of temperature (20 ± 1) and humidity (55 ± 10)%, and were euthanized by injection of an overdose of sodium pentobarbitone (150) in the marginal ear vein, in accordance with the Italian regulation on animal experimentation (Authorisation 956/2015-PR) in compliance with Decreto Legislativo 26/2014 and EU directive 2010/63. 
Three rabbits were used for the experiments. 
HMM fragments of myosin were purified from rabbit soleus and psoas muscles as reported previously in pertici2018myosin, pertici2020myosin. 
The functionality of the purified motors was always preliminarily checked with in vitro motility assay. 
Actin was prepared from leg muscles of the rabbits according to pardee1982mechanism, and polymerised F-actin was fluorescently labelled by incubating it overnight at 4 with an excess of phalloidin-tetramethyl rhodamine isothiocyanate kron199133.
For the mechanical measurements in the nanomachine, the correct polarity of the actin filament was pursued by attaching the + end of the filament to a polystyrene bead (3 diameter) (Bead-Tailed Actin, BTA, suzuki1996preparation) with either the  Ca^2+ -sensitive capping protein gelsolin pertici2018myosin or the  Ca^2+  insensitive gelsolin fragment TL40 (Hypermol, Germany) pertici2020myosin, pertici2021muscle). 




 §.§ Mechanical experiments


The mechanical apparatus, described in detail in pertici2018myosin, is depicted in Figure <ref>. HMM fragments of myosin were deposited randomly on the lateral surface of a glass pipette pulled to a final diameter of ∼ 3–4 and functionalised with nitrocellulose 1 (w/v). 
The glass pipette was mounted in the flow chamber carried on a three–way piezoelectric nanopositioner (nano-PDQ375, Mad City Lab, Madison WI, USA) that acts as a displacement transducer, and was brought to interact with a BTA trapped in the focus of a Dual Laser Optical Tweezers (DLOT) that acts as a force transducer. 
The DLOT system has a dynamic range for both force (0-200pN, resolution 0.3pN) and displacement (0-75, resolution 1.1) adequate for the measuring the output of the nanomachine. 
The buffer solutions used for all the experiments are already reported in pertici2018myosin 
and contained physiological concentrations of ATP (2) unless differently specified. 
0.5 methylcellulose was added to the running buffer in order to inhibit the lateral diffusion of F-actin uyeda1990myosin and minimise the probability of loss of acto–myosin interaction. 
The concentration of HMM from soleus and psoas muscle used for the experiments was defined by the concentration at which the number of rupture events in rigor attained a saturating value.

The mechanical apparatus, as already reported in pertici2018myosin, can be operated either in position clamp (Figure <ref>, red branch), when the feedback signal is the position of the nanopositioner carrying the motor array (x), or in force clamp (green branch), when the feedback signal is the force (F), calculated as the product of the stiffness of the trap (e) times the change in position of the bead in the laser trap (x_bead). 
Recording of the nanomachine performance in true isometric condition, however, cannot be achieved in position clamp, due to the large trap compliance (∼4), which implies both several tens of nanometres movement to develop the maximum steady force and blunting of the force of individual attachment–detachment events (Supplementary Figure 7 in pertici2018myosin). 
To eliminate the trap compliance the system has been implemented with a length clamp (blue branch in Figure <ref>), which uses as a feedback signal the change in distance (L) between the position of the actin attached bead in the laser trap (x_bead) and that of the nanopositioner (x), so that the movement of the bead with the force change is counteracted by the movement of the nanopositioner. 
In this way the effective trap compliance is reduced to 0.2. 

In length clamp the frequency response of the system is reduced by the propagation time of the mechanical signal through the loop from the force transducer to the nanopositioner, which also includes the array of actin attached myosin motors. 
The power density spectrum (PDS) of the system, measured with sinusoidal oscillations at different frequencies, changes depending on the selected feedback mode: in position clamp the PDS shows an upper -3 frequency (or corner frequency f_c) of 59 (Figure <ref>, red); in length clamp, when the feedback loop is closed with the array of actin–attached myosin motors in rigor, f_c decreases to 32 with HMM from fast muscle (violet) and to 17 with HMM from slow muscle (cyan). 


The mass of the system (m) is the same with either isoform array thus the different corner frequency between the two nanomachines should almost in part depend on the different stiffness of the two arrays in rigor. 

The architecture of the machine (with the length of the motor array much shorter than the length of the overlapping actin filament) implies that, for a given HMM concentration, the measured number of rupture events does not significantly change from experiment to experiment, therefore there is no need to normalise the mechanical response obtained in different experiments at physiological [ATP] by the actin–filament length. 
All the experiments were conducted at room temperature (24).





 §.§ Statistics and reproducibility

Data are expressed as mean ± standard deviation unless otherwise stated. The number of replicates is defined in the text and in the figure legends.



 §.§ Stochastic model: on the governing master equation


The master equation can be cast in the general form:

    ∂ P(n,t)/∂ t= ∑_n'n[ T(n|n') P(n',t)- T(n'|n)P(n,t) ]

where  T(n'|n) represent the transition rates from the state n to a new state n', compatible with the former. 
In the following, to identify the arrival/departure state n' we solely highlight the individual component of the vector n that changes due to the considered reaction. 
The explicit expression for the transition rates that originates from the chemical equations (<ref>) is given in the annexed Supplementary Note 1. 

In the Supplementary Note 1 are also presented the details of the numerical simulation of a single stochastic orbit of the considered dynamics, obtained via the celebrated Gillespie algorithm gillespie1976general, gillespie1977exact. 
In Supplementary Figure 3 it is shown the time evolution of the (discrete) concentration of molecular motors in each configuration, while in Supplementary Figure 4 the results of the stochastic simulations for the probability distributions of the fractions of motors in the force-generating configurations are compared with the theoretical prediction. 





 §.§ The deterministic limit


From the master equation one can readily derive the mean field equations that governs the deterministic dynamics for the continuous concentrations of the molecular motors in configurations A_1 and A_2:

    dy/dt    = k_1 -(k_1+k_-1+k_2) y -(k_1-k_-2) z 
    
    dz/dt    = k_2 y- (k_-2+k_3) z

where y and z identify the averaged fraction of the molecular motors in states A_1 and A_2, respectively. 
Equations (<ref>) can be studied at equilibrium, yielding the fixed point solutions:  

    y^*=   (k_1/k_1+G) k_-2+k_3/k_2+k_-2+k_3
       
    
         z^*=   (k_1/k_1+G) k_2/k_2+k_-2+k_3

where:

    G= k_-1(k_-2+k_3)+k_2k_3/k_2+k_-2+k_3
 
We define the duty ratio r as the fraction of attached motors (or the fraction of the ATPase cycle time a motor spends attached). 
It can be computed as:

    r= y^*+z^*= k_1/k_1+G .

The temporal evolution of the mean field concentrations of motors in different configurations, for a suitable choice of the kinetics parameters is shown in the Supplementary Figure 2, in Supplementary Note 1. 
A straightforward calculation can be performed to show that z^* ≃ r=k_1/k_1+G and y^* ≪ 1, when k_-2/k_2, k_3/k_2 ≪ 1. 
In practical terms, under this operating assumption, which for the mammalian muscle myosin under consideration is approached at temperature T ≃24, motors are solely found in state A_2. 
As discussed earlier, this is the relevant setting for the specific case study at hand. 

Equations (<ref>) can be drastically simplified by performing a self-consistent elimination of the variable y.  To this end we set dy/dt=0 in the first of equations (<ref>) to eventually express y as a function of z. This procedure is customarily invoked to carry out the so called adiabatic elimination, which proves correct when there is a clear separation of time scales between co–evolving variables.  Although this is not a priori the case for the system at hand, we will postulate the validity of the aforementioned condition and operate with the ensuing approximation that, as we shall prove, will materialise in an accurate interpretative framework. Further details can be found in the Supplementary Note 1 (see Supplementary Figure 1).
Plugging the expression for y as a function of z into the second of equations (<ref>) and solving the ensuing differential equation readily yields:

    dz/dt = k_1 k_2/k_1+k_-1+k_2-z [k_-2+k_3+ k_2 (k_1- k_-2)/k_1+k_-1+k_2] ≡ b-a z

which immediately yields solution (<ref>), as reported in the Results.
From equation (<ref>) we can write z^*= b/a and this latter condition matches the homologous estimate derived from the original two dimensional model and reported in equations (<ref>). 




 §.§ Exact solution of the stochastic problem


At first, we remark that the probability distribution P(n;t)≡ P(n_1,n_2;t) can be written as a vector P(t) of dimension (N+1)× (N+1). 
This latter returns the probability at time t, of finding the system in the state characterised by n_1 motors in configuration A_1 and n_2 motors in configuration A_2. 
Here, n_1 and n_2 can in principle assume every integer values in the range [0,N], i.e. a total of N+1 values each. 
Observe however that the populations of motors in the configurations A_1 and A_2, must satisfy the obvious constraint that reflects the conservation law, i.e. n_1+n_2≤ N. 
A simple way to express the condition above is to consider that for each possible value of n_1, n_2 can assume values in the range [0,N-n_1]. 
This readily implies that the total number of possible states for the system is identically equal to M= (N+1)(N+2)/2.  
The number of allowed states are hence smaller than what anticipated above. 
Indeed the non trivial entries of P(t) are M=(N+1)(N+2)/2. 
We will consequently focus on the non trivial elements of vector P(t) which we shall denote as P_m(t) with m=1, …, M. 
For the relevant case of N=16 molecular motors, instead of (N+1)× (N+1)=289 configurations we only have M=153 possible states that can be eventually visited by the system, and that we explicitly list in the Supplementary Note 1. 
As we shall also discuss in the same Supplementary Note 1, the stationary probability distribution P^st defines the kernel of a M × M matrix ℚ and can be hence computed as the eigenvector of ℚ relative to the null eigenvalue. 
The entries of the matrix ℚ can be computed from the transition rates of the underlying master equation, as highlighted in the Supplementary Note 1. 

The marginal probability ρ_k to find k ≤ N motors in A_2 can be extracted from the stationary probability distribution P^st, the stationary solution of the master equation.
This is done by summing the elements of P^st that refer to the selected k, and that account for all possible partitioning of the remaining N-k motors among configurations D and A_1. 
The knowledge of the stationary probabilities (ρ_0,ρ_1,ρ_2,...,ρ_N) opens up the perspective to calculate the stationary state distribution of the applied force F, an important asset when aiming at a refined fitting scheme which meticulously accounts for the role played by fluctuations.

To work along these lines we shall assume that the contribution to the force (including fluctuations) of the motors in the state  A_1 is always negligible. 
This assumption is motivated by the fact that, for the experimental setting here explored, only a tiny fraction of motors is found to populate state A_1, at any time t. 
In the Supplementary Note 1, we will however relax this working assumption so as to provide a rigorous theoretical framework that extends to account for the relevant setting where the population of A_1 motors is instead significant in size. 

Let us focus on k ≤ N distinct motors in state A_2. 
As postulated earlier, each motor can exert a constant random force f, uniformly spanning the assigned interval ℐ_2. 
For each choice of k, we can compute the distribution of the forces Π_k(f) applied by the selected k motors, by combining independent and identically uniformly distributed random variables drawn for the interval of pertinence ℐ_2 (see the Supplementary Note 1 for further technical information, Supplementary Figure 6 and Supplementary Figure 7). 
As stated in the Results, functions Π_k(f) need to be combined together with proper weighting factors that reflect the stationary probability ρ_k of having exactly k motors in the force-generating state A_2, namely P(F)= ∑_k=0^N ρ_k Π_k (f).

The relevant steps of the fitting strategy are listed below.
We begin by focusing on the average force profile, hence disregard the impact of finite size fluctuations. 
As mentioned above, the time evolution of the recorded force can be approximated by an effective, two–parameters model (see Supplementary Figure 5 in the Supplementary Note 1).  
The latter parameters – respectively denoted F_0 and a – can be estimated via a direct fit.
Having accessed to preliminary estimated values for the average force at the stationary plateau F_0 and for the rate of isometric force development a, one can then set forth to characterise the other kinetic parameters by analysing the distribution of the fluctuations of the force around the asymptotic plateau. 
To this end we note that f_0, one of the unknown of the model, can be written as: 

    f_0=20/11F_0/Na/b

where a is constrained to the value determined above while b=k_1 k_2 / (k_1+k_-1 +k_2) as defined by equation (<ref>). 

Armed with the above knowledge, we can proceed further by comparing the probability density function of the force fluctuations as obtained analytically to the homologous histogram computed from the stochastic simulations. 
The former is adjusted to the latter by modulating the free parameters k_1, k_-1, k_2, k_-2 and k_3, for a fixed choice of N. 
As discussed in the Supplementary Note 1, testing the method against synthetic data generated in silico enables us to conclude that parameters f_0 and r=k_1/(k_1+G) can be correctly estimated, following the above fitting scheme (see Supplementary Table 2). 
Also the estimated b and a (recomputed from the best fitted values for the kinetic constants) are pretty close to their nominal values as imposed in the simulations. 
Remarkably ϕ, the rate of motors completing the interacting cycle with the actin, is also correctly recovered. 
A graphic comparison between estimated and exact parameters (i.e. those employed in the inspected simulations) is also shown in Supplementary Figure 8 of the Supplementary Note 1. 
Notice however that multiple combinations of the parameters k_1, k_-1, k_2, k_-2 and k_3 exists that yields the same fitted profile (with almost identical estimates for the relevant quantities f_0, r, a and b). 

While the kinetics of the scrutinised model cannot be solved unequivocally, we are in a position to accurately determine crucial parameters – as e.g. the maximum force exerted by a single motor and the associated duty ratio of the ensemble –  which proved elusive under the deterministic viewpoint, as can be appreciated in Supplementary Figure 9 of the Supplementary note 1. 

The above analysis refers to a fixed value of N, the size of the system that we assumed (from the experiment results shown in Figure <ref>) N=16. 
In principle the correct value of N is not a priori known. 
To overcome this intrinsic limitation, one could repeat the  analysis by varying N and recording the parameter estimated as follows the fitting scheme. 
Here, we will consider the simplified setting where a and b are frozen to the values determined for N=16 (so that z^* stays unchanged). 
This is implemented by removing two parameters from the pool of quantities to be fitted. 
Specifically k_-1 and k_3 are constrained to match two constitutive relations, that involve k_1, k_2 and k_-2, in addition to a and b. 
The parameters to be fitted are hence k_1, k_2 and k_-2, while k_-1, k_3 and f_0 can be self–consistently determined from the their best fit values.
Notice that f_0 is expected to change as a function of N as specified by relation (<ref>). 
The result of the analysis are reported in Supplementary Figure 10 in the Supplementary Note 1: the fitting procedure converges (with the requested limit of precision) only over a finite range of values of N, centred around the value adopted when performing the simulations. 
This observation implies in turn that we are in a position to obtain a reasonable estimate for the interval of pertinence of N, as follows the procedure outlined above.   

The introduced theoretical framework and the ensuing fitting strategy, thoroughly validated against synthetic data, can be hence applied to the analysis of the experimental data so to yield a self–consistent estimate of the underlying mechanokinetic parameters. 
For a detailed validation of the fitting scheme against synthetically generated data refer to the Supplementary Note 1. 
 

Data collection and analysis: A custom built program written in LabVIEW (National Instruments) was used for signal recording. Data analysis was carried out using LabVIEW (National Instruments) and MATLAB (MathWorks) dedicated scripts, and Origin 2018 (OriginLab Corp., Northampton, MA, USA) and Igor Pro 8 (WaveMetrics, Portlan, OR, USA) software.


Data availability: The source data for all figures and tables are provided as Supplementary Data 1. All remaining data will be available from the corresponding authors upon reasonable request.







Acknowledgments

We thank Gabriella Piazzesi for critical evaluation of this work. We thank the staff of the mechanical workshop of the Department of Physics and Astronomy (University of Florence) for mechanical engineering support. This work was supported by Fondazione Cassa di Risparmio di Firenze (2020.1582 (Italy), the Italian Ministry of Education, Universities and Research (PRIN2020, CUP: B55F22000540001), the European Joint Program on Rare Diseases 2019 (PredACTINg, EJPRD19–033), and the Next Generation EU programme [DM 1557 11.10.2022] in the context of the National Recovery and Resilience Plan, Investment PE8–Project Age–It: “Ageing Well in an Ageing Society", Italy.



Author contributions

PB, VL and DF designed the research. IP, PB, ML, MC, MM, IM performed the experiments and analysed the data. VB and DF developed the stochastic model.  VL, DF, VB, PB and MR wrote the paper or revised it critically for important intellectual content. All authors participated in discussions on this work and approved the final version of the manuscript.


Competing interests: The authors declare no competing interests.



[figure]labelfont=bf,labelformat=default,labelsep=period,name=Supplementary Figure	
[table]labelfont=bf,labelformat=default,labelsep=period,name=Supplementary Table







§ SUPPLEMENTARY INFORMATION






§ 












§ 




 §.§ Additional information on the stochastic model
  

The quantities T(n'|n), as introduced in the main body of the paper, stand for the transition probabilities from the state n to the state n'. 
In the following, to identify the arrival/departure state n' we solely highlight the individual component of the vector n that changes due to the considered reaction.
The transition rates as stemming from the chemical equations that define the stochastic model under investigation take the following explicit form:

    (1) ATTACHMENT         T(n_1+1 |n)= k_1  n_D/N= k_1 [1- 1/N (n_1+n_2)] 
       (2) DETACHMENT           T(n_1-1 |n)= k_-1 n_1/N
       (3) CONVERSION            T(n_1-1, n_2+1 |n)= k_2 n_1/N
       (4) CONVERSION           T(n_1+1, n_2-1 |n)= k_-2 n_2/N
       (5) DETACHMENT           T(n_2-1 |n)= k_3  n_2/N

The governing master equation can be hence written in the following explicit form:

    ∂ P(n,t)/∂ t = 
    		 [T(n| n_1-1) P(n_1-1;t)- T(n_1+1 |n) P(n,t)+ 
                  + T(n| n_1+1) P(n_1+1;t)- T(n_1-1 |n) P(n,t)+ 
                  + T(n| n_1+1,n_2-1) P(n_1+1,n_2-1;t)+ 
                  - T(n_1-1,n_2+1 |n) P(n,t)+ 
                  + T(n| n_1-1,n_2+1) P(n_1-1,n_2+1;t)+ 
                  - T(n_1+1,n_2-1 |n) P(n,t)+ 
                   + T(n| n_2+1) P(n_2+1;t) - T(n_2-1 |n) P(n,t)  ]  .







 §.§ Additional information on the adiabatic approximation
 

As anticipated in the main text, the validity of the
adiabatic elimination was postulated even in the absence of a clear time scales separation of the dynamics of the system, and here we test the validity of this approach, for the operating conditions, in reproducing the correct time scale of the dynamics of the concentration for motors in A_2.
In the Supplementary Figure <ref> it is shown the evolution of the concentration of the motors in the force generating configuration A_2, from a initial condition with all the motors detached from the actin, from which it is possible to estimate the time scale on which the stationary state is approached. 







 §.§ Additional information on the structure of the stochastic matrix
 

As remarked in the main body of the paper for each possible value of n_1, n_2 can assume values in the range [0,N-n_1]. 
Hence, the total number of possible states of the system is M= (N+1)(N+2)/2. 
Consider the relevant setting N=16. We thus have to deal with M=153 possible states as listed in the following: 

    [ m ∈ [1,153]  (n_1, n_2);           1       (0,0);           2       (0,1);           ⋮           ⋮;          17      (0,16);          18       (1,0);           ⋮           ⋮;          33      (1,15);          34       (2,0);           ⋮           ⋮;         150      (14,2);         151      (15,0);         152      (15,1);         153      (16,0);             ][           ;        k=0;        k=1;           ;          ⋮;           ; .
    		} k=15; .
    		} k=16 ]

Focus now on the generic element W_lm that enters the definition of matrix ℚ. 
Assume in particular m to label the reference initial state; l identifies the state that can be eventually reached through the chemical dynamics. 
Five possible types of transitions exist, organised in k=N+1 blocks, which corresponds to the selected value of n_1, while n_2 can freely varies within [0, N-n_1]:

    l_1   = m^max(k+1)+ n_2+1 
    
    		l_2   = m^max(k)- n_2^max+ n_2+1 
    
    		l_3   = m^max(k)- n_2^max+ n_2+2 
    
    		l_4   = m^max(k+1)+ n_2 
    
    		l_5   = m-1

where we denote m^max(k) the largest possible index as associated to 
block k, for the selected choice of m; n_2^max stands the largest values that can eventually take the discrete variable n_2. Hence:

    n_2^max   =N-n_1 
    
    		m^max(k+1)   = m^max(k)+N+1-n_1

The non trivial elements  W_m,l are hence: 


    W_l_1,m   ≡ T(n_1+1,n_2 | n_1,n_2)= k_1/N(N-n_1-n_2)     n_2 ∈[0,n_2^max-1]
     n_1 ∈[0,N-1]
    
    	W_l_2,m   ≡ T(n_1-1,n_2 | n_1,n_2)=k_-1/N n_1      n_2 ∈[0,n_2^max]
     n_1 ∈[1,N]
    
    	W_l_3,m   ≡ T(n_1-1,n_2+1 | n_1,n_2)=k_2/Nn_1     n_2 ∈[0,n_2^max]
     n_1 ∈[1,N]
    
    	W_l_4,m   ≡ T(n_1+1,n_2-1 | n_1,n_2)=k_-2/N n_2    n_2 ∈[1,n_2^max]
     n_1 ∈[0,N-1]
    
    	W_l_5,m    ≡ T(n_1,n_2-1 | n_1,n_2)= k_3/N n_2     n_2 ∈[1,n_2^max]
     n_1 ∈[0,N]

Given the above structure, it is possible to identify for every choice of m, the corresponding combination of n_1 and n_2, and associate the m-component of  vector P(t) to a specific state (n_1,n_2). That is because the m^max(k) are the partial sums of the finite sequence:

    m^max(k)= ∑_i=0^k N+1-i              i ∈{0, …, N}

for k=0,…,N.

For a given m, we thus identify the index k that matches the relation:

    m^max(k)≥ m

and then set:

    n_1= k 
    
    		n_2= m-m^max(k-1)-k

The stationary solution of the stochastic dynamics is found as the eigenvector of ℚ associated to the null eigenvalue. 







 §.§ Deterministic and stochastic simulations of the underlying  model
 

Consider first the set of differential equations that represents the mean field approximation for the examined system. 
These equations can be numerically integrated, for a representative choice of the chemical reaction parameters, so as to resolve the temporal evolution of the concentrations of the motors, in configurations D, A_1 and A_2.
As shown in Supplementary Figure <ref>, after a transient, the concentration of the motors in the various populations approach their stationary state x^*, y^*, z^*. 

We now turn to studying the stochastic model that implements the microscopic dynamics described by the chemical equations (1) in the main text. 
It is possible to numerically simulate a single stochastic orbit of the considered dynamics (working with the same kinetic constants as assumed in the the mean field simulations) via the celebrated Gillespie algorithm. 
In doing so one eventually obtains the time evolution of the (discrete) concentration of molecular motors in each configuration, i.e. n_D(t)/N, n_1(t)/N, n_2(t)/N. A typical solution is displayed in Supplementary Figure <ref>. 
As expected the stochastic trajectories fluctuate around the corresponding deterministic orbit (solid lines). 
The observed fluctuations are a material imprint of the inherent discreteness of the simulated system.

The distribution of fluctuations can be numerically accessed from  individual stochastic simulation, by averaging over a large set of independent stochastic realisations. 
In Supplementary Figure <ref> the equilibrium distribution of the fluctuations (i.e. the fluctuations displayed around the fixed point, once the initial transient has faded away) is depicted and compared to the analytical solution obtained from the governing master equation, via the procedure discussed above. The agreement is satisfying and testifies on the correctness of the proposed analytical treatment.   

We are now in the position to simulate also the temporal series of the force of the ensemble, that is obtained by assigning to each individual motor the force that it is able to exert, based on its configuration (as stipulated by the stochastic dynamics), and following the prescriptions described in the Results.
The time series of the force exerted by motors of the populations A_1 and A_2 is displayed in Supplementary Figure <ref>, for the specific choice of the parameters here operated. 

By accessing the temporal evolution of the force, including the equilibrium fluctuations, one can recover key information on the underlying structural and chemical parameters. 
As shown in the main body of the paper, this corresponding to solving an inverse problem, from the observed force back to the relevant parameters.










 §.§ Analytical characterisation of the force probability distribution
 


As discussed in the main body of the paper, the fluctuations of the force around the average value stem from finite size corrections. 
To estimate the distribution we focus on k ∈ [1, N] force–generating distinct motors and postulate that each of them can exert a uniform, randomly selected force f. 
For each choice of k, one can compute the distribution of the force Π_k (f) exerted by the k motors. 
This is a particular case of the more general problem of calculating the probability distribution for the total force exerted by all the n_1 and n_2 molecular motors in the force–generating configurations A_1 and A_2. In this section we discuss the problem under this general perspective.
To this end we denote by f_1 the random force uniformly distributed in the interval ℐ_1 and exerted by the motors in configuration A_1 and by f_2 the one extracted from the interval ℐ_2, exerted by the motors in the configuration A_2.
As suggested in sadooghi2009distribution 
we are dealing with the problem of finding the probability distribution of the sum of n random variables Y_i, for i=1, …, n each of them uniformly distributed in the interval [b_i,c_i]. 
In our case we have only two classes of variables Y_i: the ones relative to the forces exerted by motors in the configuration A_1, and those relative to the forces exerted by motors in the configuration A_2, so that:

    ∑_i=1^n Y_i= ∑_i=1^n_1 Y_i+∑_i=n_1+1^n Y_i

and:

    Y_i=
    	 
    		f_1     and     [b_i,c_i]= [-f_0,f_0]     if     i=1, …, n_1 
    
    
    		f_2     and     [b_i,c_i]= [f_0/10,f_0]     if     i=n_1+1, …, n

We can observe that the probability distributions of the sum of the variables Y_i is the same as the probability distribution of the sum of the variables X_i=Y_i-b_i, which are defined in the intervals [0, a_i], where a_i=c_i-b_i.

Introduce the sum s:

    s=∑_i=1^n X_i
 
with n≥ 2. Then the distribution of the sum as defined above reads sadooghi2009distribution:

    Π_n(s;n_1,n_2)= 1/(n-1)!1/(a_1)^n_1 (a_2)^n_2[ s^n-1+∑_k=1^n (-1)^k(∑_J_k(s-∑_l=1^k a_j_l)_+)^n-1 ]

where we adopted the notation: (f)_+=max{0,f }.

From this expression it is possible to compute the probability distribution for the sum of the variables of our interest:

    ∑_i=1^n Y_i= ∑_i=1^n (X_i + b_i)= ∑_i=1^n X_i + ∑_i=1^n_1 b_i + ∑_i=n_1+1^n b_i= s + n_1 b_1 + n_2 b_2

where b_1= -f_0 and b_2=f_0/10.

If we consider just one class of variables Y_i, meaning if we consider only the forces exerted by one of the two force–generating populations of motors, the distributions Π_n can be computed as a specific case of the generalisation of the Irwin–Hall distribution, hall1927distribution, 
the uniform sum distribution.
These refer to the sum of n random variables x_i, each of them defined in the interval [a,b] and take the form:

    Π_n (x)= 1/b-a g(y;n)             with:         y=x-na/(b-a)

where:

    g(y,n)= 1/2(n-1)!∑_k=0^n(-1)^k nk (y-k)^n-1(y-k) .

In order to obtain the correct expression for the probability distribution of the total force exerted by both n_1 motors in the configuration A_1 and n_2 motors in the configuration A_2 at a given time, one has to weight the above expressions
with the probability to find the system in the considered condition. This latter follows the stationary solution of the master equation as discussed in the main body of the paper. From P^st, we can in particular extract the probability ρ_k to find k ≤ N motors in A_2. This is obtained by summing the elements of P^st that refer to a given k, thus accounting for all possible partitioning of the remaining N-k motors among states D and A_1 (see also discussion in the main body of the paper). 
Labelling these marginal probabilities (ρ_0,ρ_1,ρ_2,...,ρ_N) one gets:

    P(F)= ρ_1 Π_1(f) +ρ_2 Π_2(f) + …
    	+ρ_16Π_16(f)  =  ∑_i=1^N  ρ_i  Π_i (f)      .

where use has been made of the fact that Π_0=0. 

The obtained profiles are reported in Supplementary Figure <ref> for the relevant case N=16 and for i=1,2,3,4. In Supplementary Figure <ref> the global distribution of fluctuations is depicted for a specific choice of the kinetic parameters of the model.  
 
 







 §.§ Parameters estimation
 

The fitting procedure described in Results and applied to the experimental data set of soleus and psoas HMM has been validated against synthetic data, and the details are presented in this Section. 

The parameters f_0 and r are correctly estimated, as it follows from inspection of Supplementary Table <ref>, where the error associated with the estimated value of r is the standard deviation. 
Also the estimated b and a (recomputed from the best fitted values for the kinetic constants) are pretty close to their nominal values as imposed in the simulations, as well as the value of the rate ϕ. 
Supplementary Figure <ref> shows the graphic comparison between the results of the optimisation procedure for the parameters estimation performed under the assumptions of the effective model, and the values of the parameters adopted in the stochastic simulations of the dynamics (these latter are referred to as true parameters, as they are imposed in the simulation and thus known with infinite precision). 
The symbols refers to the mean and SD of the duty ratio r, the force of a single motor f_0, the rate ϕ and the two parameters a and b that describe the dynamics of the system in the deterministic limit. 
The blue line represents the value of the corresponding parameter adopted in the simulations.

Supplementary Figure <ref> refers to the parameters estimation in the plane (f_0, r), where the solutions resulting from the analysis of the force of the ensemble in the deterministic framework are represented by the solid line, while the symbols shows the value of the parameter f_0 and r that can be estimated by taking into account the fluctuations of the force of the ensemble.

The result of the analysis for different system size N are reported in Supplementary Figure <ref>: the solid line represent f_0 as a function of N, while the symbols refers to the best fit value of f_0 as determined for different choices of N. 
We remark that the fitting procedure converges (with the requested limit of precision) only over a finite range of values of N, centred around the value adopted when performing the simulations. 
This observation implies in turn that we are in a position to obtain a reasonable estimate for the interval of pertinence of N, as follows the procedure outlined above.   















 §.§ Parameters estimation for soleus and psoas HMM data
 

A representative example of the fitting outcome for the soleus HMM ensemble is reported in the Supplementary Figure <ref>.
The histogram is generated from the experimental data series of the isometric force in the stationary state (i.e. at the isometric plateau).  
The results of the fitting procedure and the parameters estimation performed on the experimental data sets from rabbit psoas and rabbit soleus are listed in the Supplementary Table <ref> and Supplementary Table <ref> respectively. 

The result of the analysis for different system size N are reported in Supplementary Figure <ref>: the symbols refers to the best fit value of f_0 as determined for different choices of N.
The shaded region identifies the portion of the parameters plane where the solutions are expected to be found.
Specifically, it is assumed to lay in between the two curves:

    f_0 N= 20/11F_0 ± 2 Δ F_0a/b

Here the relative error associated with the average value of the quantity a/b  is assumed negligible, as compared to that  stemming from the average stationary force, Δ F_0.
The histogram computed from the collection of fitted parameters (each choice of symbols refers to a different experimental series) can be conceptualised as an indirect imprint of the degree of experimental variability as associated to f_0 and N.











§ 




 §.§ Rate of force development in situ


The time course of isometric force redevelopment following a release that drops the isometric force to 0 was determined in  Ca^2+ –activated demembranated fibres from the same slow and fast muscles of the rabbit used to extract the myosin isoforms for the nanomachine. 

As previously described linari2007stiffness, percario2018mechanical, caremani2022force, small bundles dissected from the two muscles were stored in skinning solution containing 50% glycerol at -20 for 3–4 weeks and single fibres were prepared just before the experiment. 
A fibre segment 4–6 long was clamped at its extremities by T–clips and mounted between the lever arms of a loudspeaker motor and a capacitance force transducer lombardi1990contractile. 
To prevent sliding of the ends of the fibre segment inside the clips and minimise the shortening of the activated fibre against the damaged sarcomeres at the ends of the segment during force development, the extremities of the fibre were fixed first with a rigor solution containing glutaraldehyde and then glued to the clips with shellac dissolved in ethanol. 
Fibres were activated by temperature jump using a solution exchange system as previously described linari2007stiffness. 
A striation follower huxley1981millisecond allowed nanometre–microsecond resolution recording of length changes in a selected population of sarcomeres. 

The composition of the solutions has been reported previously (caremani2022force, Supplementary Table 1, 25). 
The increase of interfilamentary distance following cell membrane permeabilisation was reversed by the addition of the osmotic agent Dextran T–500 (4% weight/volume). 

The rate of force development was determined on the isometric force rise recorded after superimposing on the isometric contraction of the maximally  Ca^2+ –activated fibre (pCa 4.5) a fast ramp shortening (5–6% of the initial fibre length) able to drop the force to zero (Supplementary Figure <ref>).  


[heading=bibintoc]







 Paper Title: Low energy excitations in bosonic quantum quasicrystals.






alejandro.mendoza@ufsc.br
Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, Brazil
Max Planck Institute for the Physics of Complex Systems, Nothnitzerstr. 38, 01187 Dresden, Germany


Max Planck Institute for the Physics of Complex Systems, Nothnitzerstr. 38, 01187 Dresden, Germany


Max Planck Institute for the Physics of Complex Systems, Nothnitzerstr. 38, 01187 Dresden, Germany
Theoretical Physics III, Center for Electronic Correlations and Magnetism,
Institute of Physics, University of Augsburg, 86135 Augsburg, Germany

§ ABSTRACT 
   We present the first principles construction of the low-energy effective action for bosonic self-organized quantum quasicrystals. Our generalized elasticity approach retains the appropriate number of phase- and corresponding conjugate density- degrees-of-freedom required for a proper description of the Goldstone modes. For the dodecagonal and decagonal quasicrystal structures we obtain collective longitudinal and transversal excitations with an isotropic speed of sound. Meanwhile, for the octagonal structure, the coupling between phononic and phasonic degrees of freedom leads in turn to hybridization of the latter with the condensate sound mode, producing collective excitations with a longitudinal and transversal component, and an anisotropic speed of sound. Finally, we discuss the fate of each excitation mode at the low and high density phase transitions limiting the quantum quasicrystal phase.  
      
    
    
     

    


Low energy excitations in bosonic quantum quasicrystals
    F. Piazza
    
=======================================================



Introduction.–Quasicrystals (QCs) are an exotic state of matter where crystalline and disordered behaviour meet. They retain long range orientational order but lack any discrete translational invariance <cit.>.  Although they were initially discovered in metallic alloys <cit.>, it is known today that this kind of exotic state can be stabilized in many different contexts <cit.>, as long as an appropriate effective competition between two or more characteristic length scales is present in the system <cit.>. One of the most intriguing versions of quasi crystalline systems  are the so-called quantum quasicrystals <cit.>, which appear as ground states of quantum many-body Hamiltonians. Such is the case of QCs produced by spin-orbit interactions in dipolar bosons <cit.>, quasicrystalline phases induced in ultracold bosonic gases by quasiperiodic optical traps <cit.> and more recently, self-organized quasicrystals in Bose-Einstein condensates produced by cavity mediated interactions <cit.>.
In this Letter, we present the first principles construction of the low energy elastic theory of bosonic quantum quasicrystals, having supersolids as a particular case. Our calculations maintain full connection with the microscopic properties of the system allowing us to make qualitative and quantitative predictions regarding the hybridization of the low momentum gapless energy modes, the corresponding propagation speed and the behavior of these excitations at the phase boundaries of the  quasicrystal phase. As particular examples we consider the three most relevant quasiperiodic structures in two dimensions, the octagonal, the decagonal and the dodecagonal quasicrystals. To our knowledge, this is the first microscopic elastic theory of bosonic quantum quasicrystals able to corroborate the symmetry arguments regarding the existence of five gapless excitation modes for these systems. Our results show that the hybridization process between phonons, phasons and the longitudinal condensate sound mode qualitatively depends on the kind of quasicrystal structure.


 Model and Method.–We consider a 2D bosonic gas at zero temperature, interacting through an isotropic non-local two-body interaction of the form U v(r), where U represents a dimensionless parameter characterizing the intensity of the non-local interaction with respect to the kinetic energy. The nature of the two-body interaction between particles is such that the system hosts a quasicrystalline pattern at high enough particle densities in the ground state. As discussed in the literature <cit.>, the key ingredient for the stabilization of such phases is the presence of competing minima in the interaction potential v̂(k). In dimensionless units, the path-integral formulation of the grand canonical partition function of the system 𝒵=∫ Dϕ Dϕ^*e^-S[ϕ,ϕ^*] <cit.> is characterized by the action: 

    S[ϕ,ϕ^*]      =∫_0^∞ dτ∫ d^2x  ϕ(x,τ)^*(∂_τ-1/2∇^2-μ.
          .+U/2∫ d^2x'v(x-x')|ϕ(x',τ)|^2 )ϕ(x,τ).


To access the low-energy properties of the system we should allow long-wavelength fluctuations of the global phase, as well as of the phase of the modulated pattern of the quasicrystal ground-state wave-function which can be expanded in a Fourier basis as ϕ_0(x)=√(ρ)∑_nc_nexp(iG_n·x). Here ρ stand for the average particle density of the system and c_n and G_n corresponds to the Fourier amplitudes and wave vectors of the pattern respectively. We thus propose a perturbed ground state wave function of the form:  

    ϕ(x,τ)=√(ρ)c_0√(1+δ n_0(x,τ)/ρ c_0^2)exp(iθ(x,τ))
       +   ∑_n≠0√(ρ)c_n√(1+G_n·Π(x,τ)/Z+G_n,⊥·Π_⊥(x,τ)/Z_⊥)
          exp(i G_n·x+i G_n·u(x,τ)
        + i G_n,⊥·w(x,τ).
       +   .iθ(x,τ)),

where the fluctuation fields u(x,τ) and w(x,τ) represent the phonon and phason fields introduced as in the classical case <cit.>. Moreover, due to the quantum nature of the system we also have to admit fluctuations in the canonical-conjugate density fields associated to these phase variables, analogously to what was proposed by Pretko et al. <cit.> for supersolids. Accordingly, we have included the fields Π(x,τ) and Π_⊥(x,τ) as conjugate to u(x,τ) and w(x,τ), respectively. Finally, we added the overall phase fluctuation field θ(x,τ) of the wave function and its canonically conjugate zero momentum density fluctuation δ n_0(x,τ). It is understood that all fluctuation fields are slowly varying functions in space and imaginary time, with zero mean over the size of the system and, as expected, in the absence of fluctuations our ansatz for ϕ(x,τ) reduces to the ground state wave function ϕ_0(x). Furthermore, the set of vectors {G_n,⊥} is constructed from a permutation of the set {G_n}, such that the extended basis G̃_n=G_n⊕G_n,⊥ is orthogonal in the sense ∑_nG̃_n,αG̃_n,β∝δ_α,β <cit.>.

It is simple to show that the proposed ansatz satisfies the normalization condition of the boson field  ∫_A d^2x<|ϕ(x,τ)|^2>=N=ρ A, as long as ∑_nc_n^2=1. Such normalization constraint allows the calculation of the chemical potential μ for a system with a given particle density ρ. Lastly, the free parameters Z and Z_⊥ in Eq. (<ref>) are chosen to enforce the expected relation between canonically conjugate fields in the low energy effective action of the system <cit.>. By replacing the above ansatz for ϕ(x,τ) in Eq. (<ref>) and expanding up to quadratic order in the fluctuation fields we obtain the following long-wavelength effective action:
 
    δ S   =∫ dτ d^2x (i δ n_0(x,τ)∂_τθ(x,τ)+i Π(x,τ)·∂_τu(x,τ).
       +.i Π_⊥(x,τ)·∂_τw(x,τ))+ ∫ dτ d^2x 1/2[ρ(∇θ)^2.
       +γ_0δ n_0(x,τ)^2+γΠ(x,τ)^2+γ_⊥Π_⊥(x,τ)^2
       +2Π·∇θ+.E^T C̃ E],

where E stands for the extended phonon-phason strain field defined as E={∂_xu_x,∂_yu_y,(∂_xu_y+∂_yu_x)/2,(∂_yu_x+∂_xu_y)/2,∂_xw_x,∂_yw_y,∂_yw_x,∂_xw_y} and C̃ represents the phonon-phason elastic tensor. Closed formulae for the elements of this tensor as well as for the γ's couplings in terms of the ground state wave function are provided in the Supp. Matt. <cit.>. The expression in Eq. (<ref>) for the low-energy effective action of a bosonic quasicrystal is one of the main results of this work. It allows qualitative and quantitative predictions since all couplings are known in terms of the ground state wave function of the system. Moreover, it generalizes the well-known density-phase action of a homogeneous condensate <cit.>, not only to the present case of quantum quasicrystals, but also to supersolids if we eliminate the phasonic degrees of freedom <cit.>.



Elementary excitations for the dodecagonal QC– The effective action in Eq. (<ref>) can be recast as δ S=1/2∫dω/(2π)d^2q/(2π)^2η̂(q,ω)^†M(q,ω)η̂(q,ω), where η̂(q,ω)=(δn̂_0(q,ω),
θ̂(q,ω),
  Π̂_x(q,ω),
  û_x(q,ω),
  Π̂_y(q,ω),
  û_y(q,ω), Π̂_x,⊥(q,ω),
  ŵ_x(q,ω),
  Π̂_y,⊥(q,ω), ŵ_y(q,ω)), the energy-momentum dispersion relations ϵ(q)=ω(q) can be computed by solving the equation [M(q,iω(q))]=0 <cit.>. The dodecagonal quasicrystal (Fig. <ref>(a)) is the simplest case due to the absence of the coupling between phonons and phasons. The latter are deformations of the density pattern which are absent in ordinary crystals. We obtain analytical expressions for the dispersion relation of the five gapless modes <cit.>. In this case there are three longitudinal modes and two transversal modes, as schematically represented in Fig. <ref>(b). Two of the longitudinal modes result from the hybridization of the condensate sound mode (corresponding to a modulation of the overall phase θ) with the longitudinal phonon mode, while the transversal phonon as well as the phason mode remain decoupled. 
  
 
  


To deepen our understanding of the excitation modes we calculate the eigenvectors η_j(q) using the charateristic equation M(q,i ω_j(q))η̂_j(q)=0 for each excitation mode. Each eigenvector corresponds to a fluctuation field in real space and time given by η_j(x,t)=Re[η̂_j(q)exp(i(q·x-ω_j(q)t))]. By expanding Eq. (<ref>) to linear order in the fluctuation fields we obtain a general expression for the correction to the ground state wave function: δψ_j,q=[u_j,q(r)exp(i(q·r-ω_j(q)t))-v_j,q(r)^*exp(-i(q·r-ω_j(q)t))], allowing us to identify the Bogoliubov modes u_j,q(r) and v_j,q(r) <cit.>. 
The local phase and density fluctuations can be then written as Δρ_j,q=| u_j,q(r)-v_j,q(r)|^2 and Δφ_j,q=| u_j,q(r)+v_j,q(r)|^2 <cit.>. In Fig. <ref>(c-l) we show the typical behavior of these quantities. Each column corresponds to a mode in Fig. <ref>(b) in increasing order of excitations energy. In agreement with previous observations for supersolids <cit.>, the hybridized  longitudinal modes present significant (O(1)) phase and density fluctuations, while the transversal phonon mode presents significant density fluctuations and  O(q) phase fluctuations. On the other hand, while showing the same scaling (O(1) vs O(q)) of the transversal phonon modes, phasons feature density and phase fluctuations patterns which presents a different behavior around the center of the quasicrystal structure [The center of the quasicrystal here is being understood as the special point where all reflections symmetry planes of the quasicrystal pattern intersects. Given the form of our anzats for ϕ_0(x) it coincides with the origin of our coordinates system.] compare e.g. panel (e) and (i) of Fig.<ref>.  Mathematically, this is ultimately a consequence of the phason's property of producing a redistribution of the QC structure destroying the center of the quasicrystal without altering the average properties of the system <cit.>. Finally, we would like to remark that differently from the case of supersolids, here all density and phase fluctuations patterns are aperiodic. 


Excitations in decagonal and octagonal QCs.–In the case of ten- and eight-fold quasicrystals, the presence of the phonon-phason coupling leads to a characteristic equation [M(q,iω(q))]=0 with convoluted analytical solutions. However, some general properties can be obtained from the numerical solution of this problem. In the case of the decagonal quasicrystal, the longitudinal and transversal modes separately hybridize among themselves, leading to two groups of three and two modes, respectively, all with isotropic dispersion relations. In contrast, for the octagonal quasicrystal, the structure of the elasticity action characterized by an anisotropic phason contribution and a finite phonon-phason coupling produces anisotropic dispersion relations ϵ(q) with propagation speed c_s that depend on the orientation of the momentum q when q→0.  Additionally, unless the momentum of the excitation q is oriented along one of the symmetry axes of the quasicrystal, every mode has a longitudinal and a transversal component. As an example, we show in Fig. <ref>(a) the anisotropic behavior of the propagation speed and hybribization of modes for an octagonal QC. The numerical evaluation was carried out considering values of the couplings for which the system displays strong anisotropy and  hybridization effects <cit.>. The red to blue color gradient indicates the longitudinal and transversal character of each mode, using the angular orientation of the field u as a measure of this property. Interestingly, the propagation of excitations along non-equivalent symmetry axes of the octagonal quasicrystal produces an exchange of the longitudinal and transversal character for some modes. Moreover, the evaluation of the local density and phase fluctuations profile in this case shows that the scaling of the fluctuations when the propagation direction of the excitation do not coincide with a symmetry axes of the quasicrystal are all O(1). Meanwhile, in the opposite case, longitudinal modes have density and phase fluctuations of amplitude O(1) and transversal modes with density fluctuations O(1) and phase fluctuations O(q). The aperiodic patterns of phase and density fluctuations for each scenario discussed can be found in <cit.>.


Instabilities of the quantum quasicrystal phase.– A first analysis of the instabilities can be based on the behaviour of the propagation speed of the gapless modes. With this aim we consider the case of the decagonal QC structure and pursue the calculation of the couplings in the action (<ref>), considering small amplitudes of the QC modulation <cit.>. 
In order to make a numerical evaluation possible without defining a particular form of the interaction potential, we assume that the QC amplitude is a concave increasing  function of ρ U [Mathematically this means ∂ c_1(Uρ)/∂ (Uρ)>0 and ∂^2 c_1(Uρ)/∂ (Uρ)^2<0, where c_1(Uρ) stand for the modulation amplitude of ϕ_0.]. Although this is the expected behavior of this quantity, such ansatz does not intend to capture the actual quantitative behavior of a certain model <cit.>. The results for the propagation speed of each mode are shown in Fig. <ref>(b) as a functon of ρ U. One recognizes a discontinuous behavior of the propagation speed at the homogeneous-superfluid-to-quantum-quasicrystal phase transition, analogous to previous observations in supersolids <cit.>, and a continuous behavior at the quantum-to-classical quasicrystal transition at high enough densities where the propagation speed of the lowest mode vanishes <cit.>,[Although Mean Field treatments of our action consistent with the Gross-Pitaevskki description are not able to reach a strictly zero propagation speed for the lowest mode, beyond Mean Field descriptions have succeeded in observing rigorously the supersolid to normal solid transition <cit.>.]. As we shall see next, this transition corresponds to the breaking of quantum phase coherence. 


To further explore the nature of the instabilities we compute the second-order correlations of the fluctuation fields. At the lower density boundary, when the QC amplitude is small and the ground state wave function is dominated by the main Fourier amplitude of the pattern (c_1), we find that Z, Z_⊥ and all elastic couplings in C̃ approaches zero as c_1^2, while γ and γ_⊥ diverges as c_1^-2. This behavior lead us to the conclusion that in this regime ⟨δ n_0^2⟩ and ⟨θ^2⟩ remain finite, while ⟨Π^2⟩∝ c_1^2, ⟨Π_⊥^2⟩∝ c_1^2, ⟨u^2⟩∝ c_1^-2  and ⟨w^2⟩∝ c_1^-2, as c_1→0, with c_1≫ c_n>1. In this scenario the average wave function can be roughly approximated as ⟨ϕ(x,τ)⟩≈√(ρ)∑_nc_nexp(iG_n·x)×exp(-G_n^2⟨u^2⟩/4-G_n,⊥^2⟨w^2⟩/4-⟨θ^2⟩/2). This result confirms that close enough to the superfluid-quantum-quasicrystal stability boundary, where we have large phonon and phason fluctuations, there is a strong non-perturbative “renormalization" of the Fourier amplitudes of the quasicrystal profile [The strong renormalization process is signaled by the non-analytic behavior of ⟨ϕ⟩ in the limit c_1→ 0 upon inclusion of phonon and phason fluctuations ]. This phenomenology suggests that in cases in which the mean field superfluid to quantum quasicrystal phase transition is weak enough, the nature of the transition could be modified to a continuous one, even with the presence of intermediate hexatic-like phases. 

At the upper boundary of the quantum quasicrystal phase (γρ→ 1^+) all couplings of the effective action in Eq. (<ref>) remain finite, however in this limit the propagation speed of the lower hybridized mode approaches zero, as observed  before. This behavior  signals the disappearance of this mode, in the same way in which the lower energy excitation mode disappears at the supersolid to normal solid phase transition <cit.>. Calculating the correlations of the fluctuation fields in this regime, we obtain to leading order that ⟨δ n_0^2⟩ ⟨u^2⟩, ⟨w^2⟩ and ⟨Π_⊥^2⟩ remain finite as the stability boundary is approached, while ⟨θ^2⟩∝(γρ-1)^-1/2 and ⟨Π^2⟩∝(γρ-1)^-1/2. This result is to be expected since in the high ρ U regime the localization of particles should eventually lead the system to the loss of quantum phase coherence. Interestingly, due to the effective Π·∇θ interaction the de-stabilization of the θ(x,τ) field also produces the destabilization of the Π(x,τ) field, indicating the natural breakdown of our theory at this phase boundary. All evidence available in this case indicates a second order phase transition in this limit, as observed in supersolid systems <cit.>.  






Conclusions.–We have developed the elastic theory of a self-organized quantum quasicrystal with 8, 10 or 12-fold rotational symmetry. We obtained the effective action of the system described in terms of the global wave function phase, the two-dimensional phonon and phason fields and their respective conjugate density fields. From here, we were able to prove from first principles the existence of five gapless extended low energy excitation modes [It is worth to mention, that due to the lack of periodicity quantum quasicrystals can also host particle-hole localized excitations with an energy arbitrarily close to zero. See Refs. <cit.>.] for the considered two-dimensional QC structures  <cit.>. We showed that the presence of the Π·∇θ coupling in our effective theory produces the hybridization of the condensate sound mode with phonon and phason modes, leading to a rich phenomenology depending on the QC structure. A particularly interesting finding was that, due to the phonon-phason coupling, the octagonal QC structure develops an anisotropic speed of sound that depends on direction of the propagation, as well as that all modes show a mixed longitudinal-transversal character. Finally, our results shed light on the behavior of the low energy fluctuation fields at the thermodynamic boundaries of the quantum quasicrystal phase, elucidating possible scenarios for phase transitions in two dimensions.            


Acknowledgments: A.M.C. acknowledge MPIPKS for financial support and hospitality.



 
ieeetr




Supplemental Materials: Low energy excitations in super quasicrystals: a generalized elasticity approach 














§ I. CONSTRUCTION OF THE EFFECTIVE ACTION

In this section we present details about the  construction of the effective action for the low energy behavior of the octagonal, decagonal and dodecagonal cluster quasicrystalline structures. We assume that the pair interaction potential v(x) is such that the model self-stabilizes one of these phases as its ground state in the quantum regime of the system <cit.>. By quantum regime, we refer to the region of parameters in which particle delocalization is strong enough to produce a finite superfluid fraction in the system. As a consequence, we consider that the stationary configuration that minimizes the action in Eq. (1) can be expressed as a Fourier expansion of the form

    ϕ_0(x)=√(ρ)∑_n c_nexp(i G_n·x),

where c_n and G_n represent the amplitudes and wave vectors of the Fourier expansion of ϕ_0. Moreover, for the kind of QCs considered in this work we have G_n=k(n_1e_1+n_2e_2+n_3e_3+n_4e_4), where k stands for the characteristic wave vector of the quasicrystal and the basis vectors e_i's are specific for each kind of pattern. The sum in the Fourier expansion is performed over a four dimensional integer vector n whose components n_i's are all integer numbers. A possible definition of the vectors e_i's for each of the cases considered is presented in Table <ref>. 


The low energy excitation properties can be accessed considering the effects of long wave-length fluctuations in the Fourier amplitudes and in its corresponding conjugate phases, in the same spirit as presented in ref <cit.>. In our case we propose a general ansatz for the perturbed wave function field of the form 

    ϕ(x,τ)=√(ρ)c_0√(1+δ n_0(x,τ)/ρ c_0^2)exp(iθ(x,τ))
        +∑_n≠0√(ρ)c_n√(1+G_n·Π(x,τ)/Z+G_n,⊥·Π_⊥(x,τ)/Z_⊥)
          exp(i G_n·x+i G_n·u(x,τ)
        + i G_n,⊥·w(x,τ)
        +iθ(x,τ)),

where the set {G_n,⊥} is constructed as a permutation of the set {G_n} such that the extended four dimensional basis G̃_n=G_n⊕G_n,⊥, is orthogonal in the sense that ∑_nG̃_n,αG̃_n,β∝δ_α,β <cit.>. For the purpose of calculations we can write G_n,⊥=k(n_1e_1,⊥+n_2e_2,⊥+n_3e_3,⊥+n_4e_4,⊥), where the basis {e_i,⊥} corresponding to our particular choice of {e_i} is chosen as shown in Table <ref>. Moreover is important to notice that the index vector {n_i} for two corresponding G_n and G_n,⊥ coincide. 


Since we are interested in the low energy behavior of the system we can work in the hydrodynamic regime, in which the fluctuation fields introduced vary slowly in space and imaginary time. Mathematically, this argument allows us to neglect all components of the effective theory containing the product of slowly varying terms with spatially oscillatory contributions.

Now we can proceed with the evaluation of each term of the original action (Eq. (<ref>)) as a function of the fluctuation fields, keeping only up to quadratic contributions. The first contribution can be recast in the symmetric form 

    δ S_1=∫ dτ d^2x1/2(ϕ(x,τ)^*∂_τϕ(x,τ)-ϕ(x,τ)∂_τϕ(x,τ)^*),

which up to quadratic order in the fluctuation fields yields

    δ S_1=∫ dτ d^2x (i δ n_0(x,τ)∂_τθ(x,τ)+i (ρ∑_nc_n^2G_n^2/2Z)Π(x,τ)·∂_τu(x,τ)+i (ρ∑_nc_n^2G_n,⊥^2/2Z_⊥)Π_⊥(x,τ)·∂_τw(x,τ)).

In this way, by choosing Z=ρ∑_nc_n^2G_n^2/2 and Z_⊥=ρ∑_nc_n^2G_n,⊥^2/2 we obtain for δ S_1, the expected structure combining the phases and the corresponding conjugate density fluctuation fields. 

The second contribution of our action can be recast as

    δ S_2=∫ dτ d^2x1/2∇ϕ(x,τ)^*·∇ϕ(x,τ).

In this case, the use of our ansatz leads us to the expression 

    δ S_2   =   δ S_2^0+∫ dτ d^2x{1/8ρ c_0^2(∇δ n_0)^2+1/8Z((∇Π_x)^2+(∇Π_y)^2)+1/8Z_⊥((∇Π_x,⊥)^2+(∇Π_y,⊥)^2).
       +   .ρ c_0^2/2(∇θ)^2+ρ∑_n≠0c_n^2/2(∇θ)^2+Z/2((∇ u_x)^2+(∇ u_y)^2)+Z_⊥/2((∇ w_x)^2+(∇w_y)^2)+Π·∇θ},

where δ S_2^0, represents a fluctuation-independent contribution. The linear order contributions in the fluctuation fields are not presented in this and the rest of the terms of the effective action because they have a zero total contribution, once we are considering fluctuations around the global minimum of our action. At this point it is worth noticing that the particular form considered for our ansatz in Eq. (<ref>) as well as the appropriate selection of Z and Z_⊥ are responsible for producing a form of δ S_1 and δ S_2 preserving the expected relation between the couplings associated to conjugate density and phase fields. Other choices for this ansatz, for instance including  a dependence of Z and Z_⊥ with |G_n|, even when preserves the symmetries of the original expression will not be able to reproduce this property.  

The third contribution of the original microscopic action, δ S_3=∫ dτ d^2xμϕ(x,τ)^*ϕ(x,τ), does not provide any quadratic contribution to the total effective action in the long wave limit. Finally, we can proceed with the contribution resulting from the non-local interaction term 

    δ S_4=∫ dτ d^2xd^2x'1/2U v(x-x')|ϕ(x,τ)|^2|ϕ(x',τ)|^2.

This last term produces two kinds of contributions, one including only density-density interactions defined as δ S_4,1 and a second one corresponding to the phonons and phasons deformation fields defined as δ S_4,2. Terms of density-phase interactions cancel out exactly due to the reflection and rotational symmetries of the modulated patterns considered. Taking into account that in the determination of δ S_4,1 and δ S_4,2 we have to deal with a convolution with a non-local interaction kernel, it is easier to perform the evaluation of these terms in momentum space. After a lengthy but strightforward calculation we obtained that δ S_4,1 in the long wave limit is given as

    δ S_4,1   =   ∫ dτd^2q/(2π)^21/2(γ_0δn̂_0(q,τ)δn̂_0(-q,τ)+γΠ̂(q,τ)·Π̂(-q,τ)+γ_⊥Π̂_⊥(q,τ)·Π̂_⊥(-q,τ)
    ),

where 

    γ_0   =   -1/2∑_n1≠0,n3,n4Uv̂(G_n1)c_n1c_n3c_n4/c_0^3δ(G_n1+G_n3-G_n4,0)+∑_nUv̂(G_n)c_n^2/c_0^2,
    γ   =   -1/4∑_n1,n2,n3,n4Uv̂(G_n1-G_n2)ρ^2Π_i=1^4 c_ni/Z^2(G_n1-G_n2)_x^2δ(G_n1-G_n2+G_n3-G_n4,0),
    γ_⊥   =   -1/4∑_n1,n2,n3,n4Uv̂(G_n1-G_n2)ρ^2Π_i=1^4 c_ni/Z_⊥^2(G_n1,⊥-G_n2,⊥)_x^2δ(G_n1-G_n2+G_n3-G_n4,0).

The obtained expression for each γ reduces to the one well known for the homogeneous scenario (c_0=1 and c_n≠0=0), γ_0=Uv̂(0), γ=0 and γ_⊥=0. It is also important to remark that since V̂(k) has a negative part responsible for exciting the main Fourier modes of the modulated pattern, we will have γ_0, γ and γ_⊥ as positive parameters, which guarantees the stability of our effective theory. We should note that in the long wave limit the terms proportional to the gradient of the density fluctuation fields in δ S_2 are subleading respect to the contribution in δ S_4,1, which is why these terms do not appear in Eq. (<ref>) of the main text.     

The last contribution in δ S_4 corresponds to the phonon and phason interaction. This term can be written in general as 

    δ S_4,2   =   ∫ dτd^2q/(2π)^2U ρ^2/4∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)
       ×   (2v̂(q-(G_n1-G_n2))-2v̂(G_n1-G_n2))((G_n1-G_n2)·û(q,τ)+(G_n1,⊥-G_n2,⊥)·ŵ(q,τ) )
          ((G_n1-G_n2)·û(-q,τ)+(G_n1,⊥-G_n2,⊥)·ŵ(-q,τ) ).

It is possible to convince ourselves that unlike all previous terms in our effective action this contribution has a different form for each quasicrystal. As expected in the long wave limit δ S_4,2 reproduces the structure of the well-known classical elastic Hamiltonian for each kind of quasicrystal structure <cit.> yielding 

    δ S_4,2   =   1/2∫ dτd^2q/(2π)^2[((2μ+λ)q_x^2+μ q_y^2)û_x(q,τ)û_x(-q,τ)+((2μ+λ)q_y^2+μ q_x^2)û_y(q,τ)û_y(-q,τ).
       +   . (2μ+2λ)q_xq_yû_x(q,τ)û_y(-q,τ)+(K_1q_x^2+(K_1+K_2+K_3) q_y^2)ŵ_x(q,τ)ŵ_x(-q,τ).
       +   . (K_1q_y^2+(K_1+K_2+K_3) q_x^2)ŵ_y(q,τ)ŵ_y(-q,τ)+2(K_2+K_3)q_xq_yŵ_x(q,τ)ŵ_y(-q,τ).
       +   .(R q_x^2-R q_y^2)(û_x(q,τ)ŵ_x(-q,τ)+û_y(q,τ)ŵ_y(-q,τ))+2Rq_xq_y(û_x(q,τ)ŵ_y(-q,τ)-û_y(q,τ)ŵ_x(-q,τ))
    ].

The expression presented corresponds to the most general scenario, consistent with the octagonal quasi-crystalline pattern. In the case of the decagonal quasi-crystal we will have K_2=-K_3, which makes zero the off-diagonal phason-phason interaction proportional to ŵ_x(q,τ)ŵ_y(-q,τ) and finally, in the case of the dodecagonal quasicrystal we will have R=0, which makes zero all elastic phonon-phason interactions. It is important to mention that the particular form obtained for our elastic phonon-phason action is a direct consequence of the particular choice we made for G_n,⊥. Different but equivalent forms can be obtained for δ S_4,2 if we consider other permutations of G_n in the definition of G_n,⊥. Furthermore, from the comparison between Eqs. <ref> and <ref> it is possible to conclude that


    (2μ+λ)   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1-G_n2)_x^2∂_qx^2v̂(q-(G_n1-G_n2))|_q=0
    μ   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1-G_n2)_x^2∂_qy^2v̂(q-(G_n1-G_n2))|_q=0
    
    K_1   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1,⊥-G_n2,⊥)_x^2∂_qx^2v̂(q-(G_n1-G_n2))|_q=0
    
    K_T   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1,⊥-G_n2,⊥)_x^2∂_qy^2v̂(q-(G_n1-G_n2))|_q=0
    
    R   =U ρ^2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1,⊥-G_n2,⊥)_x(G_n1,⊥-G_n2,⊥)_x
    ×   ∂_qx^2v̂(q-(G_n1-G_n2))|_q=0,

where K_T=K_1+K_2+K_3. These expressions complete the determination of all couplings in our effective theory from the knowledge of the ground state wave function. Finally, we can notice that the gradient terms of the phonon and phason fields in Eq. (<ref>) can be absorbed in δ S_4,2 modifying the elastic coefficients as follows (2μ̃+λ̃)=(2μ+λ)+Z, μ̃=μ+Z, K̃_1=K_1+Z_⊥ and K̃_T=K+Z_⊥. This allows us to write our total effective action in the following more compact way 

    δ S   =∫ dτ d^2x (i δ n_0(x,τ)∂_τθ(x,τ)+i Π(x,τ)·∂_τu(x,τ)+i Π_⊥(x,τ)·∂_τw(x,τ))+ ∫ dτ d^2x 1/2[ρ(∇θ)^2.
       .+γ_0δn̂_0(x,τ)^2+γΠ(x,τ)^2+γ_⊥Π_⊥(x,τ)^2+2Π·∇θ+E^T C̃ E],

where E stands for the extended phonon-phason strain field defined as E={∂_xu_x,∂_yu_y,(∂_xu_y+∂_yu_x)/2,(∂_yu_x+∂_xu_y)/2,∂_xw_x,∂_yw_y,∂_yw_x,∂_xw_y} and C̃ represents the phonon-phason elastic tensor <cit.> given by


    C̃=
    [ λ̃ +2μ̃      λ̃       0       0       R       R       0       0;      λ̃  λ̃+2μ̃       0       0      -R      -R       0       0;       0       0      μ̃      μ̃       0       0      -R       R;       0       0      μ̃      μ̃       0       0      -R       R;       R      -R       0       0  K̃_̃1̃     K_2       0       0;       R      -R       0       0     K_2  K̃_̃1̃       0       0;       0       0      -R      -R       0       0    K̃_T     K_3;       0       0       R       R       0       0     K_3    K̃_T ].







§ II. EXCITATION PROPERTIES OF THE DODECAGONAL QUASICRYSTAL

As we can notice, the effective action in Eq. (<ref>) can be written in the general form

    δ S=1/2∫dω/(2π)d^2q/(2π)^2η̂(q,ω)^†M(q,ω)η̂(q⃗,ω),

where the following definition of the Fourier transform of f(x,τ) and its inverse have been used

    f̂(q,ω)   =   ∫d^2q dω/(2π)^3e^i(q·x-ωτ)f(x,τ)
    
    f(x,τ)   =   ∫ d^2x dτ e^-i(q·x-ωτ)f̂(q,ω).

Moreover, η̂(q,ω)={δn̂_0(q,ω),θ̂(q,ω),Π̂_x(q,ω),û_x(q,ω),Π̂_y(q,ω),û_y(q,ω),Π̂_x,⊥(q,ω),ŵ_x(q,ω),Π̂_y,⊥(q,ω),ŵ_y(q,ω)} and the interaction matrix M(q,ω) is written as

    M(q,ω)=
    [                   γ_0                    -ω                     0                     0                     0                     0                     0                     0                     0                     0;                     ω                 ρ q^2                 -iq_x                     0                 -iq_y                     0                     0                     0                     0                     0;                     0                  iq_x                     γ                    -ω                     0                     0                     0                     0                     0;                     0                     0                     ω (2μ̃+λ̃)q_x^2+μ̃q_y^2                     0         (μ̃+λ̃)q_xq_y                     0      R/2(q_x^2-q_y^2)                     0               Rq_xq_y;                     0                 i q_y                     0                     0                     γ                    -ω                     0                     0                     0                     0;                     0                     0                     0         (μ̃+λ̃)q_xq_y                     ω (2μ̃+λ̃)q_y^2+μ̃q_x^2                     0              -Rq_xq_y                     0      R/2(q_x^2-q_y^2);                     0                     0                     0                     0                     0                     0                   γ_⊥                    -ω                     0                     0;                     0                     0                     0      R/2(q_x^2-q_y^2)                     0              -Rq_xq_y                     ω   K̃_1q_x^2+K̃_Tq_y^2                     0       (K_2+K_3)q_xq_y;                     0                     0                     0                     0                     0                     0                     0                     0                   γ_⊥                    -ω;                     0                     0                     0               Rq_xq_y                     0      R/2(q_x^2-q_y^2)                     0      (K_2+K_3)q_x q_y                     ω  K̃_T q_x^2+K̃_1q_y^2 ].



As we can observe the interaction matrix 𝐌(q,ω) separates in two independent blocks of 6×6 and 4×4 when R=0, i.e. in the case of the dodecagonal quasicrystal pattern. This case corresponds to the simplest scenario, in which phasons and phonons fields do not interact with each other.

Now we can proceed with our study of the low energy behavior of the system. The dispersion relation ω(q) corresponding to each excitation mode can be obtained by solving the equation [M(q,iω(q))]=0. The solution of this equation leads to two groups of solutions corresponding to the two different blocks of the matrix M(q,ω) in this case. The solutions arising from the 6×6 block of M(q,ω) are given by

    ω_1(q)   =   √(A+√(A^2+B))q
    ω_2(q)   =   √(A-√(A^2+B))q
    ω_3(q)   =   √(γμ̃)q,

where A=(ργ_0+γ(2μ̃+λ̃))/2 and B=(2μ̃+λ̃)γ_0(1-γρ). The form of the solutions indicates a hybridization process of the modes with dispersion relation ω_1(q) and ω_2(q), the analysis of eigenvectors associated to these eigenvalues allow us to conclude that these two excitation modes correspond to a hybridization of the longitudinal Bogoliubov mode with the longitudinal phonon mode, while ω_3(q) corresponds to the transversal phonon mode. A similar procedure for the 4×4 block of M(q,ω) yields the dispersion relations 

    ω_4(q)   =   √(γ_⊥K̃_1)q
    ω_5(q)   =   √(γ_⊥K̃_T)q,

corresponding to the longitudinal and transversal phason modes, respectively. 



§ III. FLUCTUATION EIGENSTATES OF THE DODECAGONAL QUASICRYSTAL

The longitudinal or transversal character of the excitation modes in the dodecagonal quasicrystal refers to the way in which the vectorial fields Π(x,τ), u(x,τ), Π_⊥(x,τ) and w(x,τ) oscillate respect to the momentum of the excitation q. For a better understanding of the nature of these modes we calculated the corresponding eigenvectors η̂_j(q) in a situation in which q is oriented along the "x"-axis, yielding 


    η̂_1(q_x)   =   { (c_1^2-γ(2μ̃+λ̃))c_1 q_x, -iγ_0(c_1^2-γ(2μ̃+λ̃)), γ_0(2μ̃+λ̃)q_x, -iγ_0c_1,0,0,0,0,0,0}
    η̂_2(q_x)   =   { -(c_2^2-γ_0ρ)c_2 q_x, iγ_0(c_2^2-γ_0ρ), γ_0(2μ̃+λ̃)q_x, -iγ_0c_2,0,0,0,0,0,0}
    η̂_3(q_x)   =   {0, 0, 0, 0, c_3 q_x, -iγ, 0, 0, 0, 0}
    η̂_4(q_x)   =   {0, 0, 0, 0, 0, 0, c_4 q_x, -i γ_⊥, 0, 0}
    η̂_5(q_x)   =   { 0, 0, 0, 0, 0, 0, 0, 0, c_5 q_x, -iγ_⊥},

where c_i=∂_q ϵ_i(q), refers to the sound velocity of the corresponding excitation mode. From here, the fluctuation fields in real space and time writes as η_j(x,t)=Re[η̂_j(q)exp(i(q·x-ω_jt))] and the correction to the ground state wave function as 

    δψ   =   √(ρ)(δ n_0(x,t)/2ρ c_0^2+ic_0θ(x,t)+1/2∑_n≠0c_n(G_n·Π(x,t)/Z+G_n,⊥·Π_⊥(x,t)/Z).
          .×exp(iG_n·x)+i∑_n≠0c_nexp(iG_n·x)× (θ(x,t)+G_n·u(x,t))+G_n,⊥·w(x,t)).).

Comparing this expression with the usual ansatz for the solution of the linearized Gross-Pitaevskii equation <cit.>, δψ_j,q=u_j,q(r)exp(i(q·r-ω_j(q)t))-v_j,q(r)^*exp(-i(q·r-ω_j(q)t)), we identify in each case the  functions u_j,q(r) and v_j,q(r). This allows us to compute the phase and density fluctuations for the excitation modes as Δρ_j,q=| u_j,q(r)-v_j,q(r)|^2 and Δφ_j,q=| u_j,q(r)+v_j,q(r)|^2 <cit.>. In Fig.<ref>(c-l) we show the behavior of these quantities considering typical parameters for the ground state wave function and for the pair interaction potential. For the numerical evaluation, the effective couplings of the theory can be roughly estimated using a single mode expansion for the wave function of the dodecagonal QC. In this case, we can write ϕ(x)=√(ρ)(c_0+c_1∑_j=1^12exp(i ke_j·x)), where {e_i} is a set of unitary vectors forming regular dodecagon. In the small c_1 limit we can reach to 

    Z   ≈   Z_⊥≈6ρ k^2c_1^2
    γ   ≈   γ_⊥≈ Uv̂(k)/(6c_1^2k^2)
    γ_0   ≈   Uv̂(0)
    
    (2μ̃+λ̃)   ≈   K̃_T≈6ρ k^2c_1^2+3Uρ^2c_1^2(kv̂'(k)+3k^2v̂”(k))
    μ̃   ≈   K̃_1≈6ρ k^2c_1^2+3Uρ^2c_1^2(3 kv̂'(k)+k^2v̂”(k)).

Considering U=1, ρ=0.4, k=1,  v̂(0)=2, v̂(k)=-1, v̂'(k)=0, v̂”(k)=1.5, c_0=1, c_1=0.15 and a momentum of the excitation q=0.1(1,0) we obtain the density and phase fluctuations patterns shown in Fig.1 (c-l).





§ IV. NUMERICAL CONSIDERATIONS FOR FIG. 2

For the numerical evaluation associated to Fig.2 (a) we considered the following set of arbitrary parameters ρ=4, γ_0=4, γ=1 γ_⊥=1, 2μ̃+λ̃=52, μ̃=20, K̃_1=24, K̃_T=56 and R=32.  

Moreover, as explained in the main text for Fig.2 (b) we need more realistic estimates of the behavior of the couplings of the effective theory in Eq.(3), these can be obtained from a single mode expansion, which for the decagonal quasicrystal, yields 

    Z   ≈   Z_⊥≈5ρ k^2c_1^2
    γ   ≈   γ_⊥≈ Uv̂(k)/(5c_1^2k^2)
    γ_0   ≈   Uv̂(0)
    
    (2μ̃+λ̃)   ≈   5ρ k^2c_1^2+5/2Uρ^2c_1^2(kv̂'(k)+3k^2v̂”(k))
    μ̃   ≈   5ρ k^2c_1^2+5/2Uρ^2c_1^2(3 kv̂'(k)+k^2v̂”(k))
    K̃_1=K̃_T   ≈   5ρ k^2c_1^2+5Uρ^2c_1^2(kv̂'(k)+k^2v̂”(k))
    
    R   ≈   5/2Uρ^2c_1^2(-kv̂(k)+k^2v̂”(k)).

Additionally, for numerical evaluations we use the anzats  c_1 (Uρ)=0.035+0.51(Uρ-0.2)/(Uρ)^0.65. This ansatz is only intended to capture the two main features of the dominant Fourier amplitude of the QC pattern, i.e. c_1(Uρ) is a strictly growing function defined from the QC phase boundary (Uρ=0.2). Secondly, c_1(Uρ) should be concave reflecting the tendency of the system to avoid collapse as Uρ is increased. Furthermore, for the pair interaction parameters we consider v̂(k)=-1, v̂'(k)=0, v̂”(k)=1 and v̂(0)=1.     




§ BEHAVIOR OF THE DENSITY AND PHASE LOCAL FLUCTUATIONS IN THE OCTAGONAL QUASICRYSTAL PATTERN


In this section we present the typical behavior of the local density and phase fluctuations for the octagonal quasicrystal for different orientations α of the propagation direction of the excitation respect to one of the main directions of the quasicrystal. For numerical calculations all the couplings of our effective theory takes the same values considered for the construction of Fig.(<ref>). 

The first case presented corresponds to the most general scenario for which all low energy modes are hybridized. As a consequence the scaling of the density and phase fluctuations is O(1) for all modes. For the numerical evaluation in this case we select α=π/16, the results are presented in Fig.(<ref>). Next we consider separately the cases of α=0 in Fig.(<ref>) and α=π/8 in Fig.(<ref>). For these special directions, as they correspond to symmetry directions of the octagonal quasicrystal structure, modes are purely longitudinal or transversal and the hybridization takes place only between modes in each of these separate groups. Because of this feature the scaling of the density and phase fluctuations for the longitudinal and transversal modes remains different in these cases.  


  









