A unified diagrammatic approach to topological fixed point models

A. Bauer1*,
J. Eisert1,
C. Wille2

1 Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

University of Cologne,
Zülpicher Straße 77,
50937 Cologne, Germany

August 27, 2024

§ ABSTRACT

We introduce a systematic mathematical language for describing fixed point models and apply it to the study to topological phases of matter. The framework established is reminiscent to that of state-sum models and lattice topological quantum field theories, but is formalized and unified in terms of tensor networks. In contrast to existing tensor network ansatzes for the study of ground states of topologically ordered phases, the tensor networks in our formalism directly represent discrete path integrals in Euclidean space-time. This language is more immediately related to the Hamiltonian defining the model than other approaches, via a Trotterization of the respective imaginary time evolution. We illustrate our formalism at hand of simple examples, and demonstrate its full power by expressing known families of models in 2+1 dimensions in their most general form, namely string-net models and Kitaev quantum doubles based on weak Hopf algebras. To elucidate the versatility of our formalism, we also show how fermionic phases of matter can be described and provide a framework for topological fixed point models in 3+1 dimensions.

§ INTRODUCTION

The phase of a physical model is central to understanding the its qualitative properties.

Studying quantum phases of matter has been a major task in physics ever since quantum many-body theory
was first formulated. This work addresses the question of classifying phases, that is, predicting which phases exist and providing models for each phase.

Intuitively speaking, a phase of matter corresponds to a parameter range
in which key physical properties of a material are essentially uniform. Over the last
more than a quarter of a century, condensed matter physicists have discovered a wealth
of new exotic phases of matter: Some of them are of emergent nature,
reflecting collective states of interacting quantum systems
that share little resemblance with the solids, liquids and gases of our commonplace
experience. The study of quantum phases of matter – and the
quest for classifying and even enumerating them in the first place – has, moreover,
gained considerable momentum since the discovery of phases other than symmetry-breaking phases, known as topologically ordered phases [1].

More concretely, two translationally invariant quantum lattice
models can be seen as being in the same phase if they are equivalent up to locally restructuring
their degrees of freedom. The fact that many properties are uniform within a given phase
is not only of conceptual importance. This uniformity also reflects their
usability for, say, the storage and processing of quantum information, the
simulatability by a classical computer, or its behaviour concerning thermalization. That is to say,
the quest for a solid understanding of phases of matter draws also inspiration and motivation
from practical and technological considerations.

Efforts aimed
at studying phases of matter can be roughly divided into three areas of research:
The first area is concerned with the
study of microscopic (commuting-projector) fixed-point models, which has been most successful for non-chiral phases
[2]. Such fixed-point models are paradigmatic models characteristic for a quantum phase as
fixed points of renormalization group transformations.
The second field is aimed at the
study of abstract invariant data describing phases, known as (non-fully extended axiomatic) TQFT, including anyon data for topological phases in $2+1$ dimensions, or cobordism data for invertible phases. Those approaches have been specifically
successful in targeting also chiral phases. However, they have the problem that it is hard to know whether given invariant data extends uniquely to a phase of microscopic models.
The third field concerns the detailed
study of specific microscopic models that are solvable by other means, such as quadratic fermionic Hamiltonians, or models solvable by means of perturbation analysis. This also includes the
study of models and their phases diagrams via extensive numerical analysis.

The present work aims at introducing a new picture of fixed-point models of quantum phases of matter
from a fresh perspective, deviating from the perspective put forth in Ref. [2]. In doing so, it makes progress concerning the first two approaches. Those approaches
usually use very complex mathematical tools (namely higher order category theory), which are hardly accessible to a broader physics community. The goal of this work is to introduce a comprehensive
systematic, unified, easily accessible, and generalized language for understanding and exploring fixed-point models for
quantum phases of matter. Subsequently, we list a number of new features of our approach, and
also stress what it does and what it does not deliver.

To start with, we formalize different kinds of physical models in terms of tensor networks
[3, 4, 5, 6]. In this sense, the approach
introduced here shares some resemblance
with the approach put forth in Refs. [7, 8, 9, 10, 11], in which tensor networks – specifically projected entangled pair states (PEPS)
[3]
– for instances of Levin-Wen models
[2] have been
devised. In one spatial dimension, the classification of phases with matrix product states (MPS) [12, 13, 14]
can be
considered largely complete [15, 16, 17].

The key difference, however, is that here, we do not
construct a tensor network representation of the ground state of a quantum model, living in the physical space of the model. Instead,
the connection to the Hamiltonian formulation of quantum mechanics is made by means of a suitable Trotter approximation of imaginary time evolution, yielding a tensor network in Euclidean space-time.
Note that, unlike the common usage of tensor networks for approximating ground states (via MPS or PEPS), those tensor networks do not distinguish between “virtual” and “physical” indices, and only have open indices at their space-time boundaries.

No known approach can in practice genuinely “classify” phases straight from their definition.
In our approach, we will rely on an extendibility hypothesis. That is, we will assume that the models we want to classify can be extended to “fixed point models” with a strong notion of deformability. For example consider the toric code on a square lattice. As such it is just particular instance of a microscopical model and defies a phase-classification. However, it can naturally be extended to a class of models defined on arbitrary lattices. This kind of deformability carries over to the discretized imaginary time evolution and is the central object of our formalism. While the deformability related to topological order is clearly the proto-typical example one should have in mind here, we emphasize, that the extendibility hypthosis and our formalism that builds upon it can be applied to various types of deformability. Each one corresponds to a different class of phase. For example apart from “topological order in $2+1$
dimensions”, other deformabilities/classes of phases would be “invertible topological order on spin manifolds”, “topological boundary of topological order”, “anyons in topological order”, or “conformal order”.

A specific notion of deformability (i.e., class of phases) will be formalized by a so-called liquid (or better, a liquid class), as we explain. A liquid can be seen as a prescription to construct combinatorial representations of space-times, called networks, together with a combinatorial representation of the space-time deformations, called moves. For large parts of this paper, the networks can be geometrically interpreted in terms of triangulations, and the moves correspond to local topology-preserving changes of the triangulations.

A network can be denoted using the same notation as it is commonly used for tensor networks and a move is formalized by a cut-and-paste operation, where a specific network is cut out from a larger network structure and replaced by another network. Invariance under such a deformation for any surrounding network structure implies that a move can also be interpreted as an equation between the cut and pasted tensor network. A solution to those equations will be called a model of the liquid. Note that a model in this sense is not a Hamiltonian, but a list of tensors that constitute a tensor network formalizing a discrete (Euclidean) path integral in space-time. However, for the liquids in the scope of this paper (but not always), we can obtain commuting-projector Hamiltonian models from the liquid models via standard constructions. The resulting models are known as fixed point models. If we want the models to have a standard quantum mechanical interpretation, we need to impose an additional Hermiticity constraint, which can also be interpreted as a move.

There are different types of matter formalized as “spin systems”, “systems with fermions”, “free fermionic systems”, “spin systems with time-reversal symmetry”, and different physical theories such as classical versus quantum physics. The type of matter and the interpretation as classical or quantum mechanical system is also part of the class of the phase. In our formalism, it is incorporated by using different a so-called tensor types. Different tensor types are different interpretations of the graphical calculus of tensor networks. A tensor type consists of a “data type” describing the tensors and two operations on this data, tensor contraction and tensor product. The standard interpretation of tensor networks, via Kronecker products and Einstein summations over indices of arrays, is one particular tensor type. It is the tensor type used to describe spin systems. In this introduction, we mostly focus on this tensor type, but we also encounter tensors with symmetries, fermionic tensors, and projective tensors.
The idea of tensor types is reminiscent of concepts in category theory: There, string diagrams can be interpreted in terms of an arbitrary monoidal category. In fact, tensor types are similar to compact closed categories.

We do not show how to find concrete models of a liquid, or how to “classify” (e.g., efficiently enumerate) them. This is a difficult mathematical problem that also is not addressed in other approaches either: Let us not forget that there is no  “classification of fusion categories”
known to date. However, we would like to stress that  most likely only the simplest phases will be realizable in physical systems and therefore practically relevant. For those it might be possible to find and explore them via exhaustive numerical search. In the simple case of conventional tensors, one might use a gradient descent, or a Gauss-Newton method to find solutions to the corresponding polynomial equations.

If the moves and the resulting equations are restrictive enough, the liquid models fall into discrete families (of models related by basis changes), and can be classified practically. Often we will find that that the different families are in different phases, though this is not always the case. We give a natural definition for when two models are in the same phase. It is a theoretically hard (and maybe even undecidable) problem to find out whether two models are in the same phase. However, in the world of fixed point models, it is often possible to “confirm by looking” that this is the case, and otherwise prove that it is not the case by computing invariants.
The following relational diagram summarizes the approach that we take here. In the course of this
work, the precise meaning of the terms used will become clear.

\begin{equation}
\begin{tikzpicture}
\node(mod)at(0,0){model};
\node(net)at(-1,2){network};
\node(mov)at(1,2){move};
\node(typ)at(-5,0){\breakcell{tensor\\type}};
\node(cla)at(-5,4){\breakcell{class of\\phases}};
\node(liq)at(0,4){liquid};
\node(pha)at(-3.3,1.6){phase};
\node(alg)at(4,4){\breakcell{algebraic\\structure}};
\node(ins)at(4,0){\breakcell{instance of\\alg. struct.}};
\begin{scope}[font=\small]
\draw (typ)edge[->]node[midway,below]{\breakcell{type of\\data}}(mod);
\draw (cla)edge[->]node[midway,left]{\breakcell{type of\\matter/description}}(typ);
\draw (cla)edge[->]node[midway,above]{\breakcell{represents\\deformability}}(liq);
\draw (liq)edge[->]node[midway,left]{\breakcell{building\\blocks}}(net);
\draw (liq)edge[->]node[midway,right,yshift=0.1cm]{\breakcell{contains\\different}}(mov);
\draw (net)edge[<-]node[midway,left,xshift=-0.15cm,yshift=0.2cm]{\breakcellr{associates\\data to}}(mod);
\draw (mov)edge[->]node[midway,right]{\breakcell{strongly constrains\\by equations}}(mod);
\draw (mod)edge[->]node[midway,left,yshift=-0.3cm]{\breakcell{modulo local\\restructuring}}(pha);
\draw (pha)edge[->]node[midway,right,yshift=0.2cm]{\breakcell{classify\\according to}}(cla);
\draw (liq)edge[->]node[midway,above]{similar to}(alg);
\draw (ins)edge[->]node[midway,right]{\breakcell{solution\\to axioms}}(alg);
\draw (mov)edge[->]node[midway,below,xshift=0.3cm]{\breakcell{similar\\to axioms}}(alg);
\draw (ins)edge[->]node[midway,below]{\breakcell{often\\inspire}}(mod);
\draw (mov)edge[->]node[midway,below]{deforms}(net);
\end{scope}
\end{tikzpicture}
\end{equation}

§ PHYSICAL SYSTEMS AS TENSOR NETWORKS

At the heart of our approach are tensor networks.
A tensor (in the conventional sense) is simply
a multi-dimensional array $A_{i,j,l,\ldots}$. Each of the indices $i,j,l,\ldots$ can take a finite number of values, e.g., $i\in \{0,\ldots,n_i-1\}$, where $n_i$ is called the dimension of the index. The values can be real or complex numbers. There are two operations we apply to tensors as

On the one hand,
the tensor product is the entry-wise product of two arrays, yielding an array with indices from both arrays,
acting as

\begin{equation}
(A\otimes B)_{i,j,\ldots,k,l,\ldots}=A_{i,j,\ldots}\cdot B_{k,l,\ldots}\;.
\end{equation}

On the other hand, the contraction is the Einstein summation over two indices with the same bond dimension, yielding a tensor with those two indices removed, acting as

\begin{equation}
A_{i,j,l,\ldots} \quad\mapsto
\quad ([A]_{i,l})_{j,\ldots}=\sum_x A_{x,j,x,\ldots}\;.
\end{equation}

Following the familiar Penrose notation,
a tensor can be graphically represented by a box with one line sticking out for each index, e.g.,

\begin{equation}
\begin{tikzpicture}
\node[box](x) at (0,0){$A$};
\draw (x.east)edge[dind=i]++(0:0.4) (x.north)edge[dind=j]++(90:0.4) (x.west)edge[dind=k]++(180:0.4);
\node at (0,-0.5){$\ldots$};
\end{tikzpicture}\;.
\end{equation}

A computation using tensor products and contractions can be represented by a network-like diagram: For every (copy of a) tensor, we draw (a copy of) the corresponding shape at an arbitrary location (possibly rotated or reflected). For every contraction between two indices, we connect the corresponding lines. E.g., the computation

\begin{equation}
\sum_{x,y,z,w} A_{x,y,i,z} A_{j,y,x,w} B_{w,z,k}\;
\end{equation}

could be represented by

\begin{equation}
\begin{tikzpicture}
\node[box](0) at (-0.8,0){$A$};
\node[box](1) at (0.8,0){$A$};
\node[circle,draw,inner sep=0.07cm](2) at (0,-0.8){$B$};
\draw (0)--(1) (0.north)to[out=90,in=90](1.north) (2)to[out=0,in=-90](1.south) (2)to[out=180,in=-90](0.south) (0.west)edge[dind=i]++(180:0.4) (1.east)edge[dind=j]++(0:0.4) (2.south)edge[dind=k]++(-90:0.4);
\end{tikzpicture}\;.
\end{equation}

In general, we might also use shapes without labels for tensors. For a more detailed introduction into tensor networks and the corresponding notation, see, e.g.,
Section 2.1 in Ref. [18].

The language of tensor networks can be used to formalize local physical models. Readers familiar with tensor networks might think of MPS or PEPS approximating the ground states of gapped local quantum Hamiltonians. It is neither proven in full generality nor completely clear that ground states can be approximated by MPS or PEPS. Also, being gapped is a strong constraint. Thus, this is not the way in which tensor networks are used to describe quantum systems in the approach taken here. Instead we use a simpler and more direct translation by writing the (imaginary) time evolution of a system as a tensor network in space time.

In order to get a tensor network from a continuous (imaginary) time evolution of a quantum many-body system with a local translation-invariant Hamiltonian we have to discretize it using a procedure known as Trotterization. As a simplest non-trivial example, let us consider a $1$-dimensional quantum spin chain with a translation-invariant nearest-neighbour Hamiltonian

\begin{equation}
H=\sum_i h_{i,i+1},
\end{equation}

where $h_{i,i+1}$ is the $2$-spin Hamiltonian acting on the spins $i,i+1$. Global properties such as e.g. the boundary conditions are not relevant for the following considerations, which are applied to some small patch in the bulk of the translation-invariant model.

We can divide the Hamiltonian terms into the ones acting on even-odd site pairs and those acting on odd-even site pairs

\begin{equation}
H=\sum_i h_{2i,2i+1} + \sum_i h_{2i+1,2i+2}=H_1+H_2.
\end{equation}

All terms within $H_1$ act non-trivially only on non-overlapping sets of spins and therefore commute. The same holds for $H_2$. $H_1$ and $H_2$ however do not commute, and therefore

\begin{equation}
e^{it(H_1+H_2)}\neq e^{itH_1}e^{itH_2} .
\end{equation}

We can still use the following Suzuki-Trotter expansion

\begin{equation}
e^{it(H_1+H_2+\ldots)}= \lim_{n\rightarrow\infty} \left(e^{i\frac{t}{n}H_1}e^{i\frac{t}{n}H_2}\ldots\right)^n
\end{equation}

to obtain

\begin{equation}
\begin{aligned}
e^{itH}= \lim_{n\rightarrow\infty} \left(\prod_ie^{i\frac{t}{n}h_{2i,2i+1}} \prod_i e^{i\frac{t}{n}h_{2i+1,2i+2}}\ldots\right)^n .
\end{aligned}
\end{equation}

Consider the expression on the right-hand side for a fixed $n$. It is a product of operators acting on different degrees of freedom. $e^{ith/n}$ is a linear operator acting on two spins, so it is a tensor with $4$ indices, two for both input and output of the operator

\begin{equation}
e^{i\frac{t}{n}h}\quad \rightarrow\quad
\begin{tikzpicture}
\atoms{trotstep}{0/}
\draw (0)--++(60:0.5) (0)--++(120:0.5) (0)--++(-60:0.5) (0)--++(-120:0.5);
\end{tikzpicture}\;.
\end{equation}

With this interpretation, the product of operators becomes a tensor network. E.g.,
for $n=3$, we get the network

\begin{equation}
\begin{tikzpicture}
\foreach \y in {0,1,2}{
\foreach \x in {0,1,2,3}{
\atoms{trotstep}{{a\x\y/p={\x*0.8,\y*0.8}}}
\foreach \y in {0,1,2}{
\foreach \x in {0,1,2,3}{
\atoms{trotstep}{{b\x\y/p={\x*0.8+0.4,\y*0.8+0.4}}}
\draw (a00)--(b00) (a10)--(b00) (a10)--(b10) (a20)--(b10) (a20)--(b20) (a30)--(b20) (a30)--(b30);
\draw (a01)--(b00) (a11)--(b00) (a11)--(b10) (a21)--(b10) (a21)--(b20) (a31)--(b20) (a31)--(b30);
\draw (a01)--(b01) (a11)--(b01) (a11)--(b11) (a21)--(b11) (a21)--(b21) (a31)--(b21) (a31)--(b31);
\draw (a02)--(b01) (a12)--(b01) (a12)--(b11) (a22)--(b11) (a22)--(b21) (a32)--(b21) (a32)--(b31);
\draw (a02)--(b02) (a12)--(b02) (a12)--(b12) (a22)--(b12) (a22)--(b22) (a32)--(b22) (a32)--(b32);
\draw (a00)--++(-135:0.3) (a00)--++(-45:0.3) (a10)--++(-135:0.3) (a10)--++(-45:0.3) (a20)--++(-135:0.3) (a20)--++(-45:0.3) (a30)--++(-135:0.3) (a30)--++(-45:0.3);
\draw (b02)--++(135:0.3) (b02)--++(45:0.3) (b12)--++(135:0.3) (b12)--++(45:0.3) (b22)--++(135:0.3) (b22)--++(45:0.3) (b32)--++(135:0.3) (b32)--++(45:0.3);
\draw (a00)edge[enddots]++(135:0.2) (a01)edge[enddots]++(-135:0.2) (a01)edge[enddots]++(135:0.2) (a02)edge[enddots]++(-135:0.2) (a02)edge[enddots]++(135:0.2);
\draw (b30)edge[enddots]++(45:0.2) (b30)edge[enddots]++(-45:0.2) (b31)edge[enddots]++(-45:0.2) (b31)edge[enddots]++(45:0.2) (b32)edge[enddots]++(-45:0.2);
\end{tikzpicture}\;.
\end{equation}

The tensor network we are looking for should have the same notion of locality structure as the continuum time evolution. That is, every tensor should correspond to a finite space-time volume $\Delta x\times \Delta t$. So it does not make sense to directly take the above tensor network, as the time interval corresponding to a tensor scales like $1/n$. In addition, this tensor network has a trivial limit for $n\rightarrow\infty$

\begin{equation}
\begin{tikzpicture}
\atoms{trotstep}{{0/}}
\draw (0)--++(60:0.5) (0)--++(120:0.5) (0)--++(-60:0.5) (0)--++(-120:0.5);
\end{tikzpicture}=
e^{i\frac{t}{n}h}\quad \xrightarrow{n\rightarrow \infty}\quad
\mathbb{1}=
\begin{tikzpicture}
\draw (0,0)--(0,0.8) (0.4,0)--(0.4,0.8);
\end{tikzpicture}\;.
\end{equation}

Instead, we have to pick a fixed $\Delta t$, Trotterize $e^{i\Delta tH}$ for some fixed $n$ and divide the resulting tensor networks into spatial unit cells.
Evaluating the whole tensor-network patch inside the unit cell yields the tensor $P_n$ of the tensor network we are looking for. E.g., for $n=3$ and $\Delta x$ consisting of $4$ sites, we can choose

\begin{equation}
\begin{tikzpicture}
\atoms{trottens}{t/lab={t=P_3,p=-45:0.5}}
\draw (t-t)edge[dind=v'w'x'y]++(0,0.5);
\draw (t-b)edge[dind=vwxy]++(0,-0.5);
\draw (t-l)edge[dind=abcdef]++(-0.5,0);
\draw (t-r)edge[dind=a'b'c'd'e'f']++(0.5,0);
\end{tikzpicture}=
\begin{tikzpicture}
\foreach \y in {0,1,2}{
\foreach \x in {0,1}{
\atoms{trotstep}{{a\x\y/p={\x*0.8,\y*0.8}}}
\foreach \y in {0,1,2}{
\foreach \x in {0,1}{
\atoms{trotstep}{{b\x\y/p={\x*0.8+0.4,\y*0.8+0.4}}}
\draw (a00)--(b00) (a10)--(b00) (a10)--(b10);
\draw (a01)--(b00) (a11)--(b00) (a11)--(b10);
\draw (a01)--(b01) (a11)--(b01) (a11)--(b11);
\draw (a02)--(b01) (a12)--(b01) (a12)--(b11);
\draw (a02)--(b02) (a12)--(b02) (a12)--(b12);
\draw[dind=v,rounded corners] (a00)--++(-135:0.3)--++(-90:0.2);
\draw[dind=w,rounded corners] (a00)--++(-45:0.3)--++(-90:0.2);
\draw[dind=x,rounded corners] (a10)--++(-135:0.3)--++(-90:0.2);
\draw[dind=y,rounded corners] (a10)--++(-45:0.3)--++(-90:0.2);

\draw[dind=w',rounded corners] (b02)--++(135:0.3)--++(90:0.2);
\draw[dind=x',rounded corners] (b02)--++(45:0.3)--++(90:0.2);
\draw[dind=y',rounded corners] (b12)--++(135:0.3)--++(90:0.2);
\draw[dind=v',rounded corners] ($(0,3*0.8)+(-135:0.3)+(180:0.2)$)-|($(-0.4,0.4*5)+(45:0.3)+(90:0.2)$); % both labels at once doesn't work due to tikz bug
\draw[mark={lab=$f'$,b},rounded corners] ($(0,3*0.8)+(-135:0.3)+(180:0.2)$)-|($(-0.4,0.4*5)+(45:0.3)+(90:0.2)$);

\draw[dind=a,rounded corners] (a00)--++(135:0.3)--++(180:0.2);
\draw[dind=b,rounded corners] (a01)--++(-135:0.3)--++(180:0.2);
\draw[dind=c,rounded corners] (a01)--++(135:0.3)--++(180:0.2);
\draw[dind=d,rounded corners] (a02)--++(-135:0.3)--++(180:0.2);
\draw[dind=e,rounded corners] (a02)--++(135:0.3)--++(180:0.2);

\draw[dind=a',rounded corners] (b10)--++(-45:0.3)--++(0:0.2);
\draw[dind=b',rounded corners] (b10)--++(45:0.3)--++(0:0.2);
\draw[dind=c',rounded corners] (b11)--++(-45:0.3)--++(0:0.2);
\draw[dind=d',rounded corners] (b11)--++(45:0.3)--++(0:0.2);
\draw[dind=e',rounded corners] (b12)--++(-45:0.3)--++(0:0.2);
\draw[dind=f',rounded corners] (b12)--++(45:0.3)--++(0:0.2);
\end{tikzpicture}\;.
\end{equation}

This blocked tensor does not have a sensible large $n$-limit as well. If we let $n\rightarrow \infty$, the number of indices to block on the right and left, and therefore the bond dimension of those indices, grows with $n$. More precisely, the number of blocked indices grows linearly, and thus the bond dimension increases exponentially with $n$.

It turns out, though, that the tensor can be well approximated by a tensor with much lower bond dimension. We conjecture the following. There is a sequence of tensors $P^x$ of (horizontal) bond dimension $x$, numbers numbers $\alpha_x$, and integers $N_x$. Additionally, for every $x$, there is a sequence of left-invertible matrices $I^x_n$ (i.e., there is a matrix $(I^x_n)^{-1}$ such that $(I^x_n)^{-1}I^x_n=\mathbb{1}$), such that for all $x$,
we have

\begin{equation}
\label{eq:trotter_truncation}
\begin{gathered}
% not needed?
% \left\|
% \begin{tikzpicture}
% \piclab{t}{square}{(0,0)}{(-45:0.5)}{$P^x$};
% \draw (t-t)edge[dind=v']++(0,0.5);
% \draw (t-b)edge[dind=v]++(0,-0.5);
% \draw (t-l)edge[dind=a]++(-0.5,0);
% \draw (t-r)edge[dind=a']++(0.5,0);
% \end{tikzpicture}-
% \begin{tikzpicture}
% \piclab{t}{square}{(0,0)}{(-50:0.5)}{$P_n$};
% \piclab{i0}{square}{(0.7,0)}{(90:0.4)}{$I^x_n$};
% \piclab{i1}{square}{(-0.7,0)}{(90:0.4)}{$(I^x_n)^{-1}$};
% \draw (t-r)--(i0-l) (t-l)--(i1-r);
% \draw (t-t)edge[dind=v']++(0,0.5);
% \draw (t-b)edge[dind=v]++(0,-0.5);
% \draw (i1-l)edge[dind=a]++(-0.3,0);
% \draw (i0-r)edge[dind=a']++(0.3,0);
% \end{tikzpicture}
% \right\|<\alpha_x\\
\left\|
\begin{tikzpicture}
\atoms{square}{{t/lab={t=$P_n$,p=-45:0.5}}}
\draw (t-t)edge[dind=v']++(0,0.5);
\draw (t-b)edge[dind=v]++(0,-0.5);
\draw (t-l)edge[dind=a]++(-0.5,0);
\draw (t-r)edge[dind=a']++(0.5,0);
\end{tikzpicture}-
\begin{tikzpicture}
\atoms{trottens}{t/lab={t=P^x,p=-50:0.5}, {i0/p={0.7,0},lab={t=(I^x_n)^{-1},p=90:0.4}}, {i1/p={-0.7,0},lab={t=I^x_n,p=90:0.4}}}
\draw (t-r)--(i0-l) (t-l)--(i1-r);
\draw (t-t)edge[dind=v']++(0,0.5);
\draw (t-b)edge[dind=v]++(0,-0.5);
\draw (i1-l)edge[dind=a]++(-0.3,0);
\draw (i0-r)edge[dind=a']++(0.3,0);

\end{tikzpicture}
\right\|<\alpha_x, \quad \forall n>N_x,\\
\left\|
\begin{tikzpicture}
\atoms{trottens}{{t/lab={t=P^x,p=-45:0.5}}}
\draw (t-t)edge[dind=v']++(0,0.5);
\draw (t-b)edge[dind=v]++(0,-0.5);
\draw (t-l)edge[dind=a]++(-0.5,0);
\draw (t-r)edge[dind=a']++(0.5,0);
\end{tikzpicture}-
\begin{tikzpicture}
\atoms{trottens}{{t/lab={t=P^{x+1},p=-40:0.55}}}
\draw (t-t)edge[dind=v']++(0,0.5);
\draw (t-b)edge[dind=v]++(0,-0.5);
\draw (t-l)edge[dind=a]++(-0.5,0);
\draw (t-r)edge[dind=a']++(0.5,0);
\end{tikzpicture}
\right\|<\alpha_x,\\
\alpha_x\sim e^{-x}.
\end{gathered}\;
\end{equation}

To be concrete, we fix the norm above to be the $2$-norm of the tensors (when reshaped into vectors), though we do not provide a particularly good reason for this choice, and we
would expect other reasonable norms to work as well. This conjecture is based on observations in MPS algorithms like iTEBD, where we apply the Trotterized time-evolution to an MPS. In general, the bond dimension of the MPS might grow exponentially with time. However, the rate of growth does not depend on the Trotterization step we choose. After time evolution for a fixed time interval we see an eigenvalue spectrum in the MPS transfer operator that decays exponentially.

After picking a bond dimension which we consider a sufficient approximation $P$, the (imaginary) time evolution is given by a square lattice tensor network

\begin{equation}
\label{eq:trotter_resulting_network}
\begin{tikzpicture}
\atoms{square,lab={t=$P$,p=-135:0.45}}{0/p={0.5,0.5}, 1/p={1.5,0.5}, 2/p={0.5,1.5}, 3/p={1.5,1.5}}
\draw (0)--(1) (0)--(2) (2)--(3) (1)--(3) (0)edge[enddots]++(180:0.5) (0)edge[enddots]++(-90:0.5) (1)edge[enddots]++(0:0.5) (1)edge[enddots]++(-90:0.5) (2)edge[enddots]++(180:0.5) (2)edge[enddots]++(90:0.5) (3)edge[enddots]++(0:0.5) (3)edge[enddots]++(90:0.5);
\end{tikzpicture} \;.
\end{equation}

It is easy to see how to generalize the Trotterization procedure to other geometric setups, such as higher dimensions, higher spatial support of the Hamiltonian terms, or presence of boundaries or defects of any kind. First we divide the terms into a constant (system-size independent) number of subsets, such that the terms in one subset all commute with each other. Then we proceed using the Suzuki-Trotter expansion applied to the division into subsets, resulting in a tensor network, which we block and truncate into finite unit cells.

Also local classical/quantum thermal models can be written as tensor networks. In the classical case, the partition function is represented by a tensor network made of Boltzmann weights and delta tensors, without any approximation. In the quantum thermal case, we Trotterize the imaginary time evolution, with imaginary time compactified into a loop of circumference $\beta$, where $\beta$ is the inverse temperature. In both cases, we get a tensor network living in space instead of space time.

Quantum phases of matter are phases of ground states. Thus, the relevant tensor network to consider for the study of quantum phases is the one arising from the imaginary, not the real, time evolution: The operator $e^{-\beta H}$ converges to the ground state projector for a gapped system, if we scale both the inverse temperature/imaginary time $\beta$ and the system size simultaneously.

The evaluation of a tensor network can be seen as a “computation” in the following way. The input data to the computation are the tensors. The “network” itself describes the combinatorial nature of the computation, i.e., in which order and to which components the individual operations (such as copying tensors, evaluating Kronecker products, or Einstein summations) are applied. The graphical notation makes sense, because the steps of the computation obey certain “axioms”, e.g., different contractions commute with each other, or, the Kronecker product is associative.

There are types of data other than arrays, and types of operations other than Kronecker products and Einstein summations, which obey the same axioms. In other words, there are other structures which yield different interpretations for the same graphical network notation. We will call such other structures tensor types [19]. We will later see that, e.g., models with fermions can be formalized by using a different tensor type in Section <ref>. In fact, we have already encountered two different tensor types, namely complex tensors and real tensors.

§ PHASES OF MATTER AND THE EXTENDIBILITY HYPOTHESIS

Conventionally, quantum phases of matter are equivalence classes of local translation-invariant gapped Hamiltonians $H\in \mathcal{H}$. By gapped, we mean that there is an integer $g\geq 0$ called ground state degeneracy and a real number $\epsilon>0$ called the gap, such that for every system size $n$ (greater than some $n_0$), the $g$ lowest eigenvalues of $H$ are separated from the rest of the spectrum by $\epsilon$, and among each other by $\beta_n$ such that $\beta_n\rightarrow 0$ for $n\rightarrow \infty$. Two gapped Hamiltonians $H_1,H_2\in \mathcal{H}$ are considered equivalent if there is a continuous path of gapped Hamiltonians connecting $H_1$ and $H_2$ [1]:

\begin{equation}
\begin{gathered}
\widetilde{H}: [0,1] \rightarrow \mathcal{H} , \\
\widetilde{H}(0)=H_1,\quad \widetilde{H}(1)=H_2 .
\end{gathered}
\end{equation}

If we aim at comparing two Hamiltonians with different local Hilbert spaces, we can arbitrarily embed both into the same local Hilbert space and use the same definition. Note that all Hamiltonians $\widetilde{H}$ are supposed to be gapped, otherwise one speaks of a “gap closing” inducing a “phase transition”.

This definition is clear but not very constructive: Checking all possible paths between two Hamiltonians is practically infeasible, and the fact that a path is something continuous does not make it easier. Moreover, there is evidence that it might
not even be decidable in general, whether a family of Hamiltonians is uniformly gapped for different system sizes [20].

Another definition, which is different in its formulation but equivalent to the previous one has been identified via the concept of quasi-adiabatic evolution [21]: Roughly, two Hamiltonians are considered to be in the same phase if they are related via a finite-depth local (generalized) unitary circuit [22]. However, two “generic” Hamiltonians will never be exactly related by such a circuit. Starting from $H_1$, we can only get closer and closer to $H_2$, if we increase the locality of the circuit (in fact one typically expects that one can get exponentially close in the locality). Of course, apart from those additional technicalities related to exponential tails, we would still have to search all possible unitary circuits in order to find out whether two Hamiltonians are in the same phase. However, we have replaced finding a continuous path to finding a discrete circuit, which makes the problem more accessible for exhaustive numerical approaches in practice.

Quantum phases, once again, describe the ground state properties of Hamiltonians. Such ground states can be obtained directly from the Hamiltonian by applying the imaginary time evolution to some initial state vector $\ket{x}$ as

\begin{equation}
\lim_{t\rightarrow\infty} e^{-tH} \ket{x}.
\end{equation}

However, for any finite system size, the lowest eigenvalue will not generically have a $g$-fold degeneracy, but the “ground states” will have slightly different energies. So, at a particular system size, $e^{-tH}$ will not converge to a ground state projector with $g$-dimensional support, but to the projector on the lowest eigenstate only. We see that in order to talk about ground states, we do not only have to scale the imaginary time $t$, but also simultaneously the system size $n$. Thus, it is natural to represent the model as something living in euclidean space-time. We consequently choose to represent models by their imaginary-time evolution tensor network instead of their Hamiltonians.

The tensor network representation comes with a very natural definition of a phase. If we conjugate Hamiltonian terms by a unitary operator, the tensors in the imaginary-time evolution get conjugated in the same way. So, e.g., if we conjugate a $1+1$-dimensional system as in Section <ref> by an on-site unitary $U^{\otimes N}$, the tensor $P$ in Eq. (<ref>) gets conjugated by $U$,

\begin{equation}
\label{eq:conjugated_tensor}
\begin{tikzpicture}
\atoms{square}{0/lab={t=$\widetilde{P}$,p=-135:0.45}}
\draw (0)--++(180:0.5) (0)--++(90:0.5) (0)--++(-90:0.5) (0)--++(0:0.5);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{square}{0/lab={t=$P$,p=-135:0.45}}
\atoms{circ,small}{{2/lab={t=$U$,p=0:0.3},p={0,-0.5}}, {1/lab={t=$U^\dagger$,p=0:0.35},p={0,0.5}}}
\draw (0)--(1) (0)--(2) (0)--++(180:0.5) (1)--++(90:0.5) (2)--++(-90:0.5) (0)--++(0:0.5);
\end{tikzpicture}\;.
\end{equation}

$U$ is unitary, which can be denoted in network notation by

\begin{equation}
\label{eq:unitary}
\begin{tikzpicture}
\atoms{circ,small}{0/lab={t=$U$,p=0:0.3}, {1/lab={t=$U^\dagger$,p=0:0.35},p={0,0.5}}}
\draw (0)--(1) (0)--++(0,-0.4) (1)--++(0,0.4);
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,0)--(0,0.7);
\end{tikzpicture}\;.
\end{equation}

Imagine starting from the conjugated square lattice tensor network. Now, replace every occurrence of the conjugated tensor $\widetilde{P}$ by its definition in terms of Eq. (<ref>). This will create a pair of $U$ and $U^\dagger$, that is, an occurrence of the left hand side of Eq. (<ref>), at every bond of the original network. We can remove those pairs by replacing each occurrence with the right hand side of Eq. (<ref>), yielding the non-conjugated network

\begin{equation}
\label{eq:conjugated_tensor_network}
\begin{tikzpicture}
\atoms{square,lab={t=$\widetilde{P}$,p=-135:0.45}}{0/p={0.5,0.5}, 1/p={1.5,0.5}, 2/p={0.5,1.5}, 3/p={1.5,1.5}}
\draw (0)--(1) (0)--(2) (2)--(3) (1)--(3) (0)edge[enddots]++(180:0.5) (0)edge[enddots]++(-90:0.5) (1)edge[enddots]++(0:0.5) (1)edge[enddots]++(-90:0.5) (2)edge[enddots]++(180:0.5) (2)edge[enddots]++(90:0.5) (3)edge[enddots]++(0:0.5) (3)edge[enddots]++(90:0.5);
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}
\atoms{square}{0/p={0,0}, 1/p={1,0}, 2/p={0,1.5}, 3/p={1,1.5}}
\atoms{circ,small,lab={t=$U$,p=0:0.3}}{u0/p={0,0.5}, u1/p={1,0.5}, u2/p={0,2}, u3/p={1,2}}
\atoms{circ,small,lab={t=$U^\dagger$,p=0:0.35}}{u0d/p={0,-0.5}, u1d/p={1,-0.5}, u2d/p={0,1}, u3d/p={1,1}}
\draw (0)--(1) (0)--(u0)--(u2d)--(2) (2)--(3) (1)--(u1)--(u3d)--(3) (0)--(u0d) (1)--(u1d) (2)--(u2) (3)--(u3);
\draw (0)edge[enddots]++(180:0.5) (u0d)edge[enddots]++(-90:0.4) (1)edge[enddots]++(0:0.5) (u1d)edge[enddots]++(-90:0.4) (2)edge[enddots]++(180:0.5) (u2)edge[enddots]++(90:0.4) (3)edge[enddots]++(0:0.5) (u3)edge[enddots]++(90:0.4);
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}
\atoms{square,lab={t=$P$,p=-135:0.45}}{0/p={0.5,0.5}, 1/p={1.5,0.5}, 2/p={0.5,1.5}, 3/p={1.5,1.5}}
\draw (0)--(1) (0)--(2) (2)--(3) (1)--(3) (0)edge[enddots]++(180:0.5) (0)edge[enddots]++(-90:0.5) (1)edge[enddots]++(0:0.5) (1)edge[enddots]++(-90:0.5) (2)edge[enddots]++(180:0.5) (2)edge[enddots]++(90:0.5) (3)edge[enddots]++(0:0.5) (3)edge[enddots]++(90:0.5);
\end{tikzpicture}\;
\end{equation}

in Eq. (<ref>).
In the conventional Hamiltonian picture, it is clear that conjugating a model by an on-site unitary does not change the phase. As we have seen, in our tensor-network picture such an operation corresponds to “rewriting” the tensor network by replacing subnetworks with other subnetworks which evaluate to the same tensor. We will call such a replacement operation a circuit move, and define two tensor networks to be in the same phase if they are related by a circuit move.

The above reasoning also works if $U$ is not an on-site operator, but is only a product of operators acting on constant-size non-overlapping patches. Furthermore, it suffices if $U$ is an isometry rather than a unitary, such that it can also change the local Hilbert space dimension. Last, we can conjugate by more than one layer of unitaries. So for any two Hamiltonians related by a finite-depth generalized local unitary circuit, the corresponding tensor networks are related by a circuit move. But circuit moves also go beyond conjugation by unitaries. Consider the following example for a different circuit move acting on square lattices. First, we split each tensor into a network consisting of $4$ tensors:

\begin{equation}
\begin{tikzpicture}
\atoms{square,lab={t=$\widetilde{P}$,p=-135:0.45}}{0/}
\draw (0)--++(0:0.5) (0)--++(180:0.5) (0)--++(90:0.5) (0)--++(-90:0.5);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{small,circ}{a/p={-0.4,0}, b/p={0.4,0}, c/p={0,-0.4}, d/p={0,0.4}}
\draw[line width=2] (a)--(c) (c)--(b) (b)--(d) (d)--(a);
\draw (b)--++(0:0.4) (a)--++(180:0.4) (d)--++(90:0.4) (c)--++(-90:0.4);
\end{tikzpicture}\;.
\end{equation}

The dimension of the bonds between the new tensors can be different, which we reflect by using a different line style. At every bond of the original network, we will get two of the new $3$-index tensors. We replace those two by two other tensors, connected by a bond perpendicular to the old bond

\begin{equation}
\begin{tikzpicture}
\atoms{small,circ}{0/p={-0.3,0}, 1/p={0.3,0}}
\draw[line width=2] (0)--++(135:0.4) (0)--++(-135:0.4) (1)--++(45:0.4) (1)--++(-45:0.4);
\draw (0)--(1);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{small,circ,dcross}{0/p={0,-0.3}, 1/p={0,0.3}}
\draw[line width=2] (0)--++(-45:0.4) (0)--++(-135:0.4) (1)--++(45:0.4) (1)--++(135:0.4);
\draw[densely dotted,line width=1.5] (0)--(1);
\end{tikzpicture}\;.
\end{equation}

Now, at every vertex of the square lattice there are $4$ tensors on the adjacent edges. We can block those into a single tensor again according to

\begin{equation}
\begin{tikzpicture}
\atoms{small,circ,dcross}{a/p={-0.4,0}, b/p={0.4,0}, c/p={0,-0.4}, d/p={0,0.4}}
\draw[line width=2] (a)--(c) (c)--(b) (b)--(d) (d)--(a);
\draw[densely dotted,line width=1.5] (b)--++(0:0.4) (a)--++(180:0.4) (d)--++(90:0.4) (c)--++(-90:0.4);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{square,lab={t=$P$,p=-135:0.45}}{0/}
\draw[densely dotted,line width=1.5] (0)--++(0:0.5) (0)--++(180:0.5) (0)--++(90:0.5) (0)--++(-90:0.5);
\end{tikzpicture}\;.
\end{equation}

Applying this circuit move, we obtain a network whose tensors are now located at the positions where the vertices of the old square lattice have been previously,

\begin{equation}
\begin{tikzpicture}
\draw[dashed,orange] (-0.2,0)--(2.2,0) (-0.2,1)--(2.2,1) (-0.2,2)--(2.2,2);
\draw[dashed,orange] (0,-0.2)--(0,2.2) (1,-0.2)--(1,2.2) (2,-0.2)--(2,2.2);
\atoms{square,lab={t=$\widetilde{P}$,p=-135:0.45}}{0/p={0.5,0.5}, 1/p={1.5,0.5}, 2/p={0.5,1.5}, 3/p={1.5,1.5}}
\draw (0)--(1) (0)--(2) (2)--(3) (1)--(3) (0)--++(180:0.7) (0)--++(-90:0.7) (1)--++(0:0.7) (1)--++(-90:0.7) (2)--++(180:0.7) (2)--++(90:0.7) (3)--++(0:0.7) (3)--++(90:0.7);
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}
\draw[dashed,orange] (-0.2,0)--(2.2,0) (-0.2,1)--(2.2,1) (-0.2,2)--(2.2,2);
\draw[dashed,orange] (0,-0.2)--(0,2.2) (1,-0.2)--(1,2.2) (2,-0.2)--(2,2.2);
\foreach \a/\b in {0/0, 0/1, 1/0, 1/1}{
\atoms{small,circ}{a\a\b/p={\a+0.25,\b+0.5}, b\a\b/p={\a+0.75,\b+0.5}, c\a\b/p={\a+0.5,\b+0.25}, d\a\b/p={\a+0.5,\b+0.75}}
\draw (a\a\b)--(c\a\b) (c\a\b)--(b\a\b) (b\a\b)--(d\a\b) (d\a\b)--(a\a\b);
\draw (d00)--(c01) (b00)--(a10) (d10)--(c11) (b01)--(a11);
\draw (a00)--++(180:0.45) (c00)--++(-90:0.45) (c10)--++(-90:0.45) (b10)--++(0:0.45) (a01)--++(180:0.45) (d01)--++(90:0.45) (d11)--++(90:0.45) (b11)--++(0:0.45);
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}
\draw[dashed,orange] (-0.2,0.45)--(2.2,0.45) (-0.2,1.45)--(2.2,1.45);
\draw[dashed,orange] (0.45,-0.2)--(0.45,2.2) (1.45,-0.2)--(1.45,2.2);
\foreach \a/\b in {0/0, 0/1, 1/0, 1/1}{
\atoms{small,circ,dcross}{a\a\b/p={\a+0.25,\b+0.5}, b\a\b/p={\a+0.75,\b+0.5}, c\a\b/p={\a+0.5,\b+0.25}, d\a\b/p={\a+0.5,\b+0.75}}
\draw (a\a\b)--(c\a\b) (c\a\b)--(b\a\b) (b\a\b)--(d\a\b) (d\a\b)--(a\a\b);
\draw (d00)--(c01) (b00)--(a10) (d10)--(c11) (b01)--(a11);
\draw (a00)--++(180:0.45) (c00)--++(-90:0.45) (c10)--++(-90:0.45) (b10)--++(0:0.45) (a01)--++(180:0.45) (d01)--++(90:0.45) (d11)--++(90:0.45) (b11)--++(0:0.45);
\end{tikzpicture}
\rightarrow
\begin{tikzpicture}
\draw[dashed,orange] (-0.2,0.45)--(2.2,0.45) (-0.2,1.45)--(2.2,1.45);
\draw[dashed,orange] (0.45,-0.2)--(0.45,2.2) (1.45,-0.2)--(1.45,2.2);
\atoms{square,lab={t=$P$,p=-135:0.45}}{0/p={0.5,0.5}, 1/p={1.5,0.5}, 2/p={0.5,1.5}, 3/p={1.5,1.5}}
\draw (0)--(1) (0)--(2) (2)--(3) (1)--(3) (0)--++(180:0.7) (0)--++(-90:0.7) (1)--++(0:0.7) (1)--++(-90:0.7) (2)--++(180:0.7) (2)--++(90:0.7) (3)--++(0:0.7) (3)--++(90:0.7);
\end{tikzpicture}\;.
\end{equation}

Note that also the definition via circuit moves has the difficulty of having to deal with exponential tails, and we have not found the most natural way to do this yet.

No known approach can classify general phases starting from a fundamental definition. In our approach, we will instead rely on an observation which we call the extendibility hypothesis: Models coming from condensed matter physics are typically defined on regular spatial grids, times a linear time. That is, they are defined in a flat Euclidean space time. Sometimes one can extend the definition of such a model from only flat space times to arbitrary curved space time. Moreover, there is a very powerful set of deformations under which the extended model is invariant. The extendibility hypothesis is the (admittedly vague) assumption that, the more “generic” a model is, the more “powerful” are the deformations that it is invariant under after extension.

The most relevant example for extendibility are models with topological or symmetry-breaking order: Such models can be extended from flat space to arbitrary topological manifolds, such that the extended model is invariant under arbitrary homeomorphisms, i.e. deformations leaving the topology invariant. Irreducible topological phases
are known to be robust under perturbations [23] and generic in that sense. The “second most generic” type of models are those at transitions between two topological or symmetry-breaking phases. Such critical models are known to be described by CFTs. That is, they can be extended to manifolds with a conformal metric, and the extension is invariant under arbitrary transformations that leave this conformal structure invariant. On the other hand, there are also models with a more powerful extendibility than topological phases: E.g., models with invertible topological order can be extended to topological manifolds, and the extension is not only invariant under homeomorphisms but additionally under arbitrary surgery moves. Equivalently, those manifolds are not defined up to homoemorphism, but only up to cobordisms.

Extendibility helps classifying phases of matter in the following way: The invariance of the extended models under deformations is very restrictive. This makes it possible to “classify” such extended models by identifying them with instances of some algebraic structure. Those instances often fall into discrete families, which makes the problem of classifying phases practically feasible. The extended models have special properties, such as some notion of being “exactly solvable”. Models of that kind are often called “fixed-point models”.

§ EXTENDIBILITY AND LIQUIDS: TOY EXAMPLES

We can use the language of tensor networks to formalize the extendibility hypothesis. By this, we do not try to answer the question of which models are extendible in which way, or what it means for a model to be “generic” in full generality. Instead, the latter will remain the central guess that we rely on, but that we find to be true in all considered cases.
We will at this point  merely formulate what it means for a specific model to be extendible in a specific way and provide $3$ simple examples for how models extend in a topological way in the following sections.

§.§ Topological extendibility in 1+1 dimensions

As a first example we consider models with topological extendibility in $1+1$ dimensions, that is, in one
spatial dimension. Despite the fact that non-trivial intrinsic topological order in the conventional sense only exists in $2+1$ dimensions, this example is well suited to illustrate the concept. Moreover, we would like to mention that, even though they are not robust to perturbations, symmetry-breaking phases have a topological deformability as well. Also, if we impose on-site or time-reversal symmetries, there are non-trivial symmetry protected topological (SPT)
phases. For pedagogical reasons, we neglect the following two important technical details in this section. In Section <ref> we will see that we need to distinguish the indices of a tensor for a better representation of topological space-time, and in Section <ref> we will see that we need to add an orientation and Hermiticity move in order to give the models a standard quantum mechanical interpretation.

§.§.§ Square lattice model and extended model

We start with a $4$-index tensor from which we can build tensor networks on arbitrary square lattices. Such a tensor can be obtained from a quantum Hamiltonian by Trotterization or directly from a classical model, as explained in Section <ref>, as

\begin{equation}
\begin{tikzpicture}
\draw[dashed,orange] (-0.2,0)--(2,0) (-0.2,0.6)--(2,0.6) (-0.2,1.2)--(2,1.2);
\draw[dashed,orange] (0,-0.2)--(0,1.4) (0.6,-0.2)--(0.6,1.4) (1.2,-0.2)--(1.2,1.4) (1.8,-0.2)--(1.8,1.4);
\atoms{toyplaquette}{0/p={0.3,0.3}, 1/p={0.9,0.3}, 2/p={1.5,0.3}, 3/p={0.3,0.9}, 4/p={0.9,0.9}, 5/p={1.5,0.9}}
\draw (0)--(1) (1)--(2) (3)--(4) (4)--(5) (0)--(3) (1)--(4) (2)--(5) (0)edge[]++(0,-0.5) (1)edge[]++(0,-0.5) (2)edge[]++(0,-0.5) (3)edge[]++(0,0.5) (4)edge[]++(0,0.5) (5)edge[]++(0,0.5) (0)edge[]++(-0.5,0) (3)edge[]++(-0.5,0) (2)edge[]++(0.5,0) (5)edge[]++(0.5,0);
\end{tikzpicture}\;.
\end{equation}

One of the main points of this work is to keep the combinatorial structure (the “network”) of a tensor network separate from the data (the “content of the tensors”).
On the combinatorial side, the places to which we assign the tensors will be referred to as atoms, and the places where the contractions happen will be called bonds. The “graph” formed by atoms and bonds will be referred to as network. In contrast to an actual graph, a network can have bonds with “loose ends” which will be referred to as open indices.

The different types of atoms (referred to as elements) form the so-called substrate. In this case the substrate is given by only one single type of atom representing a $4$-index tensor. On the data side, the $4$-index tensor itself will be called a model of the substrate.

The “extended model” is a prescription that assigns tensor networks to arbitrary triangulations of $2$-manifolds. One of the simplest ways to arrive at such a prescription is to take a $3$-index tensor, and to associate one copy of this tensor to each triangle, with contractions between tensors at adjacent triangles as

\begin{equation}
\begin{tikzpicture}
\atoms{void}{0/, 1/p={0.8,-0.8}, 2/p={2,-1}, 3/p={3,-0.4}, 4/p={2.8,0.8}, 5/p={1.6,1.2}, 6/p={0.4,1}, 7/p={1,0.2}, 8/p={2,0}}
\draw[dashed,orange] (0)--(1)--(2)--(3)--(4)--(5)--(6)--(0) (7)--(0) (7)--(1) (7)--(6) (7)--(5) (7)--(8) (8)--(1) (8)--(2) (8)--(3) (8)--(4) (8)--(5);
\atoms{toyface}{a/p={0.6,-0.2}, i/p={1.3,-0.2}, h/p={1.7,-0.5}, g/p={2.3,-0.4}, f/p={2.6,0.1}, e/p={2.1,0.7}, d/p={1.5,0.5}, c/p={1,0.8}, b/p={0.4,0.4}}
\draw (a)--(i)--(h)--(g)--(f)--(e)--(d)--(c)--(b)--(a) (d)--(i) (a)edge[]++(-140:0.5) (h)edge[]++(-120:0.5) (g)edge[]++(-60:0.5) (f)edge[]++(20:0.5) (e)edge[]++(70:0.5) (c)edge[]++(110:0.5) (b)edge[]++(160:0.5);
\end{tikzpicture}\;.
\end{equation}

So the extended model is a model with a different substrate, as it is given by a $3$-index tensor.
There is a canonical prescription to turn models of the extended substrate into models of the square-lattice substrate: We can restrict the extended model to regular triangulations of flat space. Specifically, we can divide each square into two triangles, and use as square tensor the tensor obtained by contracting two triangle tensors according to

\begin{equation}
\label{eq:extension_mapping}
\begin{tikzpicture}
\draw[dashed,orange] (0,0)--(0.5,0)--(0.5,0.5)--(0,0.5)--cycle;
\atoms{toyplaquette}{{0/p={0.25,0.25}}}
\draw (0)--++(0,0.4) (0)--++(0,-0.4) (0)--++(-0.4,0) (0)--++(0.4,0);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\draw[dashed,orange] (0,0)--(0.7,0)--(0.7,0.7)--(0,0.7)--cycle (0,0)--(0.7,0.7);
\atoms{toyface}{0/p={0.5,0.2}, 1/p={0.2,0.5}}
\draw (1)--++(0,0.4) (0)--++(0,-0.4) (1)--++(-0.4,0) (0)--++(0.4,0) (0)--(1);
\end{tikzpicture}\;.
\end{equation}

The pure combinatorics of an equation like this will be referred to as a substrate mapping. In the above case, the substrate mapping goes from the square lattice substrate to the extended substrate, which means that, conversely, it will map models of the extended substrate to models of the square lattice substrate.

In order to define the “invariance under homoemorphisms”,
we need a combinatorial analogue of the latter in terms of triangulations. This is given by the so-called Pachner moves [24] which act as

\begin{equation}
\begin{gathered}
\begin{tikzpicture}
\atoms{void}{0/, 1/p={0.5,-0.5}, 2/p={1,0}, 3/p={0.5,0.5}}
\draw (0)--(3) (3)--(2) (2)--(1) (0)--(1) (0)--(2);
\end{tikzpicture}
\quad \longleftrightarrow \quad
\begin{tikzpicture}
\atoms{void}{0/, 1/p={0.5,-0.5}, 2/p={1,0}, 3/p={0.5,0.5}}
\draw (0)--(3) (3)--(2) (2)--(1) (0)--(1) (3)--(1);
\end{tikzpicture}\\
\begin{tikzpicture}
\atoms{void}{0/, 1/p={-30:0.6}, 2/p={90:0.6}, 3/p={-150:0.6}}
\draw (0)--(1) (0)--(2) (0)--(3) (3)--(2) (2)--(1) (3)--(1);
\end{tikzpicture}
\quad \longleftrightarrow \quad
\begin{tikzpicture}
\atoms{void}{1/p={-30:0.6}, 2/p={90:0.6}, 3/p={-150:0.6}}
\draw (3)--(2) (2)--(1) (3)--(1);
\end{tikzpicture}
\end{gathered}\;.
\end{equation}

It is known that any two (combinatorial) triangulations of the same manifold are related by Pachner moves. Conversely, it is easy to see that Pachner moves preserve the topology of a triangulation.

We can associate tensor networks to the patches on the left- and the right hand side of the Pachner moves. Topological invariance of the model means that the evaluations of the two corresponding tensor networks are equal as

\begin{equation}
\label{eq:pachner_moves}
\begin{gathered}
\begin{tikzpicture}
\atoms{void}{0/, 1/p={0.5,-0.5}, 2/p={1,0}, 3/p={0.5,0.5}}
\draw[dashed,orange] (0)--(3) (3)--(2) (2)--(1) (0)--(1) (0)--(2);
\atoms{toyface}{a/p={0.5,-0.2}, b/p={0.5,0.2}}
\draw (a)--(b) (b)--++(45:0.4) (b)--++(135:0.4) (a)--++(-45:0.4) (a)--++(-135:0.4);
\end{tikzpicture}
\quad \text{\breakcellm{$=$\\ \color{orange}{$\longleftrightarrow$}}} \quad
\begin{tikzpicture}
\atoms{void}{0/, 1/p={0.5,-0.5}, 2/p={1,0}, 3/p={0.5,0.5}}
\draw[dashed,orange] (0)--(3) (3)--(2) (2)--(1) (0)--(1) (3)--(1);
\atoms{toyface}{a/p={0.3,0}, b/p={0.7,0}}
\draw (a)--(b) (b)--++(45:0.4) (a)--++(135:0.4) (b)--++(-45:0.4) (a)--++(-135:0.4);
\end{tikzpicture}\\
\begin{tikzpicture}
\atoms{void}{0/, 1/p={-30:0.8}, 2/p={90:0.8}, 3/p={-150:0.8}}
\draw[dashed,orange] (0)--(1) (0)--(2) (0)--(3) (3)--(2) (2)--(1) (3)--(1);
\atoms{toyface}{a/p={30:0.25}, b/p={150:0.25}, c/p={-90:0.25}}
\draw (a)--(b)--(c)--(a) (a)--++(30:0.4) (b)--++(150:0.4) (c)--++(-90:0.4);
\end{tikzpicture}
\quad \text{\breakcellm{$=$\\ \color{orange}{$\longleftrightarrow$}}} \quad
\begin{tikzpicture}
\atoms{void}{1/p={-30:0.6}, 2/p={90:0.6}, 3/p={-150:0.6}}
\draw[dashed,orange] (3)--(2) (2)--(1) (3)--(1);
\atoms{toyface}{a/}
\draw (a)--++(30:0.5) (a)--++(150:0.5) (a)--++(-90:0.5);
\end{tikzpicture}
\end{gathered}\;.
\end{equation}

We will refer to the pure combinatorics of an equation like this as a move. The collection of moves will be referred to as a liquid. A model of a substrate fulfilling all the equations corresponding to the moves will be called a model of the liquid.

A square lattice model $X$ is extendible if there exists a model of a liquid (in our case the liquid Eq. (<ref>)), such that the square lattice model obtained by applying the corresponding mapping (in our case Eq. (<ref>)) is in the same phase as $X$.

§.§.§ Relation to algebraic structures

In this section, we relate the above moves to algebraic structures.
An algebra is a linear map $\cdot:V\otimes V\rightarrow V$, where $V$ is a vector space. A finite-dimensional algebra is represented by its structure coefficients, which form a $3$-index tensor,

\begin{equation}
\begin{tikzpicture}
\atoms{toyalg}{0/}
\draw (0)edge[]++(135:0.5) (0)edge[]++(45:0.5) (0)edge[]++(-90:0.5);
\end{tikzpicture}\;.
\end{equation}

Here, we think of $\cdot$ as a linear map from the top two indices to the bottom index. So, an algebra is nothing but a model of the substrate depicted above.
An algebra is associative, if

\begin{equation}
(a\cdot b)\cdot c = a\cdot (b\cdot c) \;.
\end{equation}

This can be formulated as an equation between two tensor networks, namely

\begin{equation}
\begin{tikzpicture}
\atoms{toyalg}{0/, 1/p=-45:0.8}
\draw (0)edge[dind=a]++(135:0.5) (0)edge[dind=b]++(45:0.5) (1)edge[dind=c]++(45:0.5) (0)--(1) (1)edge[dind=d]++(-90:0.5);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{toyalg}{0/, 1/p=-135:0.8}
\draw (0)edge[dind=b]++(135:0.5) (1)edge[dind=a]++(135:0.5) (0)edge[dind=c]++(45:0.5) (0)--(1) (1)edge[dind=d]++(-90:0.5);
\end{tikzpicture}
\;.
\end{equation}

So, associativity defines a move, and associative algebras are models of the liquid defined by that move. There are many other examples of algebraic structures which are models of liquids, such as Frobenius algebras, unital algebras, commutative algebras, Hopf algebras, representations of algebras, etc.

Every model of the topological liquid can be turned into an algebra by a substrate mapping from the algebra liquid to the topological liquid

\begin{equation}
\begin{tikzpicture}
\atoms{toyalg}{0/}
\draw (0)edge[dind=a]++(135:0.5) (0)edge[dind=b]++(45:0.5) (0)edge[dind=c]++(-90:0.5);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\atoms{toyface}{a/}
\draw (a)edge[dind=b]++(30:0.5) (a)edge[dind=a]++(150:0.5) (a)edge[dind=c]++(-90:0.5);
\end{tikzpicture}
\end{equation}

Let us substitute this definition into the associativity axiom above. We obtain

\begin{equation}
\begin{tikzpicture}
\atoms{toyface}{0/, 1/p=-45:0.8}
\draw (0)edge[dind=a]++(135:0.5) (0)edge[dind=b]++(45:0.5) (1)edge[dind=c]++(45:0.5) (0)--(1) (1)edge[dind=d]++(-90:0.5);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{toyface}{0/, 1/p=-135:0.8}
\draw (0)edge[dind=b]++(135:0.5) (1)edge[dind=a]++(135:0.5) (0)edge[dind=c]++(45:0.5) (0)--(1) (1)edge[dind=d]++(-90:0.5);
\end{tikzpicture}
\end{equation}

This is nothing but the 2-2 Pachner move of the extended liquid. Thus, every model of the extended liquid yields an algebra which is automatically associative. On the level of combinatorics, we have a mapping between two substrates which are also liquids. Each move of the algebra liquid yields a move for the topological liquid by substituting the mapping. We found that each of these mapped moves is also a move of the topological liquid. We call such a substrate mapping which is compatible with the moves of two liquids a liquid mapping.

We often observe that topological liquids have liquid mappings from well-known algebraic structures. However, usually there is no inverse liquid mapping from the topological liquid to the algebraic liquid. This means that models of topological liquids define some algebraic structure, but the algebraic structure misses some additional axioms which are needed for a topological fixed point model. This is mostly due to the fact that the networks in the moves of algebraic structures always allow for a global “flow of time” (in case of the associativity above from the top to the bottom), whereas the topological liquid includes moves with “closed time-like loops”.

§.§.§ Models

As we have seen in the paragraph above, models of the extended liquid correspond to associative algebras, for which a few additional axioms hold [We are not yet fully precise here, but in the end the models will be equivalent to something like commutative special Frobenius algebras.]. Such algebras fall into discrete families, up to basis changes. One family of algebras which also yield topological models is given by the algebra of complex functions over an $x$-element set under point-wise multiplication, for arbitrary $x$. This corresponds to the

\begin{equation}
\label{eq:delta_model}
\begin{tikzpicture}
\atoms{toyface}{a/}
\draw (a)edge[dind=b]++(30:0.4) (a)edge[dind=a]++(150:0.4) (a)edge[dind=c]++(-90:0.4);
\end{tikzpicture}=
\begin{tikzpicture}
\atoms{delta}{0/}
\draw (0)edge[dind=b]++(30:0.4) (0)edge[dind=a]++(150:0.4) (0)edge[dind=c]++(-90:0.4);
\end{tikzpicture}
\begin{cases}
1& \text{if } a=b=c\\
0& \text{otherwise}
\end{cases}\;,
\end{equation}

for $0\leq a,b,c<x$, which will also be referred to as the delta tensor, and denoted by a small dot. Delta tensors can be defind for an arbitrary number of indices, with entry $1$ if all the index values are equal and $0$ otherwise.

If we evaluate such a model on a network representing a triangulation of a sphere, we get the number $x$. So we see that every family yields a different topological invariant and thus all families correspond to different phases.
Physically speaking, these phases are symmetry-breaking phases.
E.g., for $x=2$, the liquid is equivalent to the ordered-phase fixed point of the $1+1$-dimensional Ising model. This is a chain of qubits with a nearest-neighbour Hamiltonian

\begin{equation}
H=\sum_i h_i= -\sum_i Z_iZ_{i+1} \;.
\end{equation}

If we apply the Trotterization procedure in Section <ref> to any Hamiltonian, we never obtain a topological tensor liquid. This is because the first excited state always has a finite energy, corresponding to a finite “correlation length in time direction”. In contrast, a topological tensor liquid has zero correlation length in any direction due to its topological deformability. Luckily, the Hamiltonian above has the special property that its terms are commuting, so we do not need to divide time into small Trotter steps. In this case,
we can directly take the limit $\beta\rightarrow \infty$ for the individual terms, which corresponds to taking the local ground state projector

\begin{equation}
\lim_{\beta \rightarrow \infty} e^{-\beta h} \sim P= \frac{1}{2}(\mathbb{1}-Z_0Z_1)\;.
\end{equation}

The imaginary time evolution is represented by a product of these operators $P$ at different places. If we write $P$ as a tensor

\begin{equation}
\label{eq:ising_projector}
\begin{tikzpicture}
\atoms{toycomproj}{0/}
\draw (0-tr)edge[dind=b]++(45:0.3) (0-tl)edge[dind=a]++(135:0.3) (0-br)edge[dind=d]++(-45:0.3) (0-bl)edge[dind=c]++(-135:0.3);
\end{tikzpicture}
\begin{cases}
1 & \text{if}\quad a=b=c=d\\
0 & \text{otherwise}
\end{cases}\;,
\end{equation}

the imaginary time evolution is represented by a network of the  form

\begin{equation}
\begin{tikzpicture}
\begin{scope}[yshift=-0.5cm]
\atoms{void}{0/, 1/p={1,0}, 2/p={-0.5,0.5}, 3/p={0.5,0.5}, 4/p={1.5,0.5}, 5/p={0,1}, 6/p={1,1}, 7/p={-0.5,1.5}, 8/p={0.5,1.5}, 9/p={1.5,1.5}, 10/p={0,2}, 11/p={1,2}, 12/p={2,0}, 13/p={2.5,0.5}, 14/p={2,1}, 15/p={2.5,1.5}, 16/p={2,2}}
\end{scope}
\draw[dashed,orange] (0)--(2)--(5)--(7)--(10) (0)--(5)--(10) (0)--(3)--(5)--(8)--(10) (1)--(3)--(6)--(8)--(11) (1)--(4)--(6)--(9)--(11) (1)--(6)--(11) (3)--(8) (2)--(7) (4)--(9) (12)--(4)--(14)--(9)--(16) (12)--(14)--(16) (12)--(13)--(14)--(15)--(16) (13)--(15);
\atoms{toycomproj}{0/, 1/p={1,0}, 2/p={2,0}, 3/p={0.5,0.5}, 4/p={1.5,0.5}, 5/p={2.5,0.5}, 6/p={0,1}, 7/p={1,1}, 8/p={2,1}, 9/p={-0.5,0.5}}
\draw (0-tr)--(3-bl) (1-tl)--(3-br) (1-tr)--(4-bl) (2-tl)--(4-br) (2-tr)--(5-bl) (3-tl)--(6-br) (3-tr)--(7-bl) (4-tl)--(7-br) (4-tr)--(8-bl) (5-tl)--(8-br) (0-tl)--(9-br) (6-bl)--(9-tr) (0-bl)--++(-135:0.2) (0-br)--++(-45:0.2) (1-bl)--++(-135:0.2) (1-br)--++(-45:0.2) (2-bl)--++(-135:0.2) (2-br)--++(-45:0.2) (2-tr)--++(45:0.2) (5-br)--++(-45:0.2) (6-tl)--++(135:0.2) (6-tr)--++(45:0.2) (7-tl)--++(135:0.2) (7-tr)--++(45:0.2) (8-tl)--++(135:0.2) (8-tr)--++(45:0.2) (8-br)--++(-45:0.2) (5-tr)--++(45:0.2) (9-tl)--++(135:0.2) (9-bl)--++(-135:0.2);
\end{tikzpicture}\;.
\end{equation}

In order to compare the above model (given by a $4$-index tensor) with our topological model (given by a $3$-index tensor),
we have to choose an according substrate mapping which transforms the network above into the network representing a triangulation of the plane. One such mapping is, e.g., given by

\begin{equation}
\label{eq:projector_mapping}
\begin{tikzpicture}
\atoms{toycomproj}{0/}
\draw (0-tr)edge[dind=b]++(45:0.3) (0-tl)edge[dind=a]++(135:0.3) (0-br)edge[dind=d]++(-45:0.3) (0-bl)edge[dind=c]++(-135:0.3);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\atoms{void}{a/p={0,-0.5}, b/p={0,0.5}, c/p={-0.5,0}, d/p={0.5,0}}
\draw[orange,dashed] (a)--(b) (a)--(c)--(b) (a)--(d)--(b);
\atoms{toyface}{0/p={-0.2,0}, 1/p={0.2,0}}
\draw (0)--(1) (1)edge[dind=b]++(45:0.4) (0)edge[dind=a]++(135:0.4) (1)edge[dind=d]++(-45:0.4) (0)edge[dind=c]++(-135:0.4);
\end{tikzpicture} \;.
\end{equation}

If we evaluate the right hand side, we get indeed the tensor $P$ from Eq. (<ref>).

§.§.§ Commuting-projector Hamiltonians and stacking

We have already introduced the notion of a liquid mapping to formalize the relation between the “original” and the “extended” model, and between the algebraic liquid and the topological liquid. Here, we will give two more examples for operations which can be neatly formalized by a liquid mapping, namely the construction of a commuting-projector Hamiltonian from the section before, and the operation of embedding two copies of a model into the same space-time. Both will lead to generalizations of the notion of liquid mapping we introduced so far.

$H$ is a commuting-projector Hamiltonian, meaning that $P$ is a projector, and operators $P$ acting on different pairs of sites commute. I.e., the following two equations are fulfilled.

* The projector property is

\begin{equation}
\label{eq:projector_move}
\begin{tikzpicture}
\atoms{toycomproj}{0/, 1/p={0,0.8}}
\draw (1-tr)edge[dind=b]++(45:0.3) (1-tl)edge[dind=a]++(135:0.3) (0-br)edge[dind=d]++(-45:0.3) (0-bl)edge[dind=c]++(-135:0.3);
\draw (0-tr)to[bend right=40](1-br) (0-tl)to[bend left=40](1-bl);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{toycomproj}{0/}
\draw (0-tr)edge[dind=b]++(45:0.3) (0-tl)edge[dind=a]++(135:0.3) (0-br)edge[dind=d]++(-45:0.3) (0-bl)edge[dind=c]++(-135:0.3);
\end{tikzpicture}\;,
\end{equation}

* the commutativity is

\begin{equation}
\begin{tikzpicture}
\atoms{toycomproj}{0/, 1/p={0.6,0.6}}
\draw (1-tr)edge[dind=b]++(45:0.3) (1-tl)edge[dind=a]++(135:0.3) (0-br)edge[dind=d]++(-45:0.3) (0-bl)edge[dind=c]++(-135:0.3) (0-tl)edge[dind=e]++(135:0.3) (1-br)edge[dind=f]++(-45:0.3);
\draw (0-tr)--(1-bl);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{toycomproj}{0/, 1/p={0.6,-0.6}}
\draw (1-tr)edge[dind=b]++(45:0.3) (0-tr)edge[dind=a]++(45:0.3) (1-bl)edge[dind=d]++(-135:0.3) (0-bl)edge[dind=c]++(-135:0.3) (0-tl)edge[dind=e]++(135:0.3) (1-br)edge[dind=f]++(-45:0.3);
\draw (0-br)--(1-tl);
\end{tikzpicture}\;.
\end{equation}

So “commuting-projector Hamiltonians” themselves are models of the above liquid.
Now, if we plug the mapping Eq. (<ref>) into the move Eq. (<ref>), we get

\begin{equation}
\begin{tikzpicture}
\atoms{toyface}{0/, 1/p={0.8,0}, 2/p={0,0.8}, 3/p={0.8,0.8}}
\draw (0)--(1) (1)--(3) (2)--(3) (0)--(2) (0)edge[dind=c]++(0,-0.4) (1)edge[dind=d]++(0,-0.4) (2)edge[dind=a]++(0,0.4) (3)edge[dind=b]++(0,0.4);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{toyface}{0/, 1/p={0.8,0}}
\draw (0)--(1) (0)edge[dind=c]++(0,-0.4) (1)edge[dind=d]++(0,-0.4) (0)edge[dind=a]++(0,0.4) (1)edge[dind=b]++(0,0.4);
\end{tikzpicture}\;.
\end{equation}

This is not a move of the topological liquid. However, it is equivalent to a
sequence of moves

\begin{equation}
\begin{tikzpicture}
\atoms{toyface}{0/, 1/p={0.8,0}, 2/p={0,0.8}, 3/p={0.8,0.8}}
\draw (0)--(1) (1)--(3) (2)--(3) (0)--(2) (0)edge[dind=c]++(0,-0.4) (1)edge[dind=d]++(0,-0.4) (2)edge[dind=a]++(0,0.4) (3)edge[dind=b]++(0,0.4);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{toyface}{0/, 1/p={0,0.8}, 2/p={0.5,0.4}, 3/p={1,0.4}}
\draw (0)--(1) (1)--(2) (2)--(0) (3)--(2) (0)edge[dind=c]++(0,-0.4) (3)edge[dind=d]++(0,-0.4) (1)edge[dind=a]++(0,0.4) (3)edge[dind=b]++(0,0.4);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{toyface}{0/, 1/p={0.8,0}}
\draw (0)--(1) (0)edge[dind=c]++(0,-0.4) (1)edge[dind=d]++(0,-0.4) (0)edge[dind=a]++(0,0.4) (1)edge[dind=b]++(0,0.4);
\end{tikzpicture}\;.
\end{equation}

In the first step we applied the 2-2 Pachner move, and in the second step the 1-3 Pachner move. We will refer to moves equivalent to sequences of moves of a liquid as derived moves, and the corresponding sequences as derivations. We will generalize our notion of a liquid mapping by allowing the mapped moves to be derived moves of the target liquid.

For the second example, consider “stacking two copies” of a model, i.e., embedding both models in the same space(-time) in a non-interacting way. If a model has topological deformability, then so will the stacked model. Stacking is nothing but a mapping from a liquid (here the $1+1$ dimensions topological liquid) to itself

\begin{equation}
\begin{tikzpicture}
\atoms{toyface}{a/}
\draw (a)edge[dind=bb']++(30:0.4) (a)edge[dind=aa']++(150:0.4) (a)edge[dind=cc']++(-90:0.4);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\atoms{toyface}{a/}
\draw (a)edge[dind=b]++(30:0.4) (a)edge[dind=a]++(150:0.4) (a)edge[dind=c]++(-90:0.4);
\atoms{toyface}{a/p={1.5,0}}
\draw (a)edge[dind=b']++(30:0.4) (a)edge[dind=a']++(150:0.4) (a)edge[dind=c']++(-90:0.4);
\end{tikzpicture}\;.
\end{equation}

This example shows a slight generalization of the concept of a liquid mapping introduced so far, as every open index on the left hand side corresponds to two open indices on the right hand side.

If we apply this mapping to any network, we will obtain two copies of that network. E.g., the mapped 2-2 Pachner move gets mapped to a move relating two disconnected networks consisting of two atoms each on each if its sides. Obviously, this mapped move can be derived by applying the original move to both copies independently.

§.§.§ Phases

In this section, we will illustrate how the definition of a phase in terms of circuit moves of tensor networks from Section <ref> applies to models of liquids. The algebra of functions over a 2-element set yields a model as we have seen above. Another model of the liquid is given by the $\zz_2$ group algebra

\begin{equation}
\begin{tikzpicture}
\atoms{toyface}{{a/lab={p=-150:0.35,t=\zz_2}}}
\draw (a)edge[dind=b]++(30:0.4) (a)edge[dind=a]++(150:0.4) (a)edge[dind=c]++(-90:0.4);
\end{tikzpicture}=
\begin{cases}
\frac{1}{\sqrt{2}} & \text{if } a+b+c=0 \mod 2\\
0& \text{otherwise}
\end{cases}\;.
\end{equation}

The two algebras are isomorphic via a basis change known as the Hadamard transformation

\begin{equation}
\begin{tikzpicture}
\atoms{hadamard}{h/lab={p=-90:0.35}}
\draw (h-r)edge[dind=b]++(0.3,0) (h-l)edge[dind=a]++(-0.3,0);
\end{tikzpicture}
\frac{1}{\sqrt{2}}
\begin{pmatrix}
\end{pmatrix}\;.
\end{equation}

Basis changes are a very specific example of circuit moves, and thus the two liquid models are in the same phase. Concretely, we have

\begin{equation}
\label{eq:z2_basis_change}
\begin{tikzpicture}
\atoms{toyface}{{0/lab={t=\zz_2,p=-140:0.3}}}
\atoms{hadamard}{{h0/p=30:0.5,rot=30,lab={p=-60:0.3}}, {h1/p=150:0.5,rot=150,lab={p=-120:0.3}}, {h2/p=-90:0.5,rot=-90,lab={p=0:0.3}}}
%\atoms{small, rotate=150,square}{{h1/p={150:0.5},lab={t=H,p=-120:0.3})}}
%\atoms{small, rotate=-90,square}{{h2/p={-90:0.5},lab={t=H,p=0:0.3})}}
\draw (0)--(h0-l) (h0-r)edge[dind=b]++(30:0.3) (0)--(h1-l) (h1-r)edge[dind=a]++(150:0.3) (0)--(h2-l) (h2-r)edge[dind=c]++(-90:0.3);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{toyface}{{a/lab={t=\delta,p=-150:0.35}}}
\draw (a)edge[dind=b]++(30:0.4) (a)edge[dind=a]++(150:0.4) (a)edge[dind=c]++(-90:0.4);
\end{tikzpicture}\;.
\end{equation}

$H$ happens to be its own inverse

\begin{equation}
\label{eq:hadamard_selfinverse}
\begin{tikzpicture}
\atoms{hadamard}{{h0/lab={p=-90:0.35}}}
\atoms{hadamard}{{h1/p={0.6,0},lab={p=-90:0.35}}}
\draw (h1-r)edge[dind=b]++(0.3,0) (h0-l)edge[dind=a]++(-0.3,0) (h0-r)--(h1-l);
\end{tikzpicture}
\begin{tikzpicture}
\draw[mark={lab=$a$,b},dind=b](0,0)--(0.5,0);
\end{tikzpicture}\;.
\end{equation}

The circuit move starts with a network of the $\delta$ liquid, e.g.,

\begin{equation}
\begin{tikzpicture}
\atoms{toyface}{{0/lab={t=\delta,p=180:0.35}}, {1/p={0.8,0},lab={t=\delta,p=0:0.35}}}
\draw (0)--(1) (0)--++(120:0.5) (0)--++(-120:0.5) (1)--++(60:0.5) (1)--++(-60:0.5);
\end{tikzpicture}\;.
\end{equation}

We identify all different occurrences of the network on the right hand side of Eq. (<ref>), and replace them by the network on the left hand side. We obtain

\begin{equation}
\begin{tikzpicture}
\atoms{toyface,lab/t=\zz_2}{0/lab={p=180:0.35}, {1/p={1.2,0},lab={p=0:0.35}}}
\atoms{hadamard}{{h0/p={120:0.5},rot=120,lab={p=30:0.3}}, {h1/p={-120:0.5},rot=-120,lab={p=-30:0.3}}, {h2/p={$(1)+(60:0.5)$},rot=50,lab={p=150:0.3}}, {h3/p={$(1)+(-60:0.5)$},rot=-60,lab={p=-150:0.3}}, {h4/p={0.4,0},lab={p=-90:0.3}}, {h5/p={0.8,0},lab={p=-90:0.3}}}
\draw (0)--(h4-l) (h4-r)--(h5-l) (h5-r)--(1) (0)--(h0-l) (0)--(h1-l) (1)--(h2-l) (1)--(h3-l) (h0-r)--++(120:0.3) (h1-r)--++(-120:0.3) (h2-r)--++(60:0.3) (h3-r)--++(-60:0.3);
\end{tikzpicture}\;.
\end{equation}

Now we identify all different occurrences of the network on the left hand side of Eq. (<ref>) and replace them by the right hand side. For this to be a well-defined procedure, the different occurrences must not overlap. This is ensured by how we obtained the network in the previous step. We obtain

\begin{equation}
\begin{tikzpicture}
\atoms{toyface,lab/t=\zz_2}{{0/lab={p=180:0.35}}, {1/p={0.8,0},lab={p=0:0.35}}}
\atoms{hadamard}{{h0/p=120:0.5,rot=120,lab={p=30:0.3}}, {h1/p=-120:0.5,rot=-120,lab={p=-30:0.3}}, {h2/p=$(1)+(60:0.5)$,rot=60,lab={p=150:0.3}}, {h3/p=$(1)+(-60:0.5)$,rot=-60,lab={p=-150:0.3}}}
\draw (0)--(1) (0)--(h0-l) (h0-r)--++(120:0.3) (0)--(h1-l) (h1-r)--++(-120:0.3) (1)--(h2-l) (h2-r)--++(60:0.3) (1)--(h3-l) (h3-r)--++(-60:0.3);
\end{tikzpicture}\;.
\end{equation}

As we have seen, using Eqs. (<ref>), (<ref>), we can transform any closed $\delta$-tensor network into the according $\zz_2$-tensor network. If the network has open indices, we will end up with some residual Hadamard transformations near this open boundary. We found that the $\zz_2$-model and the $\delta$-model are related by an exact circuit move, and thus in the same phase. Note that generic square lattice tensor networks are generically only connected by circuit moves up to exponential tails. However, for liquid models as highly restricted by moves as the present one, we tend to find that there are exact, and even rather simple circuit moves connecting them.

As another example, consider the model

\begin{equation}
\begin{tikzpicture}
\atoms{toyface}{{a/lab={t=x,p=-150:0.35}}}
\draw (a)edge[dind=b]++(30:0.4) (a)edge[dind=a]++(150:0.4) (a)edge[dind=c]++(-90:0.4);
\end{tikzpicture}=
2^{(-3/2)}\quad (\forall a,b,c)\;.
\end{equation}

We will show that this model is in the same phase as the trivial model where each tensor is the number $1$. We notice that the tensor above is the tensor product of three times the same vector

\begin{equation}
\label{eq:product_tensor}
\begin{tikzpicture}
\atoms{toyface}{{a/lab={t=x,p=-150:0.35}}}
\draw (a)edge[dind=b]++(30:0.4) (a)edge[dind=a]++(150:0.4) (a)edge[dind=c]++(-90:0.4);
\end{tikzpicture}=
\begin{tikzpicture}
\atoms{whiteblack,small,circ}{0/p={30:0.3}, 1/p={150:0.3}, 2/p={-90:0.3}}
\draw (0)edge[dind=b]++(30:0.4) (1)edge[dind=a]++(150:0.4) (2)edge[dind=c]++(-90:0.4);
\end{tikzpicture}\;,
\end{equation}

\begin{equation}
\begin{tikzpicture}
\atoms{whiteblack,small,circ}{{0/}}
\draw (0)edge[dind=a]++(0.4,0);
\end{tikzpicture}
\frac{1}{\sqrt{2}}\quad (\forall a)\;.
\end{equation}

Furthermore, this vector is normalized, i.e. (note that the empty network evaluates to the scalar $1$),

\begin{equation}
\label{eq:vector_normalization}
\begin{tikzpicture}
\atoms{whiteblack,small,circ}{{0/}}
\atoms{whiteblack,small,circ}{{1/p={0.5,0}}}
\draw (0)--(1);
\end{tikzpicture}
\hspace{2cm}\;.
\end{equation}

From the above two moves, we can again construct a circuit move. We start with a tensor network of the $x$-model

\begin{equation}
\begin{tikzpicture}
\atoms{toyface}{{0/lab={t=x,p=180:0.35}}, {1/p={0.8,0},lab={t=x,p=0:0.35}}}
\draw (0)--(1) (0)--++(120:0.5) (0)--++(-120:0.5) (1)--++(60:0.5) (1)--++(-60:0.5);
\end{tikzpicture}\;,
\end{equation}

and then apply Eq. (<ref>) by replacing every occurrence of the left hand side by the right hand side

\begin{equation}
\begin{tikzpicture}
\atoms{whiteblack,small,circ}{h0/p={120:0.3}, h1/p={-120:0.3}, h4/p={0.3,0}, h5/p={0.7,0}, h2/p={$(1,0)+(60:0.3)$}, h3/p={$(1,0)+(-60:0.3)$}}
\draw (h4)--(h5) (h0)--++(120:0.4) (h1)--++(-120:0.4) (h2)--++(60:0.4) (h3)--++(-60:0.4);
\end{tikzpicture}\;.
\end{equation}

Finally, we apply Eq. (<ref>), which yields the empty network everywhere except for at a potential boundary

\begin{equation}
\begin{tikzpicture}
\atoms{whiteblack,small,circ}{h0/p={120:0.3}, h1/p={-120:0.3}, h2/p={$(1,0)+(60:0.3)$}, h3/p={$(1,0)+(-60:0.3)$}}
\draw (h0)--++(120:0.4) (h1)--++(-120:0.4) (h2)--++(60:0.4) (h3)--++(-60:0.4);
\end{tikzpicture}\;.
\end{equation}

So we see that this model is in the same phase as the trivial model, again via an exact and very simple circuit move.

Surely, in more general examples of liquid models being in the same phase, the circuit moves will not always be quite as simple as basis changes. However, we continue to observe that they are exact and consist of a small number of steps.

§.§ Topological extendibility in 2+1 dimensions

In the next example, we explore extendibility in $2+1$ dimensions. Again, we start from a tensor network on a cubic lattice obtained by Trotterization of a Hamiltonian or by other means. The extended model is a tensor network that can be defined on arbitrary triangulations of $3$-dimensional manifolds. The most straight-forward construction would be to consider cellulations into simplices and to associate one tensor to every simplex. For this liquid the topological invariance would then be formulated analogous to the $1+1$-dimensional case, i.e., as invariance under $3$-dimensional Pachner moves. In this section we will instead sketch a less standard formulation of topological invariance, which will be presented in more detail in Section <ref>. The equivalence between the two formulations will be shown in Section <ref>.

§.§.§ The face-edge liquid

For the liquid we introduce here we allow arbitrary cellulations of a 3-manifold, but we demand that every face is either a triangle or a $2$-gon and that every edge is $3$-valent or $2$-valent (i.e., it is adjacent to three or two faces). A moment of thought reveals that every cell complex can be brought into this form, e.g., a $4$-gon can be split into two triangles with a $2$-valent edge in between as

\begin{equation}
\begin{tikzpicture}
\atoms{void}{0/, 1/p={0.5,-0.5}, 2/p={1,0}, 3/p={0.5,0.5}}
\fill[orange,opacity=0.35] (0)--(3)--(2)--(1)--cycle;
\draw[dashed,orange] (0)--(3) (3)--(2) (2)--(1) (0)--(1);
\end{tikzpicture}
\quad\rightarrow\quad
\begin{tikzpicture}
\atoms{void}{0/, 1/p={0.5,-0.5}, 2/p={1,0}, 3/p={0.5,0.5}}
\fill[orange,opacity=0.35] (0)--(3)--(2)--(1)--cycle;
\draw[dashed,orange] (0)--(3) (3)--(2) (2)--(1) (0)--(1) (1)--(3);
\end{tikzpicture}\;.
\end{equation}

Dually, a $4$-valent edge can be split into two $3$-valent edges with a $2$-gon face in between,

\begin{equation}
\begin{tikzpicture}
\fill[orange,opacity=0.35] (0,0)--(1,0)--++(135:0.6)--++(-1,0)--cycle;
\fill[orange,opacity=0.35] (0,0)--(1,0)--++(-135:0.6)--++(-1,0)--cycle;
\draw[dashed,orange,actualedge] (0,0)--(1,0);
\fill[orange,opacity=0.35] (0,0)--(1,0)--++(45:0.5)--++(-1,0)--cycle;
\fill[orange,opacity=0.35] (0,0)--(1,0)--++(-45:0.5)--++(-1,0)--cycle;
\end{tikzpicture}
\quad\rightarrow\quad
\begin{tikzpicture}
\fill[orange,opacity=0.35] (0,0)to[bend left](1,0)--++(135:0.6)--++(-1,0)--cycle;
\fill[orange,opacity=0.35] (0,0)to[bend right](1,0)--++(-135:0.6)--++(-1,0)--cycle;
\fill[orange,opacity=0.35] (0,0)to[bend left](1,0)to[bend left](0,0);
\draw[dashed,orange,actualedge] (0,0)to[bend left](1,0);
\draw[dashed,orange,actualedge] (0,0)to[bend right](1,0);
\fill[orange,opacity=0.35] (0,0)to[bend left](1,0)--++(45:0.5)--++(-1,0)--cycle;
\fill[orange,opacity=0.35] (0,0)to[bend right](1,0)--++(-45:0.5)--++(-1,0)--cycle;
\end{tikzpicture}\;.
\end{equation}

There is one $3$-index tensor (or, combinatorially, a $3$-index atom) associated to every triangle and another $3$-index tensor to every $3$-valent edge. At every pair of adjacent triangle and $3$-valent edge, the corresponding tensors share a contracted index pair, i.e., the corresponding atoms are connected by a bond. The edge tensors are different from the face tensors, thus we use two different shapes to represent them

\begin{equation}
\begin{gathered}
\begin{tikzpicture}
\atoms{void}{1/p={-30:0.6}, 2/p={90:0.6}, 3/p={-150:0.6}}
\fill[orange,opacity=0.35] (3)--(2)--(1)--cycle;
\draw[dashed,orange] (3)--(2) (2)--(1) (3)--(1);
\atoms{toyface}{a/}
\draw (a)--++(30:0.5) (a)--++(150:0.5) (a)--++(-90:0.5);
\end{tikzpicture}
\\
\begin{tikzpicture}
\fill[orange,opacity=0.35] (0,0)--(1,0)--++(150:0.6)--++(-1,0)--cycle;
\fill[orange,opacity=0.35] (0,0)--(1,0)--(1,-0.4)--(0,-0.4)--cycle;
\draw[dashed,orange,actualedge] (0,0)--(1,0);
\atoms{toyedge}{a/p={0.5,0}}
\draw (a)--++(150:0.7) (a)--++(-90:0.5);
\fill[orange,opacity=0.35] (0,0)--(1,0)--++(30:0.4)--++(-1,0)--cycle;
\draw (a)--++(30:0.5);
\end{tikzpicture}
\end{gathered}\;.
\end{equation}

The combinatorial data of such tensor networks are networks with two kinds of atoms. Different kinds of atoms will be referred to as elements of the corresponding substrate. Thus, this substrate has two $3$-index elements, whereas the topological substrate from the section above had only one $3$-index element. Accordingly, a model of the face-edge substrate consists of two $3$-index tensors.

The $2$-gons and $2$-valent edges are not explicitly represented by atoms. Instead, the two edges adjacent to a $2$-gon, and likewise the two faces adjacent to a 2-valent edge are directly connected by a bond. E.g., two $3$-valent edges separated by a $2$-gon are represented as

\begin{equation}
\begin{tikzpicture}
\fill[orange,opacity=0.35] (0,0)to[bend left](1.5,0)--++(135:0.6)--++(-1.5,0)--cycle;
\fill[orange,opacity=0.35] (0,0)to[bend right](1.5,0)--++(-135:0.6)--++(-1.5,0)--cycle;
\fill[orange,opacity=0.35] (0,0)to[bend left](1.5,0)to[bend left](0,0);
\draw[dashed,orange,actualedge] (0,0)to[bend left](1.5,0);
\draw[dashed,orange,actualedge] (0,0)to[bend right](1.5,0);
\atoms{toyedge}{0/p={0.75,0.2}, 1/p={0.75,-0.2}}
\draw (0)--(1) (1)--++(-135:0.5) (1)--++(-45:0.5);
\fill[orange,opacity=0.35] (0,0)to[bend left](1.5,0)--++(45:0.5)--++(-1.5,0)--cycle;
\fill[orange,opacity=0.35] (0,0)to[bend right](1.5,0)--++(-45:0.5)--++(-1.5,0)--cycle;
\draw (0)--++(135:0.5)  (0)--++(45:0.5);
\end{tikzpicture}\;.
\end{equation}

As in $1+1$ dimensions, two combinatorial triangulations correspond to the same manifold exactly if they are related by $3$-dimensional Pachner moves. For the particular combinatorial network structure chosen here there is an equivalent set of moves, which can be divided into $3$ groups.

* Moves involving only triangles separated by $2$-valent edges, which equal the 2-dimensional Pachner moves for the face atoms only, namely

\begin{equation}
\begin{tikzpicture}
\atoms{void}{0/, 1/p={0.5,-0.5}, 2/p={1,0}, 3/p={0.5,0.5}}
\fill[orange,opacity=0.35] (0)--(3)--(2)--(1)--cycle;
\draw[dashed,orange] (0)--(3) (3)--(2) (2)--(1) (0)--(1) (0)--(2);
\atoms{toyface}{a/p={0.5,-0.2}, b/p={0.5,0.2}}
\draw (a)--(b) (b)--++(45:0.4) (b)--++(135:0.4) (a)--++(-45:0.4) (a)--++(-135:0.4);
\end{tikzpicture}
\quad \text{\breakcellm{$=$\\ \color{orange}{$\longleftrightarrow$}}} \quad
\begin{tikzpicture}
\atoms{void}{0/, 1/p={0.5,-0.5}, 2/p={1,0}, 3/p={0.5,0.5}}
\fill[orange,opacity=0.35] (0)--(3)--(2)--(1)--cycle;
\draw[dashed,orange] (0)--(3) (3)--(2) (2)--(1) (0)--(1) (3)--(1);
\atoms{toyface}{a/p={0.3,0}, b/p={0.7,0}}
\draw (a)--(b) (b)--++(45:0.4) (a)--++(135:0.4) (b)--++(-45:0.4) (a)--++(-135:0.4);
\end{tikzpicture}\;
\end{equation}

and the same for the 1-3 Pachner move.

* Moves involving only $3$-valent edges separated by $2$-gon faces. In terms of cell complexes, those moves are Poincaré dual to the moves above. In network notation they look the same apart from that we have to use filled circles instead of empty circles.

* The most powerful move is the following move relating face and edge elements. It merges two triangles with two shared $3$-valent edges into a single triangle with one adjacent $3$-valent edge, i.e.,

\begin{equation}
\label{eq:bialgebra_move}
\begin{tikzpicture}
\atoms{void}{2/p={-30:1}, 1/p={90:1}, 0/p={-150:1}}
\fill[orange,opacity=0.35] (0)to[bend right=20](2)--(1)--cycle;
\fill[orange,opacity=0.35] (1)--(2)--++(30:0.4)--($(1)+(30:0.4)$)--cycle;
\fill[orange,opacity=0.35] (0)--(1)--++(150:0.4)--($(0)+(150:0.4)$)--cycle;
\atoms{toyface}{a/p={0.1,-0.1}}
\draw[dashed,orange,actualedge] (0)--(1) (1)--(2);
\atoms{toyedge}{c/p={30:0.5}, d/p={150:0.5}}
\draw[dashed,orange] (0)to[bend left=20](2) (0)to[bend right=20](2);
\draw (a)to[bend left=20](c) (a)to[bend right=20](d) (a)--++(-90:0.7);
\fill[orange,opacity=0.35] (0)to[bend left=20](2)--(1)--cycle;
\atoms{toyface}{b/p={-0.2,-0.1}}
\draw (b)to[bend left=20](c) (b)to[bend right=20](d) (b)--++(-90:0.4);
\draw (c)--++(30:0.4) (d)--++(150:0.4);
\end{tikzpicture}
\quad \text{\breakcellm{$=$\\ \color{orange}{$\longleftrightarrow$}}} \quad
\begin{tikzpicture}
\atoms{void}{2/p={-30:1}, 1/p={90:1}, 0/p={-150:1}}
\fill[orange,opacity=0.35] (0)--(2)--(1)--cycle;
\fill[orange,opacity=0.35] (0)--(2)--++(-150:0.5)--($(0)+(-150:0.6)$)--cycle;
\draw[dashed,orange] (0)--(1) (1)--(2);
\draw[dashed,orange,actualedge] (0)--(2);
\atoms{toyedge}{b/p={-90:0.5}}
\draw (b)--++(-150:0.7);
\fill[orange,opacity=0.35] (0)--(2)--++(-30:0.4)--($(0)+(-30:0.4)$)--cycle;
\atoms{toyface}{a/}
\draw (a)--(b) (a)--++(30:0.6) (a)--++(150:0.6) (b)--++(-30:0.6);
\end{tikzpicture}\;.
\end{equation}

§.§.§ Bi-algebras

As in the $1+$-dimensional case, the present liquid has a great similarity to a well known algebraic structure. To see this, we first note that there is an obvious liquid mapping from the $1+1$-dimensional liquid to the present liquid, in which the triangle (as part of the 2-dimensional cell complex) is mapped to the triangle (as part of the 2-dimensional cell complex). Dually to that, there is a liquid mapping in which the triangle is mapped to the $3$-valent edge. Thus, by the means of these two mappings every model of the present liquid contains two associative algebras.

Two associative algebras (more precisely, an algebra and a co-algebra) are called a bi-algebra, if they fulfil certain additional axioms. The main axiom (which states that the co-algebra is an algebra homomorphism) is precisely the move Eq. (<ref>).

Thus, the present liquid is basically the bi-algebra liquid, together with a few additional moves which make it “more topological”. This observation could be formalized as a liquid mapping from bi-algebras to the present liquid.

§.§.§ Models

Specifying the tensor type to array tensors, we look for models of the liquid, i.e., solutions to the move equations. The similarity to bi-algebras greatly helps assessing the situation: First, we know that bi-algebras fall into a discrete set of families related by basis changes, and so do the models of the present liquid. Second, there are many known examples for bi-algebras, many of which also yield models of the present liquid. Thus, in practice, we can look at the simplest examples of bi-algebras, see whether they can be turned into models of the present liquid, and check whether some of the models are in the same phase. Moreover, we will see that it is “rather unusual” for different models to be in the same phase, and that one can usually show their distinctness by evaluating closed networks.

As a particular example we recall that every group defines a bi-algebra, which can be turned into a model of the present liquid. Those models are equivalent to the Kitaev quantum double models [25], which are models for intrinsic topological order in $2+1$ diensions. E.g., if we pick the group $\mathbb{Z}_2$, the index configurations are the group elements $\{0,1\}$, and the edge and face tensors are

\begin{equation}
\label{eq:toric_code}
\begin{gathered}
\begin{tikzpicture}
\atoms{toyface}{a/}
\draw (a)edge[dind=b]++(30:0.4) (a)edge[dind=a]++(150:0.4) (a)edge[dind=c]++(-90:0.4);
\end{tikzpicture}=
\begin{cases}
1& \text{if } a+b+c=0 \mod 2\\
0& \text{otherwise}
\end{cases}\;,\\
\begin{tikzpicture}
\atoms{toyedge}{a/}
\draw (a)edge[dind=b]++(30:0.4) (a)edge[dind=a]++(150:0.4) (a)edge[dind=c]++(-90:0.4);
\end{tikzpicture}=
\begin{cases}
1& \text{if } a=b=c\\
0& \text{otherwise}
\end{cases}\;.
\end{gathered}
\end{equation}

It is easy to see that each tensor satisfies the 2-2 and 1-3 Pachner move, and both tensors together fulfil the move in Eq. (<ref>). Actually, for the sake of simplicity, we are ignoring a global factor of $1/2$ missing in the 1-3 Pachner move for face atoms. This will be fixed in Section <ref>.

This model corresponds to a commuting-projector Hamiltonian model known as the toric code
[25] defined for qubits on the edges of a square lattice and Hamiltonian given by

\begin{equation}
H=\sum_i A_i+\sum_j B_j \;.
\end{equation}

Here, $i$ runs over all plaquettes of the lattice and
each $A_i$ is defined as (suppressing the site index)

\begin{equation}
A= -Z_0Z_1Z_2Z_3\;,
\end{equation}

where $0,1,2,3$ label the $4$ edges adjacent to the corresponding plaquette. Dually, $j$ runs over all vertices and

\begin{equation}
B= -X_0X_1X_2X_3\;,
\end{equation}

where $0,1,2,3$ label the $4$ edges adjacent to the corresponding vertex.
The commuting projectors themselves can be written as $8$-index tensors

\begin{equation}
\label{eq:toric_code_projector}
\begin{gathered}
\quad \rightarrow\quad
\begin{tikzpicture}
\node[roundbox,wid=1.7](p)at(0,0){$P_A$};
\draw ([sx=-0.6]p.north)edge[dind=a']++(0,0.3) ([sx=-0.2]p.north)edge[dind=b']++(0,0.3) ([sx=0.2]p.north)edge[dind=c']++(0,0.3) ([sx=0.6]p.north)edge[dind=d']++(0,0.3) ([sx=-0.6]p.south)edge[dind=a]++(0,-0.3) ([sx=-0.2]p.south)edge[dind=b]++(0,-0.3) ([sx=0.2]p.south)edge[dind=c]++(0,-0.3) ([sx=0.6]p.south)edge[dind=d]++(0,-0.3);
\end{tikzpicture}\;,
\\
\quad \rightarrow\quad
\begin{tikzpicture}
\node[roundbox,wid=1.7](p)at(0,0){$P_B$};
\draw ([sx=-0.6]p.north)edge[dind=a']++(0,0.3) ([sx=-0.2]p.north)edge[dind=b']++(0,0.3) ([sx=0.2]p.north)edge[dind=c']++(0,0.3) ([sx=0.6]p.north)edge[dind=d']++(0,0.3) ([sx=-0.6]p.south)edge[dind=a]++(0,-0.3) ([sx=-0.2]p.south)edge[dind=b]++(0,-0.3) ([sx=0.2]p.south)edge[dind=c]++(0,-0.3) ([sx=0.6]p.south)edge[dind=d]++(0,-0.3);
\end{tikzpicture}\;.
\end{gathered}
\end{equation}

They are commuting because adjacent plaquettes and vertices always share two adjacent edges, and $Z_0Z_1$ commutes with $X_0X_1$. A tensor network representing the imaginary time evolution of the model is given by stacking layers of those commuting projectors.

To compare our topological model with the given commuting-projector model, we need a liquid mapping from the commuting-projector liquid to the topological liquid. A little bit of geometric imagination shows that the replacement

\begin{equation}
\label{eq:faceedge_commproj_mapping}
\begin{gathered}
\begin{tikzpicture}
\node[roundbox,wid=1.7](p)at(0,0){$P_A$};
\draw ([sx=-0.6]p.north)edge[dind=a']++(0,0.3) ([sx=-0.2]p.north)edge[dind=b']++(0,0.3) ([sx=0.2]p.north)edge[dind=c']++(0,0.3) ([sx=0.6]p.north)edge[dind=d']++(0,0.3) ([sx=-0.6]p.south)edge[dind=a]++(0,-0.3) ([sx=-0.2]p.south)edge[dind=b]++(0,-0.3) ([sx=0.2]p.south)edge[dind=c]++(0,-0.3) ([sx=0.6]p.south)edge[dind=d]++(0,-0.3);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\atoms{toyface}{0/, 1/p={0.5,0}}
\atoms{toyedge}{2/p={-0.5,-0.5}, 3/p={1,-0.5}, 4/p={1,0.5}, 5/p={-0.5,0.5}}
\draw (0)--(1) (0)--(2) (0)--(5) (1)--(3) (1)--(4) (2)edge[dind=a]++(-90:0.4) (2)edge[dind=a']++(180:0.4) (3)edge[dind=b]++(-90:0.4) (3)edge[dind=b']++(0:0.4) (5)edge[dind=c]++(90:0.4) (5)edge[dind=c']++(180:0.4) (4)edge[dind=d]++(90:0.4) (4)edge[dind=d']++(0:0.4);
\end{tikzpicture}\;,
\\
\begin{tikzpicture}
\node[roundbox,wid=1.7](p)at(0,0){$P_B$};
\draw ([sx=-0.6]p.north)edge[dind=a']++(0,0.3) ([sx=-0.2]p.north)edge[dind=b']++(0,0.3) ([sx=0.2]p.north)edge[dind=c']++(0,0.3) ([sx=0.6]p.north)edge[dind=d']++(0,0.3) ([sx=-0.6]p.south)edge[dind=a]++(0,-0.3) ([sx=-0.2]p.south)edge[dind=b]++(0,-0.3) ([sx=0.2]p.south)edge[dind=c]++(0,-0.3) ([sx=0.6]p.south)edge[dind=d]++(0,-0.3);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\atoms{toyedge}{0/, 1/p={0.5,0}}
\atoms{toyface}{2/p={-0.5,-0.5}, 3/p={1,-0.5}, 4/p={1,0.5}, 5/p={-0.5,0.5}}
\draw (0)--(1) (0)--(2) (0)--(5) (1)--(3) (1)--(4) (2)edge[dind=a]++(-90:0.4) (2)edge[dind=a']++(180:0.4) (3)edge[dind=b]++(-90:0.4) (3)edge[dind=b']++(0:0.4) (5)edge[dind=c]++(90:0.4) (5)edge[dind=c']++(180:0.4) (4)edge[dind=d]++(90:0.4) (4)edge[dind=d']++(0:0.4);
\end{tikzpicture}
\end{gathered}
\end{equation}

will turn a stack of commuting projectors into a cellulation of the same space-time. Indeed, we find that via this mapping, the topological model Eq. (<ref>) is mapped to the commuting-projector model Eq. (<ref>).

§.§ Topological models with boundary in 1+1 dimensions

§.§.§ Regular-lattice and extended model

As a third example, let us look at models in $1+1$ dimensions with physical boundary, where both the bulk and the boundary have a topological deformability. After Trotterization we obtain a tensor network like the following

\begin{equation}
\begin{tikzpicture}
\draw[dashed,orange] (-0.2,0)--(2,0) (-0.2,0.6)--(2,0.6) (-0.2,1.2)--(2,1.2);
\draw[dashed,orange,actualedge] (-0.3,-0.2)--(-0.3,1.4);
\draw[dashed,orange] (0.6,-0.2)--(0.6,1.4) (1.2,-0.2)--(1.2,1.4) (1.8,-0.2)--(1.8,1.4);
\atoms{toyplaquette}{0/p={0.3,0.3}, 1/p={0.9,0.3}, 2/p={1.5,0.3}, 3/p={0.3,0.9}, 4/p={0.9,0.9}, 5/p={1.5,0.9}}
\draw (0)--(1) (1)--(2) (3)--(4) (4)--(5) (0)--(3) (1)--(4) (2)--(5) (0)edge[]++(0,-0.5) (1)edge[]++(0,-0.5) (2)edge[]++(0,-0.5) (3)edge[]++(0,0.5) (4)edge[]++(0,0.5) (5)edge[]++(0,0.5) (2)edge[]++(0.5,0) (5)edge[]++(0.5,0);
\atoms{toybdplaquette}{a/p={-0.2,0.3}, b/p={-0.2,0.9}}
\draw[bound] (a-t)--(b-b) (b-t)edge[]++(0,0.4) (a-b)edge[]++(0,-0.4);
\draw (0)--(a-r) (3)--(b-r);
\end{tikzpicture}\;
\end{equation}

on a regular square lattice with boundary.
So, formally, we have a model of a substrate with two elements, the $4$-index circle-shaped element associated to the bulk plaquettes, and the $3$-index square-shaped element associated to the boundary edges. The degrees of freedom at the boundary (as well as the Trotterization procedure) can be different from the bulk. Thus, the dimension of the indices contracted between boundary tensors can be different from those of the indices contracted between the bulk tensors, which we denoted by using thicker lines.

Combinatorially, there are two different kinds of bonds, which we will refer to as bindings. We are generalizing our notion of substrate by associating to each index of each element a binding, and allowing only bonds between indices with the same binding.

The extended model is a model of a substrate representing arbitrary triangulations of $1+1$ dimensions-manifolds with boundary. Again, there is one tensor associated to each boundary edge and one tensor associated to each bulk triangle, given by

\begin{equation}
\begin{tikzpicture}
\atoms{void}{0/, 1/p={0.9,-0.9}, 2/p={2,-0.9}, 3/p={3,-0.4}, 4/p={2.8,0.8}, 5/p={1.6,1.2}, 6/p={0.5,1}, 7/p={1,0.2}, 8/p={2,0}}
\draw[dashed,orange] (2)--(3)--(4)--(5)--(6) (7)--(0) (7)--(1) (7)--(6) (7)--(5) (7)--(8) (8)--(1) (8)--(2) (8)--(3) (8)--(4) (8)--(5);
\draw[dashed,orange,line width=2] (0)--(1)--(2) (6)--(0);
\atoms{toyface}{a/p={0.7,-0.1}, i/p={1.3,-0.2}, h/p={1.7,-0.5}, g/p={2.3,-0.4}, f/p={2.6,0.1}, e/p={2.1,0.7}, d/p={1.5,0.5}, c/p={1,0.8}, b/p={0.6,0.4}}
\atoms{small, rot=-45,square}{{x/p={0.45,-0.45}}}
\atoms{small, rot=60,square}{{y/p={0.25,0.5}}}
\atoms{small,square}{{z/p={1.45,-0.9}}}
\draw (a)--(i)--(h)--(g)--(f)--(e)--(d)--(c)--(b)--(a) (d)--(i) (b)--(y-b) (a)--(x-t) (h)--(z-t) (g)--++(-60:0.5) (f)--++(20:0.5) (e)--++(70:0.5) (c)--++(110:0.5);
\draw[bound] (x-l)to[out=135,in=-120](y-l) (x-r)to[out=-45,in=180](z-l) (y-r)--++(60:0.3) (z-r)--++(0:0.3);
\end{tikzpicture}\;.
\end{equation}

The mapping from the square-lattice substrate to the topological substrate is obvious.

The substrate is turned into a liquid by adding the following moves: First, we have the 2-2 and 1-3 Pachner moves for the triangle element, which makes the liquid from Section <ref> a sub-liquid of the current liquid. Second, we add the following additional move

\begin{equation}
\label{eq:boundary_pachner_move}
\begin{tikzpicture}
\atoms{void}{0/p={0,-0.1}, 1/p={0.6,0.8}, 2/p={1.2,-0.1}}
\draw[dashed,orange] (0)--(1) (1)--(2);
\draw[dashed, orange,line width=2] (0)--(2);
\atoms{toyface}{a/p={0.6,0.35}}
\atoms{small,square}{b/p={0.6,0}}
\draw (a)--++(30:0.5) (a)--++(150:0.5) (a)--(b-t);
\draw[bound] (b-l)--++(-0.6,0) (b-r)--++(0.6,0);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{void}{0/p={0,-0.1}, 1/p={0.8,0.2}, 2/p={1.6,-0.1}}
\draw[dashed,orange,line width=2] (0)--(1) (1)--(2);
\atoms{small, rot=30,square}{{a/p={0.4,0.15}}}
\atoms{small, rot=-30,square}{{b/p={1.2,0.15}}}
\draw[bound] (a-r)to[out=30,in=150](b-l) (a-l)--++(-150:0.4) (b-r)--++(-30:0.4);
\draw (a-t)--++(120:0.3) (b-t)--++(60:0.3);
\end{tikzpicture}\;.
\end{equation}

The geometric interpretation of this move is to attach/remove a triangle to/from the boundary, which allows us to arbitrarily deform the boundary without changing the topology.

§.§.§ Representations and models

As the liquids above, our boundary liquid is again very similar to a very well-known algebraic structure, namely representations.
A representation of an algebra $A$ is a linear map

\begin{equation}
R: V\otimes A\rightarrow V\;,
\end{equation}

\begin{equation}
R(R(x,a),b)=R(x,a\cdot b)\;.
\end{equation}

This equation can written in tensor-network notation, and looks exactly like Eq. (<ref>).

In order to find models of the present liquid, we start with models of the $1+1$-dimensional liquid in Section <ref>, and extend them by a choice of boundary tensor. Let's start with the model given in Eq. (<ref>) related to the algebra of functions over some finite set $B$. For every $x\in B$, there is the corresponding irreducible representation, which defines a choice of boundary tensor:

\begin{equation}
\begin{tikzpicture}
\atoms{small,square}{{0/}}
\draw[bound] (0-r)--++(0.4,0) (0-l)--++(-0.4,0);
\draw (0-t)edge[dind=a]++(0,0.4);
\end{tikzpicture}=
\begin{cases}
1 &\text{if}\quad a=x\\
0 &\text{otherwise}
\end{cases}\;.
\end{equation}

The boundary indices without labels are trivial, that is, they have dimension $1$. For a $2$-element set $B$, this corresponds to a boundary condition of the Ising model, where any spin near the boundary is fixed to the value $x$.

§.§.§ Bulk-to-boundary mapping

A 2-manifold with boundary might also be interpreted as a manifold with one puncture for every boundary circle. Imagine filling each such a puncture with a disk. On the combinatorial level, we can do this by adding one additional vertex corresponding to the centre of the disk, and one additional triangle for every boundary edge, spanned by this boundary edge and the central vertex. Consider the boundary-less network for the filled, and the network-with-boundary for the non-filled triangulation. They can be mapped onto each other by reinterpreting the triangle atoms for the additional triangles as the boundary edge atoms for the boundary edges.
Such a reinterpretation can be formalized as the  liquid mapping

\begin{equation}
\begin{tikzpicture}
\atoms{small,square}{{0/}}
\draw[bound] (0-r)--++(0.4,0) (0-l)--++(-0.4,0);
\draw (0-t)--++(0,0.4);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\atoms{toyface}{{0/}}
\draw (0)--++(0.4,0) (0)--++(-0.4,0) (0)--++(0,0.4);
\end{tikzpicture}\;,
\hspace{1cm}
\begin{tikzpicture}
\atoms{toyface}{{0/}}
\draw (0-r)--++(0.4,0) (0-l)--++(-0.4,0) (0-t)--++(0,0.4);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\atoms{toyface}{{0/}}
\draw (0)--++(0.4,0) (0)--++(-0.4,0) (0)--++(0,0.4);
\end{tikzpicture}\;.
\end{equation}

If we apply this mapping to the move Eq. (<ref>) of the boundary liquid, it turns directly into the 2-2 move in Eq. (<ref>) of the bulk liquid. This example shows two new features of liquid mappings. 1) For a mapping from a substrate (or liquid) with different elements, we have to give one network for each element. 2) If the substrate (or liquid) has different bindings, then each binding of the source liquid is mapped to a binding of the target liquid. In the present example, both the bulk and boundary binding are mapped to the bulk binding,

\begin{equation}
\begin{tikzpicture}
\draw (0,0)--(0,0.4);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\draw (0,0)--(0,0.4);
\end{tikzpicture}\;,
\hspace{1cm}
\begin{tikzpicture}
\draw[bound] (0,0)--(0,0.4);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\draw (0,0)--(0,0.4);
\end{tikzpicture}\;.
\end{equation}

In general, each binding of the source substrate can be associated with a collection of bindings of the target substrate, which can contain the same binding multiple times.

§ TOPOLOGY AND NON-COMMUTATIVITY

In this section we will revisit the example of topological order in $1+1$ dimensions from Section <ref> and discuss an important issue that we have not addressed so far. If we want the liquid to represent topological manifolds, we need to add more structure to the network. In particular we will motivate that it is necessary to distinguish the different indices of an element and show how this can be implemented concretely. The additional structure makes the liquid more complicated than the liquid from the previous example. To handle this complexity in the most efficient way, we seek a way to simplify the liquid without losing its ability to describe topological phases. In doing so, we invoke the concept of liquid mappings and introduce the notion of equivalent tensor liquids. We present what we believe to be the simplest representative of a topological liquid in $1+1$ dimensions and classify its phases.

§.§ Distinguishing indices

In Section <ref>, we have represented the triangulation of a manifold by a graph that we referred to as network, with vertices referred to as atoms and edges referred to as bonds. The graph is dual to the triangulation, that is, for every triangle there is one atom, and between every pair of adjacent triangles there is a bond

\begin{equation}
\label{eq:graph_liquid}
\begin{tikzpicture}
\atoms{vertex}{0/, 1/p={1,0}, 2/p={60:1}}
\draw (0)--(1) (0)--(2) (1)--(2);
\end{tikzpicture}
\quad\rightarrow\quad
\begin{tikzpicture}
\atoms{toyface}{0/}
\draw (0)--++(30:0.5) (0)--++(150:0.5) (0)--++(-90:0.5);
\end{tikzpicture}\;.
\end{equation}

We would like the combinatorics of networks and moves to reflect continuum manifolds and homeomorphisms in a faithful way. However, the network combinatorics introduced so far does not uniquely encode the full combinatorial information of the triangulation. Imagine rebuilding the triangulation from the network's graph by replacing each vertex with a triangle and gluing the triangles associated to connected vertices along common edges. We encounter two problems: 1) The combinatorial structure of the graph does not distinguish between the three adjacent bonds, so we cannot tell which edges of the triangles we have to glue together. 2) Two edges can be glued in two opposite ways. E.g., consider the following graph that corresponds to two triangles with all edges glued together pairwise as

\begin{equation}
\begin{tikzpicture}
\atoms{toyface}{0/, 1/p={1,0}}
\draw (0)edge[bend left=50](1) (0)edge(1) (0)edge[bend right=50](1);
\end{tikzpicture} \;.
\end{equation}

This graph does not determine the topology of the resulting manifold. If we glue one of the three edge pairs, we obtain a $4$-gon. Depending on how we glue the remaining edges of the $4$-gon, we can obtain a sphere, a real projective plane, a torus, or a Klein bottle.

The second problem can be solved by giving each edge an orientation and demanding that those orientations match when we glue two edges of two triangles. The first problem is solved by realizing that any manifold can be triangulated using only triangles with non-cyclic edge orientations [This can be seen by refining a non-oriented triangulation via a construction known as barycentric subdivision, which can be equipped with a canonical non-cyclic edge orientation.]. This is also known as a triangulation with a branching structure. For a fixed triangle, the non-cyclic edge orientations induce an ordering of its vertices,

\begin{equation}\label{eq:branching}
\begin{tikzpicture}
\atoms{vertex}{{0/lab={ang=-150,t=0}}, {1/p={1,0},lab={ang=-30,t=2}}, {2/p={60:1},lab={ang=90,t=1}}}
\draw (0)edge[or,redlabr=02](1) (0)edge[or,redlab=01](2) (2)edge[or,redlab=12](1);
\end{tikzpicture}\;.
\end{equation}

This allows us to distinguish the three edges and refer to them by their source and target vertex.
In our network notation, we allow rotating/reflecting the shapes of individual atoms, which makes it impossible to distinguish the three indices if the shape is a small circle. The shape for the element representing a branching-structure triangle should have less symmetry, which we implement by next-to-shape markers as

\begin{equation}
\label{eq:branching_triangle_shape}
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=+,redlabe=02]++(-90:0.6) (0)edge[flags=-,redlabe=01]++(150:0.6) (0)edge[flags=-,redlabe=12]++(30:0.6);
\end{tikzpicture} \;.
\end{equation}

The clockwise or counter-clockwise flags allow to uniquely identify the $3$ indices of the element with the edges $01$, $02$ or $12$ of the branching structure triangle, as indicated by the red labels. Note that here and in the subsequent, such red labels are not part of the formal graphical notation, but serve as a help to identify the network notation with its geometric interpretation in terms of cell complexes.
Networks using the new shape representing a branching structure triangle uniquely specify the triangulation and thus the topology. E.g., the network

\begin{equation}
\begin{tikzpicture}
\atoms{face}{0/, 1/p={1.5,0}}
\draw (0)edge[flage=-,flags=-,bend left=60](1) (0)edge[flage=-,flags=-](1) (0)edge[flage=+,flags=+,bend right=60](1);
\end{tikzpicture} \;
\end{equation}

represents a sphere unambiguously.

In our new (and final) notion of networks, the indices of an element are always distinct. However, we can still interpret the old notion, where (some of) the indices have not been distinguished. Indistinguishability of indices means that we are allowed to permute them, which is nothing but a move. Now, whenever we choose a shape for an element which has rotation/reflection symmetries due to which we cannot distinguish some of the indices, we implicitly assume that all the corresponding index permutation moves (or better, a set of moves generating all the permutation moves) are part of the liquid. Explicitly, we will denote such permutation moves using cycle notation, e.g.,

\begin{equation}
\begin{tikzpicture}
\atoms{toyface}{{0/}}
\draw (0)edge[dind=c]++(-90:0.5) (0)edge[dind=a]++(150:0.5) (0)edge[dind=b]++(30:0.5);
\end{tikzpicture}
\symeq
(ab)\;, \qquad
\begin{tikzpicture}
\atoms{toyface}{{0/}}
\draw (0)edge[dind=c]++(-90:0.5) (0)edge[dind=a]++(150:0.5) (0)edge[dind=b]++(30:0.5);
\end{tikzpicture}
\symeq
\end{equation}

If we use a shape without any symmetries, such as the one in Eq. (<ref>), the index permutations can be denoted as ordinary moves

\begin{equation}
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=+,dind=c]++(-90:0.6) (0)edge[flags=-,dind=a]++(150:0.6) (0)edge[flags=-,dind=b]++(30:0.6);
\end{tikzpicture}=
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=+,dind=c]++(-90:0.6) (0)edge[flags=-,dind=b]++(150:0.6) (0)edge[flags=-,dind=a]++(30:0.6);
\end{tikzpicture}\;,
\hspace{1cm}
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=+,dind=c]++(-90:0.6) (0)edge[flags=-,dind=a]++(150:0.6) (0)edge[flags=-,dind=b]++(30:0.6);
\end{tikzpicture}=
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=+,dind=b]++(-90:0.6) (0)edge[flags=-,dind=a]++(150:0.6) (0)edge[flags=-,dind=c]++(30:0.6);
\end{tikzpicture}\;.
\end{equation}

If we interpret those moves in terms of triangulations, they correspond to cutting out a triangle and gluing it in a different way. Such an operation generally changes the topology of the triangulation. So the liquid we introduced in Section <ref> has moves which are sufficient to have topological deformability, but also additional moves which go beyond topological deformability. We therefore expect that models of this liquid are too restricted and do not contain the most general fixed point models for topological order.

§.§ Non-simplified liquid

The branching structure/flags also need to be incorporated into the moves of the liquid. There are different ways a branching structure can be added to the Pachner moves. For the 2-2 Pachner moves (keeping in mind moves are not actually different if they are just rotated/reflected or we exchanged the left and right side), we count $3$ different versions. One of them is

\begin{equation}
\label{eq:22pachner_move}
\begin{tikzpicture}
\atoms{vertex}{{0/lab={t=0,p=180:0.25}}}
\atoms{vertex}{{1/p={0.5,-0.5},lab={t=3,p=-90:0.25}}}
\atoms{vertex}{{2/p={1,0},lab={t=2,p=0:0.25}}}
\atoms{vertex}{{3/p={0.5,0.5},lab={t=1,p=90:0.25}}}
\draw (0)edge[or](3) (3)edge[or](2) (2)edge[or](1) (0)edge[or](1) (0)edge[or](2);
\end{tikzpicture}
\quad\leftrightarrow\quad
\begin{tikzpicture}
\atoms{vertex}{{0/lab={t=0,p=180:0.25}}}
\atoms{vertex}{{1/p={0.5,-0.5},lab={t=3,p=-90:0.25}}}
\atoms{vertex}{{2/p={1,0},lab={t=2,p=0:0.25}}}
\atoms{vertex}{{3/p={0.5,0.5},lab={t=1,p=90:0.25}}}
\draw (0)edge[or](3) (3)edge[or](2) (2)edge[or](1) (0)edge[or](1) (3)edge[or](1);
\end{tikzpicture} \;.
\end{equation}

Another one can obtained by, e.g., inverting the orientation of the $2-3$ edge. Note that if we glue the two patches above at their boundary, we obtain the surface of a branching-structure tetrahedron. In general, every Pachner move corresponds to a decomposition of that tetrahedron into two parts, and the $3$ versions of the 2-2 Pachner move correspond to the $3$ different decompositions of the tetrahedron into two faces on each side.

In the new network notation, the move becomes

\begin{equation}
\label{eq:22pachner_branching}
\begin{tikzpicture}
\atoms{face}{{0/lab={t=012,p=-150:0.35}}, {1/p={0,-0.8},lab={t=023,p=150:0.35}}}
\draw (0)edge[flage=+,flags=+](1) (0)edge[flags=-,sdind=01]++(135:0.5) (0)edge[flags=-,sdind=12]++(45:0.5) (1)edge[flags=-,sdind=23]++(-45:0.5) (1)edge[flags=+,sdind=03]++(-135:0.5);
\end{tikzpicture}=
\begin{tikzpicture}
\atoms{face}{{0/lab={t=013,p=90:0.35}}, {1/p={0.8,0},lab={t=123,p=90:0.35}}}
\draw (0)edge[flage=-,flags=-](1) (0)edge[flags=-,sdind=01]++(135:0.5) (1)edge[flags=-,sdind=12]++(45:0.5) (1)edge[flags=-,sdind=23]++(-45:0.5) (0)edge[flags=+,sdind=03]++(-135:0.5);
\end{tikzpicture} \;.
\end{equation}

The red labels identify the atoms in the network with the triangles in the geometric interpretation. E.g., $023$ refers to the triangle in Eq. (<ref>) whose $0$-vertex is the vertex $0$, $1$-vertex is $2$, and $2$-vertex is $3$. Note again that the red labels are only hints for the reader and not part of the actual notation. Also, the open index labels were chosen in accordance with the names of the corresponding edges.

Analogously, there are now $4$ different versions of the 1-3 Pachner move, corresponding to the $4$ decompositions of the branching structure tetrahedron into two patches with $1$ and $3$ triangles each. One of them is

\begin{equation}
\begin{tikzpicture}
\atoms{vertex}{0/, 1/p={-30:0.6}, 2/p={90:0.6}, 3/p={-150:0.6}}
\draw (0)edge[or](1) (0)edge[or](2) (0)edge[or](3) (3)edge[or](2) (2)edge[or](1) (3)edge[or](1);
\end{tikzpicture}
\quad\leftrightarrow\quad
\begin{tikzpicture}
\atoms{vertex}{1/p={-30:0.6}, 2/p={90:0.6}, 3/p={-150:0.6}}
\draw (3)edge[or](2) (2)edge[or](1) (3)edge[or](1);
\end{tikzpicture} \;.
\end{equation}

As some sort of convention, we might also want to introduce the following triangle cancellation move

\begin{equation}
\label{eq:triangle_cancellation_geo}
\begin{tikzpicture}
\atoms{vertex}{0/lab={t=0,ang=-90}, {1/p={0,0.6},lab={t=1,ang=0}}, {2/p={0,1.2},lab={t=2,ang=90}}}
\draw (0)edge[or](1) (1)edge[or](2) (0)edge[bend right=60, or,looseness=1.3](2) (0)edge[bend left=60, or,looseness=1.3](2);
\end{tikzpicture}
\quad\leftrightarrow\quad
\begin{tikzpicture}
\atoms{vertex}{1/lab={t=0,ang=-90}, {2/p={0,1},lab={t=2,ang=90}}}
\draw (1)edge[or](2);
\end{tikzpicture} \;,
\end{equation}

implying that a non-cyclic 2-gon can be shrunk to a single edge, which is represented by a free bond in network notation as

\begin{equation}
\label{eq:triangle_cancellation}
\begin{tikzpicture}
\atoms{face}{0/lab={t=012,p=-120:0.35}, {1/p={0.8,0},lab={t=012,p=-60:0.35}}}
\draw[rounded corners,flage=+,flags=+] (0)to[out=45,in=135](1);
\draw[rounded corners,flage=+,flags=+] (0)to[out=-45,in=-135](1);
\draw (0)edge[flags=-,sdind=02l]++(180:0.5) (1)edge[flags=+,sdind=02r]++(0:0.5);
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,0)edge[mark={lab=$\scriptstyle{02l}$,b},sdind=02r](0.5,0);
\end{tikzpicture}\;.
\end{equation}

If we glue one edge of the left hand side of Eq. (<ref>) to one boundary edge of any patch of a triangulation (including itself), this can be undone with Pachner moves. So the Pachner moves imply that the corresponding tensor is a projector, and contracting any index of any other tensor of the model with this projector yields the same tensor again. However, they do clearly not imply the triangle-cancellation move, and formally, the liquids with and without that move are inequivalent (in a sense that we will make precise soon).

However, when considering ordinary models of the liquid (with real or complex tensors), Eq. (<ref>) can be viewed as a convention that does not hurt to impose. The projector in Eq. (<ref>) has a $n$-dimensional support, and there exists an isometry which identifies this $n$-dimensional support with an $n$-dimensional vector space. Applying this isometry to every index of every tensor yields a model which is equivalent, as the tensors are invariant under applying the corresponding projector. In doing so, the tensor corresponding to the projector itself becomes the identity matrix.

In total, we end up with a liquid with $8$ moves that we refer to as the “non-simplified liquid”. As the moves correspond to equations between tensor networks that we need to solve in order to find models, it is important that the moves of a liquid are as simple as possible. In the following, we will find a “simplified liquid” which is equivalent to the non-simplified liquid, but has less and simpler moves.

§.§ Simplified liquid

The simplified liquid has one additional element whose geometric interpretation is a $2$-gon cell with cyclic edge orientations,

\begin{equation}
\label{eq:auxiliary_element}
\begin{tikzpicture}
\atoms{vertex}{1/lab={t=0,ang=-90}, {2/p={0,0.8},lab={t=1,ang=90}}}
\draw (1)edge[or,bend left=45](2) (1)edge[ior,bend right=45](2);
\end{tikzpicture}
\quad\rightarrow\quad
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=-,redlabe=01]++(180:0.6) (0)edge[flags=-,redlabe=10]++(0:0.6);
\end{tikzpicture} \;.
\end{equation}

The new element will be denoted by a circle as well, however, it can be distinguished from the triangle element due to the different number of indices.
Of course, a $2$-gon cannot be embedded non-degenerately into Euclidean space without bending its edges. But this is no cause of a problem as we are talking about combinatorial/topological cell complexes and not geometric ones. The 2-gon is rotation symmetric which corresponds to a move

\begin{equation}
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=-,dind=a]++(180:0.6) (0)edge[flags=-,dind=b]++(0:0.6);
\end{tikzpicture}
\symeq (ab)
\end{equation}

justifying the choice of shape. This move can be derived from the moves below, however.

The moves of the simplified liquid only contain one single Pachner move, namely the 2-2 Pachner move in Eq. (<ref>). All other 2-2 Pachner moves can be derived via additional moves of the simplified liquid related to symmetries of the triangle. In contrast to the liquid in Section <ref>, rotating or reflecting the triangle would change the branching structure. However, the changes of the edge orientations can be undone by gluing the cyclic 2-gon to the involved edges. This yields, e.g., the (12)-triangle symmetry move

\begin{equation}
\begin{tikzpicture}
\atoms{vertex}{v1/lab={t=2,ang=-90}, {v2/p={0,0.8},lab={t=1,ang=90}}, {v5/p={-0.5,0.4},lab={t=0,ang=180}}}
\draw (v1) edge[ior] (v5) (v2) edge[ior](v5) (v1)edge[ior](v2) (v1) edge[or, bend right=60] (v2);
\end{tikzpicture}
\quad\leftrightarrow\quad
\begin{tikzpicture}
\atoms{vertex}{{v1/lab={t=2,ang=-90}}, {v2/p={0,0.8},lab={t=1,ang=90}}, {v5/p={-0.5,0.4},lab={t=0,ang=180}}}
\draw (v1) edge[ior] (v5) (v2) edge[ior](v5) (v1)edge[or](v2);
\end{tikzpicture}\;,
\end{equation}

where the nomenclature refers to effectively interchanging the role of the vertices 1 and 2. In network notation, this is

\begin{equation}
\label{eq:12_triangle_symmetry}
\begin{tikzpicture}
\atoms{face}{{0/lab={t=012,p=-60:0.35}}, {1/p={0.8,0},lab={t=12,p=-90:0.35}}}
\draw (0)edge[flags=-,flage=-](1) (0)edge[flags=-,sdind=01]++(120:0.5) (0)edge[flags=+,sdind=02]++(-120:0.5) (1)edge[flags=+,sdind=21]++(0:0.5);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{face}{{0/lab={t=021,p=-60:0.35}}}
\draw (0)edge[flags=-,sdind=01]++(120:0.5) (0)edge[flags=+,sdind=02]++(-120:0.5) (0)edge[flags=+,sdind=21]++(0:0.5);
\end{tikzpicture}\;.
\end{equation}

In order to generate the full symmetry group $S_3$ of the triangle, one only needs one further move, the (01) triangle symmetry move

\begin{equation}
\label{eq:01_triangle_symmetry_geo}
\begin{tikzpicture}
\atoms{vertex}{{v1/lab={t=1,ang=-90}}, {v2/p={0,0.8},lab={t=0,ang=90}}, {v5/p={-0.5,0.4},lab={t=2,ang=180}}}
\draw (v1) edge[or] (v5) (v2) edge[or](v5) (v1)edge[ior](v2) (v1) edge[or, bend right=60] (v2);
\end{tikzpicture}
\quad\leftrightarrow\quad
\begin{tikzpicture}
\atoms{vertex}{{v1/lab={t=1,ang=-90}}, {v2/p={0,0.8},lab={t=0,ang=90}}, {v5/p={-0.5,0.4},lab={t=2,ang=180}}}
\draw (v1) edge[or] (v5) (v2) edge[or](v5) (v1)edge[or](v2);
\end{tikzpicture}\;.
\end{equation}

Again, in network notation, this amounts to

\begin{equation}
\label{eq:01_triangle_symmetry}
\begin{tikzpicture}
\atoms{face}{{0/lab={t=012,p=-60:0.35}}, {1/p={0.8,0},lab={t=01,p=-90:0.35}}}
\draw (0)edge[flags=-,flage=-](1) (0)edge[flags=+,sdind=02]++(120:0.5) (0)edge[flags=-,sdind=12]++(-120:0.5) (1)edge[flags=+,sdind=10]++(0:0.5);
\end{tikzpicture}
\begin{tikzpicture}
\atoms{face}{{0/lab={t=102,p=-60:0.35}}}
\draw (0)edge[flags=+,sdind=02]++(120:0.5) (0)edge[flags=-,sdind=12]++(-120:0.5) (0)edge[flags=+,sdind=10]++(0:0.5);
\end{tikzpicture}\;.
\end{equation}

The 1-3 Pachner moves can be derived from the 2-2 Pachner moves via the triangle cancellation move in Eq. (<ref>), which is also part of the simplified liquid. Analogously, there is the 2-gon cancellation move

\begin{equation}
\begin{tikzpicture}
\atoms{vertex}{{v1/lab={t=0,ang=-90}},{v2/p={0,0.8},lab={t=1,ang=90}}}
\draw (v1) edge[or, bend left=60] (v2) (v2) edge[or] (v1) (v1) edge[or, bend right=60] (v2);
\end{tikzpicture}
\quad\leftrightarrow\quad
\begin{tikzpicture}
\atoms{vertex}{{v1/lab={t=0,ang=-90}},{v2/p={0,0.8},lab={t=1,ang=90}}}
\draw (v1) edge[or] (v2);
\end{tikzpicture}\;.
\end{equation}

In network notation, we have

\begin{equation}
\label{eq:2gon_cancellation}
\begin{tikzpicture}
\atoms{face}{0/, 1/p={0.8,0}}
\draw (0)edge[flags=-,flage=-](1) (0)edge[flags=-,dind=a]++(180:0.5) (1)edge[flags=+,dind=b]++(0:0.5);
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,0)edge[mark={lab=$a$,b},dind=b](0.5,0);
\end{tikzpicture}\;.
\end{equation}

§.§ Equivalence of the simplified and non-simplified liquid

In this section, we will motivate why the simplified and non-simplified liquids are “equivalent”. For this, we should be able to rewrite networks of the simplified liquid as networks of the non-simplified liquid, and vice versa. This can be formalized by two liquid mappings $\calm_1$ and $\calm_2$, going from the non-simplified liquid to the simplified liquid, and back.

Note that the elements of the non-simplified liquid are identified with a subset of the elements of the simplified liquid. So there is a “trivial” candidate for the mapping $\mathcal{M}_1$, mapping the triangle of the non-simplified liquid to the triangle of the simplified liquid. In order to show that this defines indeed a liquid mapping, we need to show that the mapped non-simplified moves are derived from the simplified moves. As the mapping is “trivial”, the mapped non-simplified moves just look like the non-simplified moves.

* One of the branching-structure 1-3 Pachner moves is derived from the 2-2 Pachner move in Eq. (<ref>) and the triangle cancellation move in Eq. (<ref>):

\begin{equation}
\label{eq:13pachner}
\begin{tikzpicture}
\atoms{face}{0/p=0:0.5, 1/p=-120:0.5, 2/p=120:0.5}
\draw (0)edge[flags=+,flage=+](1) (1)edge[flags=-,flage=-](2) (0)edge[flags=+,flage=+](2) (0)edge[flags=-,dind=a]++(0:0.5) (1)edge[flags=+,dind=b]++(-120:0.5) (2)edge[flags=-,dind=c]++(120:0.5);
\end{tikzpicture}
\overset{\eqref{eq:22pachner_branching}}{=}
\begin{tikzpicture}
\atoms{face}{0/p=0:0.5, 1/p=180:0.3, 2/p=180:1}
\draw (1)edge[flags=+,flage=+](2) (0)edge[flags=+,flage=+,bend left=50](1) (0)edge[flags=+,flage=+,bend right=50](1) (0)edge[flags=-,dind=a]++(0:0.5) (2)edge[flags=+,dind=b]++(-120:0.5) (2)edge[flags=-,dind=c]++(120:0.5);
\end{tikzpicture}
\overset{\eqref{eq:triangle_cancellation}}{=}
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=-,dind=a]++(0:0.5) (0)edge[flags=+,dind=b]++(-120:0.5) (0)edge[flags=-,dind=c]++(120:0.5);
\end{tikzpicture}\;.
\end{equation}

* All other versions of branching structure 2-2 Pachner moves are derived from the 2-2 Pachner move in Eq. (<ref>), together with the two triangle symmetry moves. E.g., the following 2-2 Pachner move

\begin{equation}
\begin{tikzpicture}
\atoms{vertex}{0/, 1/p={0.5,-0.5}, 2/p={1,0}, 3/p={0.5,0.5}}
\draw (0)edge[or](3) (3)edge[or](2) (2)edge[ior](1) (0)edge[or](1) (3)edge[or](1);
\end{tikzpicture}
\quad\leftrightarrow\quad
\begin{tikzpicture}
\atoms{vertex}{0/, 1/p={0.5,-0.5}, 2/p={1,0}, 3/p={0.5,0.5}}
\draw (0)edge[or](3) (3)edge[or](2) (2)edge[ior](1) (0)edge[or](1) (0)edge[or](2);
\end{tikzpicture}
\end{equation}

is derived by

\begin{equation}
\begin{gathered}
\begin{tikzpicture}
\atoms{face}{0/, 1/p={0.8,0}}
\draw (0)edge[flage=-,flags=-](1) (0)edge[flags=-,dind=a]++(135:0.5) (1)edge[flags=-,dind=b]++(45:0.5) (1)edge[flags=+,dind=c]++(-45:0.5) (0)edge[flags=+,dind=d]++(-135:0.5);
\end{tikzpicture}
\overset{\overline{\eqref{eq:12_triangle_symmetry}}}{=}
\begin{tikzpicture}
\atoms{face}{0/, 1/p={0.8,0}, 2/p={1.3,-0.5}}
\draw (0)edge[flage=-,flags=-](1) (0)edge[flags=-,dind=a]++(135:0.5) (1)edge[flags=-,dind=b]++(45:0.5) (1)edge[flags=-,flage=-](2) (2)edge[flags=+,dind=c]++(-45:0.5) (0)edge[flags=+,dind=d]++(-135:0.5);
\end{tikzpicture}
\\\overset{\overline{\eqref{eq:22pachner_branching}}}{=}
\begin{tikzpicture}
\atoms{face}{0/p={0,0.8}, 1/, 2/p={0.5,-0.5}}
\draw (0)edge[flage=+,flags=+](1) (0)edge[flags=-,dind=a]++(135:0.5) (0)edge[flags=-,dind=b]++(45:0.5) (1)edge[flags=-,flage=-](2) (2)edge[flags=+,dind=c]++(-45:0.5) (1)edge[flags=+,dind=d]++(-135:0.5);
\end{tikzpicture}
\overset{\eqref{eq:12_triangle_symmetry}}{=}
\begin{tikzpicture}
\atoms{face}{0/p={0,0.8}, 1/}
\draw (0)edge[flage=+,flags=+](1) (0)edge[flags=-,dind=a]++(135:0.5) (0)edge[flags=-,dind=b]++(45:0.5) (1)edge[flags=+,dind=c]++(-45:0.5) (1)edge[flags=+,dind=d]++(-135:0.5);
\end{tikzpicture}\;.
\end{gathered}
\end{equation}

The bar over the referenced equation denotes that the move is applied from right to left.

* Similarly, all other 1-3 Pachner moves are derived from the move above in Eq. (<ref>), together with the 2-gon cancellation move and the triangle symmetry moves.

The mapping $\mathcal{M}_2$ is only slightly more complicated.

* The triangle part of both liquids and accordingly mapped onto the itself (as part of the non-simplified liquid).

* A $2$-gon cell can be triangulated using two triangles

\begin{equation}
\begin{tikzpicture}
\atoms{vertex}{{1/lab={t=0,ang=-90}}, {2/p={0,1},lab={t=2,ang=90}}}
\draw (1)edge[or,bend left=45](2) (1)edge[ior,bend right=45](2);
\end{tikzpicture}
\quad\rightarrow\quad
\begin{tikzpicture}
\atoms{vertex}{{0/lab={t=0,ang=-90}}, {1/p={0,0.6},lab={t=1,ang=0}}, {2/p={0,1.2},lab={t=2,ang=90}}}
\draw (0)edge[or](1) (1)edge[ior](2) (0)edge[bend right=60, ior,looseness=1.3](2) (0)edge[bend left=60, or,looseness=1.3](2);
\end{tikzpicture}\;.
\end{equation}

Accordingly, the mapping for the $2$-gon is given by

\begin{equation}
\label{eq:2gon_mapping}
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=-,dind=a]++(180:0.5) (0)edge[flags=-,dind=b]++(0:0.5);
\end{tikzpicture}
\coloneqq
\begin{tikzpicture}
\atoms{face}{0/, 1/p={0.8,0}}
\draw[rounded corners,flage=-,flags=-] (0)to[out=45,in=135](1);
\draw[rounded corners,flage=+,flags=+] (0)to[out=-45,in=-135](1);
\draw (0)edge[flags=-,dind=a]++(180:0.5) (1)edge[flags=-,dind=b]++(0:0.5);
\end{tikzpicture}\;.
\end{equation}

Again, we have to find derivations for the mapped simplified moves from the non-simplified moves. E.g., if we plug the mapping Eq. (<ref>) into the 2-gon cancellation move Eq. (<ref>), we obtain

\begin{equation}
\begin{tikzpicture}
\atoms{face}{0/, 1/p={0.8,0}, 2/p={1.6,0}, 3/p={2.4,0}}
\draw (0)edge[out=45,in=135,flage=-,flags=-](1) (0)edge[out=-45,in=-135,flage=+,flags=+](1) (2)edge[out=45,in=135,flage=-,flags=-](3) (2)edge[out=-45,in=-135,flage=+,flags=+](3) (1)edge[flags=-,flage=+](2);
\draw (0)edge[flags=-,dind=a]++(180:0.5) (3)edge[flags=-,dind=b]++(0:0.5);
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,0)edge[mark={lab=$a$,b},dind=b](0.5,0);
\end{tikzpicture}\;.
\end{equation}

This can be derived by 1) a 2-2 Pachner move, 2) a 1-3 Pachner move, and 3) the triangle cancellation move. We will not explicitly give derivations for each mapped move here. Instead, we would like to remark that the mapped moves (except for the 2-gon and triangle cancellation moves) correspond to re-triangulations of a disk. It is known that any two triangulations of the same (piece-wise linear) manifold are related by a sequence of Pachner moves [24]. So, if we rely on this statement about the geometric interpretation, we know that derivations for all mapped moves must exist.

So, we have found two liquid mappings going from the non-simplified liquid to the simplified liquid and back. However, this alone does not really mean anything, e.g., between any two liquids there's the trivial mapping which maps every binding to the empty collection of bindings, and every element to the empty network. What we additionally need is that that if we go from the non-simplified liquid to the simplified liquid and back, we end up with the same network. In other words, $\mathcal{M}_2\circ\mathcal{M}_1$ should be the identity, and the same should hold for $\mathcal{M}_1\circ\mathcal{M}_2$.

We find indeed that $\mathcal{M}_2\circ\mathcal{M}_1$ is the identity on the triangle, and also $\mathcal{M}_1\circ\mathcal{M}_2$ is the identity on the triangle. However, if we apply $\mathcal{M}_1\circ\mathcal{M}_2$ to the cyclic $2$-gon

\begin{equation}
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=-,dind=a]++(180:0.5) (0)edge[flags=-,dind=b]++(0:0.5);
\end{tikzpicture}
\overset{\calm_2}{\coloneqq}
\begin{tikzpicture}
\atoms{face}{0/, 1/p={0.8,0}}
\draw[rounded corners,flage=-,flags=-] (0)to[out=45,in=135](1);
\draw[rounded corners,flage=+,flags=+] (0)to[out=-45,in=-135](1);
\draw (0)edge[flags=-,dind=a]++(180:0.5) (1)edge[flags=-,dind=b]++(0:0.5);
\end{tikzpicture}
\overset{\calm_1}{\coloneqq}
\begin{tikzpicture}
\atoms{face}{0/, 1/p={0.8,0}}
\draw[rounded corners,flage=-,flags=-] (0)to[out=45,in=135](1);
\draw[rounded corners,flage=+,flags=+] (0)to[out=-45,in=-135](1);
\draw (0)edge[flags=-,dind=a]++(180:0.5) (1)edge[flags=-,dind=b]++(0:0.5);
\end{tikzpicture}\;,
\end{equation}

we find that it does not map the $2$-gon to itself. This is again fine,
as the equation between the very left and the very right is a derived move of the
simplified liquid,

\begin{equation}
\begin{gathered}
\begin{tikzpicture}
\atoms{face}{0/}
\draw (0)edge[flags=-,dind=a]++(180:0.5) (0)edge[flags=-,dind=b]++(0:0.5);
\end{tikzpicture}
\overset{\overline{\eqref{eq:triangle_cancellation}}}{=}
\begin{tikzpicture}
\atoms{face}{0/, 1/p={0.8,0}, 2/p={1.6,0}}
\draw (0)edge[out=45,in=135,flage=+,flags=+](1) (0)edge[out=-45,in=-135,flage=+,flags=+](1) (1)edge[flags=+,flage=+](2) (0)edge[flags=-,dind=a]++(180:0.5) (2)edge[flags=-,dind=b]++(0:0.5);
\end{tikzpicture}
\\
\overset{\overline{\eqref{eq:2gon_cancellation}}}{=}
\begin{tikzpicture}
\atoms{face}{0/, 1/p={1.6,0}, 2/p={2.2,0}, x/p={0.5,0.4}, y/p={1.1,0.4}}
\draw (0)edge[out=-45,in=-135,flage=+,flags=+](1) (0)edge[flage=+,flags=+](x) (x)edge[flage=-,flags=-](y) (y)edge[flage=+,flags=+](1) (1)edge[flags=+,flage=+](2) (0)edge[flags=-,dind=a]++(180:0.5) (2)edge[flags=-,dind=b]++(0:0.5);
\end{tikzpicture}
\\\overset{\eqref{eq:12_triangle_symmetry}^2}{=}
\begin{tikzpicture}
\atoms{face}{0/, 1/p={0.8,0}, 2/p={1.6,0}}