Highly complex novel critical behavior from the intrinsic randomness of quantum mechanical measurements on critical ground states -- a controlled renormalization group analysis
Rushikesh A. Patil Andreas W. W. Ludwig


















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



§ ABSTRACT 


We consider the effects of weak measurements on the quantum critical ground state
of  the  one-dimensional (a)  tricritical and (b) critical quantum Ising model, by measuring in (a) the local energy and in (b) the local spin operator in a lattice formulation. By employing a controlled renormalization group (RG) analysis
we find that 
each
problem exhibits highly complex novel scaling behavior, arising from the intrinsically indeterministic (`random') nature of quantum mechanical measurements,  which is  governed by a measurement-dominated RG fixed point that we study within an ϵ expansion. In the tricritical  Ising case (a) we find (i): multifractal scaling behavior of energy and spin correlations in the measured groundstate, corresponding to an infinite hierarchy of independent critical exponents and,  equivalently, to a continuum of universal scaling exponents for each of these correlations; (ii): the presence of logarithmic factors multiplying powerlaws in correlation functions, a hallmark of `logarithmic conformal field theories' (CFT); (iii): universal `effective central charges' c^( eff)_n for the prefactors of the logarithm of subsystem size of the nth Rényi entropies, which are independent of each other for different n, in contrast to the unmeasured critical ground state, and (iv): a universal (“Affleck-Ludwig”) `effective boundary entropy' S_eff which we show, quite generally, to be  related to the system-size independent part of the Shannon entropy of the measurement record, computed explicitly here to 1-loop order.
 –  
A subset of these results have so-far also been obtained within the ϵ expansion for the measurement-dominated critical point in
the critical Ising case (b).





Highly complex novel critical behavior from the
intrinsic randomness of quantum mechanical measurements on 
critical ground states
- a controlled renormalization group
analysis
    Andreas W. W. Ludwig
    September 9, 2024
=================================================================================================================================================================================






§ INTRODUCTION


Effects of measurements have recently attracted substantial attention especially in the context of measurement-induced quantum phase transitions in deep quantum circuits, and related problems, which exhibit novel universality classes of phase transitions in such non-equilibrium quantum systems
<cit.>.
Another class of quantum systems subjected to measurements was recently introduced in Ref. [GarrattWeinsteinAltman2022], and subsequent 
works <cit.>, considering the effects of measurements on one-dimensional quantum critical ground states. Ref. [GarrattWeinsteinAltman2022] considered a Luttinger liquid, and provided a field theory formulation with measurements acting on the one-dimensional zero-time slice in space-time, exhibiting a version of the Kosterlitz-Thouless transitions, while Ref. [WeinsteinSajithAltmanGaratt,YangMaoJian,MurcianoSalaYueMongAlicea]  similarly considered several types of measurements, with and without postselection, performed on the ground state of the critical one-dimensional quantum Ising model.[Compare also Refs. LeeJianXu,ZouSangHsieh,Myerson-JainXuHughes,AshidaFurukawaOshikawa
which consider different but related problems of effects of decoherence on 1d critical ground states, which we do not study in the present paper]
The aim of the present paper is to exhibit novel universality classes of critical behavior with highly complex and novel scaling behavior that can emerge when (weak)  measurements are performed (without postselection) on quantum critical ground states. In the examples we discuss such critical behavior originates from a measurement-dominated fixed point occurring at a finite measurement strength, which we treat using a controlled
 renormalization group (RG) analysis, i.e. an ϵ expansion. Physically, the complexity of the scaling behavior originates from the intrinsic indeterministic (“random”)  nature of quantum mechanical measurements. While so-far analytically-based tools for understanding measurement-induced transitions in deep quantum circuits have been largely elusive, problems involving measurements performed on one-dimensional quantum critical ground states are typically simpler  technically, and thus more  susceptible to a controlled RG analysis, as we demonstrate in the examples we study. Yet, they exhibit similar complex scaling behavior as that in the deep circuits.

 The first problem we study, “problem (a)”,  consists of measurements with the local energy operator on the ground state of the tricritical quantum Ising model, within a lattice formulation.
 After introducing replicas this can be written as the field theory of the (1+1)-d tricritical Ising model in space-time with a perturbation acting solely on the τ=0  equal time-slice describing the measurements. This perturbation is
 relevant in the  RG sense, and it flows to an infrared fixed point at finite measurement strength which can be controlled within an ϵ expansion. This is analogous to the Wilson-Fisher ϵ=4-d expansion, except  that here the dimension of space-time is always two, and a small parameter ϵ is obtained by generalizing the tricritical Ising model to the
tricritical q-state Potts model, and expanding about q=4 where the perturbation becomes marginal [which is in essence, an expansion in the small parameter (4-q)]. This allows for a systematic calculation of all universal scaling properties at the infrared fixed point.

One reflection of the complexity of the finite measurement strength fixed point appears in correlation functions of the spin and energy operator taken in the measured ground state with measurement outcomes m⃗. When raised to the Nth power and averaged over measurement outcomes with the Born probability these Nth moments decay, for spin and energy correlations, with independent exponents, one for each moment order N. Thus, associated with each of the two observables (spin and energy) there is an infinite hierarchy of scaling exponents, in contrast to standard critical behavior.  This is referred to as multifractal scaling. Since this scaling behavior has its origin <cit.>
in a universal
scaling form of the entire probability distribution for the correlation function,
also the non-integer N moments will scale giving rise to a continuous spectrum of scaling dimensions for  each, the spin and the energy correlations.

Another exotic feature of the measurement-dominated fixed point that we find is that of a so-called “logarithmic conformal field 
theory" <cit.>.
While at RG fixed points associated with conventional critical points 
correlation functions decay with powerlaws, here we observe in certain averaged correlation functions a powerlaw multiplied by a logarithm. This arises because at the finite measurement fixed point a rescaling of distances does not act diagonally on all observables, but may  act in the form of a two-dimensional [in simplest manifestation] non-diagonalizable “Jordan-form" matrix. When translated into the behavior of averaged correlation function this amounts to the presence of the multiplicative logarithm.

The entanglement entropies exhibit further complex universal behavior at the finite measurement strength critical point. While the universal coefficients of the logarithm of subsystem size in the nth Rényi entropy of the unmeasured ground state,
1 3 c_n,
are all related to the central charge c,  i.e. c_n= c  2[1+1 n],
at the finite measurement critical point the universal coefficients 1 3 c^( eff)_n are all unrelated to each other and have a more complicated n dependence already to first order in ϵ which we calculate, and which  is further modified in higher order in ϵ. This represents another hierarchy of independent universal quantities, similar to those encountered in spin and energy correlations functions discussed above. (Specifically, these are correlation functions of the “n-twist field”, Sect. <ref>.) We furthermore show that the universal quantities c^( eff)_n also appear in the coefficient 
of the linear temperature dependence of  the extensive measurement averaged Rényi entropies of  the full mixed thermal Gibbs state of the system at finite temperature.

The problem of performing measurements on a quantum critical ground state can be viewed as a problem of the unmeasured critical (here conformally invariant) field theory in space-time with a 
defect at the zero-time slice, and the finite measurement strength fixed point represents a scale-invariant, in fact conformally invariant defect. After folding along the slice it becomes a boundary  condition on the (doubled)  unmeasured conformal field theory (CFT). In general, to any 
boundary of a CFT is associated a universal constant, the “Affleck-Ludwig”  boundary entropy <cit.>. Here we establish quite generally that for problems of measurements on 1d quantum critical ground states, the corresponding universal “effective boundary entropy” S_ eff is the constant, system size independent piece of the Shannon entropy of the measurement record. (We have computed it here explicitly to lowest order in the ϵ expansion.) 
It may be viewed as a boundary analog of the
“effective central charge”
at measurement-induced transitions in deep quantum circuits, which arises from the universal finite-size
dependence
of the Shannon entropy of the measurement record of the (bulk) space-time of the circuit <cit.>.

 Lastly, we address the problem of weak  measurements
performed on the ground state of the critical  Ising model
with the Pauli σ̂^z_i operator at lattice sites i,  via an extension of the controlled RG analysis and ϵ expansion developed for the tricritical Ising case. We find that these measurements lead to similar complex scaling behavior 
governed by another measurement-dominated RG fixed point, occurring at a finite measurement strength. In particular, we obtain (to two-loop order) an infinite hierarchy of independent multifractal critical exponents for the set of measurement averaged moments of the 
σ̂^z_i correlation function, leading again to a continuous spectrum of critical exponents and an independent scaling exponent of the typical connected correlation function. 
Moreover, we find, in analogy to the tricritical Ising case, independent coefficients 1 3 c^ (eff)_n of the logarithm of subsystem size for the measurement averaged n-th Rényi entropies for different values of n, which we compute to leading order in the ϵ expansion. The presence of multiplicative logarithms in measurement averaged correlation functions (logarithmic CFT features) is currently being studied as well in the Ising case.
The remaining parts of the paper are
structured in the following manner: In Section <ref>, we introduce the O'Brien-Fendley model and discuss its zero temperature phase diagram which has a critical point in the universality class of the tricritical Ising point. 
For this quantum tricritical ground state, in Section <ref>, we describe a measurement protocol with explicit Kraus  operators
corresponding to weak measurements. 
In Section <ref>, we develop a replica field theory to analyze the problem of described weak measurements on the tricritical ground state.
We analyze 
the
infrared behavior of the obtained replica field theory using a controlled perturbative RG  expansion and 
determine 
the new `non-trivial' fixed point 
in an ϵ-expansion.
In Section <ref>,  we determine the long distance behavior of measurement averaged moments of correlation functions for the spin σ̂^z and the energy Ê operator (defined in Section <ref>) and 
 demonstrate
the logarithmic CFT features of the measurement-dominated fixed point.
In Section <ref>, we calculate the measurement averaged n^th Rényi entanglement entropies and the von Neumann entanglement entropy.
In Section <ref>, 
we discuss the Shannon entropy of the measurement record and the relationship of its constant universal part with the `effective boundary entropy'.
In Section <ref>, we discuss the case of Ising critical point under measurements with the σ̂^z spin operator.
Section <ref>
is
reserved for conclusions and discussion of results.

.8cm
 

§ THE O'BRIEN-FENDLEY MODEL

A 
variety
of 
quantum mechanical systems
in one-dimensional space with different microscopic appearance are known to exhibit
a quantum critical point in the universality class of the tricritical Ising model,
see e.g.
Ref. <cit.>.
In 
the present paper, we will consider
one such microscopic realization convenient for our purposes,
the O'Brien-Fendley chain introduced in Ref. O'BrienFendley.
The O'Brien-Fendley chain is a 1d quantum chain with spin-1/2 
(qubit)
degrees of freedom at each site and is described by the Hamiltonian H 


    H=H_I+λ_3H_3
          H_I=-∑_j(σ̂^z_jσ̂^z_j+1+σ̂^x_j)
          H_3=∑_j (σ̂^x_jσ̂^z_j+1σ̂^z_j+2+σ̂^z_jσ̂^z_j+1σ̂^x_j+2)


where 
σ̂^a (a = x,y,z) are
the standard Pauli matrices. 
Note that at λ_3=0, the Hamiltonian H reduces to the Hamiltonian H_I of 
the critical 1d quantum Ising chain. 
As seen by inspection, the
term H_3 in the Hamiltonian is invariant under the Kramers-Wannier (K-W) transformation given by

    σ̂^z_jσ̂^z_j+1=τ^x_j+1/2
    σ̂^x_j=τ^z_j-1/2τ^z_j+1/2.

Since there are no RG relevant K-W self-dual operators
at the Ising critical point,
for sufficiently small λ_3≠ 0 the chain in Eq. <ref> 
is described by a K-W invariant line of 
second order transitions parametrized by λ_3, all in the Ising universality class.
However, as discussed in Ref. O'BrienFendley, for large enough λ_3 the spectrum becomes gapped, and since the Hamiltonian is self-dual under the K-W transformation, 
there is
a line of first order phase transitions
on the phase boundary between the ferromagnetic and the paramagnetic phase. 
The phase diagram of the chain  along the K-W line
is shown in 
Fig. <ref> 
    (from Ref. O'BrienFendley).

The renormalization group (RG) unstable critical point at λ_3=λ_tc≈ 0.856, which separates the Ising second order phase transition line from the first order phase transition line, lies in the universality class of the tricritical Ising model.
In the present paper, we will consider performing measurements on the ground state  of the
tricritical quantum Ising Hamiltonain H of Eq.(<ref>) at λ=λ_tc.
.8cm


§ MEASUREMENT PROTOCOL
AND REPLICA TRICK

Coming first back to ordinary Ising case λ_3=0, the ground state of the  critical quantum Ising chain H_I subject to measurements with operators 
σ̂^x_j
or  σ̂^z_jσ̂^z_j+1 has been investigated in Ref. WeinsteinSajithAltmanGaratt, YangMaoJian and MurcianoSalaYueMongAlicea.
Measurements with σ̂^z_jσ̂^z_j+1 and σ̂^x_j
on the Ising critical ground state
result in the same universal effects <cit.>,
 because both operators represent (after subtraction of their expectation values) the energy operator
𝔢
 of the Ising critical point 
with scaling dimension X_𝔢=1, 
 up to corrections from subleading operators which are RG-irrelevant. In other words, their connected equal time correlation functions (denoted by a subscript _c)
 in the Ising critical ground state, describing the correlations of the subtracted operators, decay asymptotically with the same exponent,
 
    ⟨σ̂^z_iσ̂^z_i+1σ̂^z_jσ̂^z_j+1⟩_c∼1/|i-j|^2X_𝔢and ⟨σ̂^x_iσ̂^x_j⟩_c∼1/|i-j|^
    2X_𝔢

as  |i-j|>>1.


We  now move  on to the O'Brien-Fendley chain which, as recalled above, has the same underlying
lattice spin-1/2 (qubit)  degrees of freedom as the ordinary Ising chain. Therefore, it is natural to study effects of measurements of similar operators for the O'Brien-Fendley chain.
Interestingly, equal-time correlation functions of the same operators
 σ̂^z_jσ̂^z_j+1 and σ̂^x_j in the ground state of
the tricritical point of the O'Brien-Fendley chain
decay asymptotically 
(after subtraction of the expectation values) 
with the critical exponent X_ℰ=
1/5 of the energy scaling operator ℰ of the Ising tricritical point <cit.>,


    ⟨σ̂^z_iσ̂^z_i+1σ̂^z_jσ̂^z_j+1⟩_c ∼1/|i-j|^2X_ℰand ⟨σ̂^x_iσ̂^x_j⟩_c ∼1/|i-j|^2X_ℰ

as |i-j| ≫ 1.
Unlike the ordinary  transverse field
quantum Ising model, the subdominant contributions to both σ̂^z_jσ̂^z_j+1 and σ̂^x_j are now not RG irrelevant.
Specifically, these two lattice operators 
are represented <cit.> by


    σ̂^z_jσ̂^z_j+1∼χ(x)≡ 
    A I +Bℰ(x)+  
    Cℰ'(x) + D ℰ^” +
    ...,
       σ̂^x_j∼χ'(x)≡ A I -B ℰ(x) + 
    Cℰ'(x) - D ℰ^” +
    ..., 
    
       
    (A,  B, C, D =non-universal constants),


where  I is the identity field with  A the tricritical expectation value of the corresponding lattice operator,  
while ℰ (with scaling dimension X_ℰ=1/5),  ℰ' (with scaling dimension X_ℰ'=6/5), and
ℰ^” (with scaling dimension X_ℰ^”=3)
are the energy, the subdominant energy, and the further subleading energy
scaling operators at the
Ising tricritical point, 
the first two of which are RG-relevant as bulk operators; the ellipses denote more subleading operators.
Forming
the difference
of σ̂^z_jσ̂^z_j+1 and σ̂^x_j, 
we define the operator Ê_j+1/2 given by

    Ê_j+1/2≡1/√(2)(σ̂^z_jσ̂^z_j+1-σ̂^x_j),

which changes sign under the K-W duality transformation, Eq. <ref>. Owing to Eq. <ref>,
this operator  
is a lattice representation of
the energy scaling operator  ℰ of the 
 tricritical Ising 
 critical point which is consequently also odd under K-W duality, with corrections from solely RG irrelevant (K-W odd) operators.
 For the same reason, the linear combination 
 
    Ê'_j+
     1/2≡1/√(2)(σ̂^z_jσ̂^z_j+1+σ̂^x_j),

 with the opposite sign than in
 Eq. <ref>, is even under K-W-duality and is (after subtraction of its expectation value) a lattice representation of the subleading energy operator ℰ'(x) which, consequently, is K-W even (together with  all occurring subleading operators) [To re-iterate, these results
can be understood by using Kramers-Wannier duality. The O'Brien-Fendley chain is invariant under Kramers-Wannier duality throughout the line in its phase diagram depicted in Fig. <ref>. 
The energy scaling operator, ℰ(x), and the subleading energy scaling operator, ℰ'(x) are, respectively, odd and even under the K-W transformation at the tricritical Ising point. 
Since σ̂^z_jσ̂^z_j+1-σ̂^x_j is odd under the K-W transformation (see Eq. <ref>), it cannot contain any contribution from the subleading energy field ℰ'(x) in the continuum limit and is given by the energy field ℰ(x) with corrections from solely RG irrelevant (K-W odd) operators.].
Note that Ê_j+1/2 lies on the link of the lattice connecting site at j and j+1, and

    (Ê_j+1/2)^2=
    
        =1/2((σ̂^z_jσ̂^z_j+1)^2+(σ̂^x_j)^2-σ̂^x_jσ̂^z_jσ̂^z_j+1-σ̂^z_jσ̂^z_j+1σ̂^x_j) = 1

implying that


    Ê_j+1/2 has eigenvalues ± 1.

We note in passing that, as verified by inspection,  the operator Ê'_i+
1/2 
also squares to the identity and therefore also has eigenvalues ± 1. [Ê'_i+1 2 does not commute with the operator Ê_i+1 2.]

In the following, we will describe a protocol for performing measurements on the ground state of the O'Brien-Fendley chain. 
.6cm


 §.§ 
Measuring Lg on Even Links 

Note that the operators  Ê_i+1/2 on neighbouring links do not commute with each other, 
because σ̂^z_i and σ̂^x_j on the same site do not commute. 
However, if we take Ê_i+1/2 operators on alternate links, say even links (those where i is even), all of them commute with each other 
(see Fig. <ref>).

Our measurement protocol consists in measuring the operators Ê_i+1/2
on even links. [
We note that the operator Ê_i+1/2 (for an even link i) has support on the two lattice sites i and i+1, and one can check that each of its eigenvalues ± 1 is two-fold degenerate.
This means that there will be two (linearly independent) eigenstates of Ê_i+1/2, which can be chosen orthogonal, that will be associated 
with each of the measurement outcomes +1 and -1.
This does not imply that the
post-measurement state
is ambiguous.
The state that results after observing
any set of measurement outcomes is uniquely obtained by acting with the projector (or more generally, in the case of weak measurements, a Kraus operator) corresponding to the eigenspace associated with the measurement outcomes on the `incoming'  state before measurement. In the case of interest to us this will be, as we will discuss below, the ground state of the tricritical O'Brien-Fendley chain.
See e.g. Eq. <ref> below. (Measurement operators with two eigenvalues which are not both non-degenerate, have also been discussed in a different context in Ref. 
MajidyAgrawalGopalakrishnanPotterVasseurHalpern2023
.)] [We note that performing weak measurements with the operator Ê_i+1/2 only on even links i does not imply that the
system will effectively collapse onto a trivial `staggered' state. 
Compare for example with
Refs. WeinsteinSajithAltmanGaratt and YangMaoJian, which consider performing measurements with the Pauli operator σ̂^x_i on  all sites i of the critical quantum Ising chain (See Eq. <ref> for the Hamiltonian). 
In terms of the Majorana formulation of the quantum Ising chain, 
this measurement operator reads
σ̂^x_i=γ̂_2iγ̂_2i+1, where γ̂_2i is a Majorana operator.
Thus measuring σ̂^x_i corresponds to performing measurements only on the even-links of the underlying Majorana chain, however a `staggered' state is not observed <cit.>.
In fact, 
even though measurements are performed only on the even links of the Majorana chain, 
it has been observed that for Born-rule measurements <cit.> the long-distance critical properties of the system are the same as that of the unmeasured state, i.e. the Ising critical ground state (which is not `staggered').
In the same spirit, in our measurement protocol of our system we also 
would not expect to see a trivial `staggered' state even though we are performing measurements only on the even-links of the chain (which are now the physical links of the spin-chain and not of the underlying Majorana chain).
In our case, the critical behaviour of the unmeasured state, however, does get modified dramatically due to the presence of measurements as we will
discuss 
in detail in the subsequent sections of the present paper. 
 – A general proof of the impossibility of obtaining a trivial `stagggered' state, for our (and also the above) system  will follow from Eq. <ref> with Ô_1 :=
Ê_i+1/2
Ê_j+1/2. This equation implies that the expectation value of Ô_1, a two point function, is unmodified by Born-rule measurements and thus exhibits the algebraic decay  with distance |i-j| characteristic of the unmeasured (tri-)critical ground state, which would be in contradiction with an exponential decay in a trivial `staggered' state.] 
Then on each even link, we can define the (weak-) measurement  Kraus
operator,

    K̂_i+1/2,± :=
    1 ±λÊ_i+1/2/√(2(1+λ^2)),

 and the measurement operators Ê_i+1/2 at different even links will commute with each other. For notational convenience, we will drop the 
 lattice-position offsets
 of +1/2 from now on and label both 
 operators as K̂_i and Ê_i, i.e. by just i, which denotes the site at the left end of the link. [
 I.e.
 K̂_i+1/2,m_i→K̂_i,m_i where m_i=±, and correspondingly for Ê]

When λ=1, the
Kraus operators 
in Eq. <ref>
reduce to projection operators
K̂_i, m_i=
1/2 ( 1 + m_iÊ_i)
onto the eigenstates 
of Ê_i with eigenvalues m_i=± 1.
The parameter 0 ≤λ≤ 1 controls the `strength' of the measurement. When λ=0 no measurements are performed at all.

When the eigenvalue m_i is measured at site i, the measurement changes a (normalized) quantum state |ψ⟩ to the following (normalized) state  after this measurement

    |ψ⟩→K̂_i,m_i|ψ⟩
    ||K̂_i,m_i|ψ⟩||
    .

Each measurement outcome m_i= ±  1  at an even link 
i
occurs with `Born-rule' probability

    p_B(m_i) = ⟨ψ| 
    (K̂_i,m_i)^†K̂_i,m_i|ψ⟩ =
    
          
    =
    1 2 (1 + λ^2) (
    1 + λ^2+ 2 m_i λ⟨ψ|Ê_i |ψ⟩ ),

which depends on the incoming  state |ψ⟩.
The measurement operators for each  even i satisfy the condition

    ∑_m_i = ± 1
    (K̂_i,m_i)^†K̂_i,m_i =  1_i,

where the right hand side denotes the identity 
operator [We note, continuing a previous footnote, that degeneracies of the measurements operators Ê_i
and thus of the  Kraus operators does not affect the identity
Eq. <ref>. For a similar situation with degeneracies of the eigenvalues of the measurement operators, see the already previously mentioned Ref.  
MajidyAgrawalGopalakrishnanPotterVasseurHalpern2023
].
This ensures the normalization of the Born-rule probabilities p_B(m_i) defined above.
 Eq. (<ref>) is referred to as the 
POVM [standing for“Positive Operator Valued Measure”] condition.


Let us take the quantum state on which we perform measurements to be the ground state
|0⟩ of the  O'Brien-Fendley chain at the tricritical point.
Since the measurement operators Ê_i  on the even links i commute with each other, the state obtained after measuring on all even links with measurement outcomes m⃗ := {m_i} is


    |Ψ_m⃗⟩=
    ∏_i∈evenK̂_i, m_i|0⟩
    ||
    ∏_i∈evenK̂_i, m_i|0⟩ ||
    
    =
    K̂_m⃗|0⟩√(⟨0|K̂^†_m⃗K̂_m⃗|0⟩),


where

    K̂_m⃗ := ∏_i∈evenK̂_i, m_i,

and 

    p_0(m⃗)= ⟨0|K̂^†_m⃗K̂_m⃗|0⟩

is 
the Born-rule probability to obtain measurement outcomes m⃗= {m_i}. 
We will refer to the state obtained upon performing measurements and corresponding to a particular set of measurement outcomes as a `quantum trajectory'.
It will be convenient to write the RHS of Eq. <ref> as

    K̂_i, ±=1 Nexp{±λ̃Ê_i},

where 0≤λ̃ = arctanh(λ)<∞, and N is 
a suitable 
normalization factor.
Then
we can write the product 
in Eq. <ref>
in the following form


    K̂_m⃗
    =
    1 N^L/2exp{λ̃∑_i = even
    m_iÊ_i},

where L denotes the number of sites.
Let us define the variable t_i s.t.

    t_i=λ̃m_i.


Since the measurement outcome 
is m_i=± 1, the variable t_i takes on
values ±λ̃.
We can reformulate the measurements by “softening” the 
variable t_i=±λ̃
to
take on continuous values  -∞  < t_i < +∞ drawn from some 
distribution P(t_i) which we take to be symmetric under t_i→ - t_i. Sometimes, it may be convenient  to choose  a Gaussian distribution whose variance is a measure of the  `strength' of measurements λ̃. 

The formulation  given
in Eqs. <ref>,<ref>
above  in terms of discrete measurement outcomes  m_i=± 1 is simply 
a special case of this where the 
distribution P(t_i) is  
the
(normalized)
sum of two delta functions peaked at t_i=±λ̃. 
It turns out that only the cumulants of  the random 
variable t_i determined by the distribution P(t_i) [We consider only such distributions P(t_i) for which all cumulants
exist, i.e. are finite.] will enter our formulation below, and the essential physics will turn out to depend only on the second cumulant and will thus be insensitive to other details of the 
distribution P(t_i).
This then also covers the case where, with some probability, no measurement is performed at a site, corresponding  to 
the
symmetric distribution  P(t_i) which is a 
 (normalized) weighted sum of three delta functions, peaked at t_i=0 and at t_i=±λ̃.

 The  corresponding reformulated Kraus operators


    K̂_t⃗≡1 ( N')^L/2exp{∑_i∈even t_iÊ_i},
      t⃗≡{t_i}_i ∈ even,


with a suitable choice of normalization factor N', 
satisfy again the  required POVM 
condition



    [ 
    ∏_i∈ even (∫_-∞^+∞ dt_i P(t_i) )
     ]
     
     (K̂_t⃗)^†K̂_t⃗
     =  1.

 This 
 follows  from Eq. <ref> for any  P(t_i) symmetric under t_i → - t_i [the role of m_iλ being played by tanh(t_i)].



 §.§ Calculation of Observables and Replica Trick


Consider now  a general  measurement average (denoted by an `overbar') of the quantum mechanical expectation of N (potentially different) operators 
Ô_1, Ô_2, ..., Ô_N, where each of these we 
consider here to be a local operator or a product of local operators. We will compute this average using the Born-rule probability distribution 
p_0(m⃗), Eq. <ref>.
We will also assume, for now, that
each
operator
𝒪̂_i 
commutes
with the Kraus operator K̂_m⃗,
but we will relax this assumption at the end of this section.
Then averaging over measurement outcomes we 
obtain the measurement-averaged expectation values

    [⟨Ô_1⟩_m⃗ ...
      ⟨Ô_N⟩_m⃗]
          =
    ∑_m⃗ p_0(m⃗)
     ⟨0|K̂^†_m⃗Ô_1 K̂_m⃗|0⟩ 
    ...
    ⟨0|K̂^†_m⃗Ô_N K̂_m⃗|0⟩
    p_0^N(m⃗)
    
          =
    lim_R→ 1∑_m⃗(⟨0|K̂^†_m⃗Ô_1 K̂_m⃗|0⟩ 
    ...
    ⟨0|K̂^†_m⃗Ô_N K̂_m⃗|0⟩×
          ×[⟨0|K̂^†_m⃗K̂_m⃗|0⟩]^R-N)



    =
    lim_R→ 1∑_m⃗(⟨0|Ô_1K̂^†_m⃗K̂_m⃗|0⟩ 
    ...
    ⟨0|Ô_NK̂^†_m⃗K̂_m⃗|0⟩×
          ×[⟨0|K̂^†_m⃗K̂_m⃗|0⟩]^R-N).

Note that when N=1 
the last factor in the above equation
disappears since (R-N )→ 0 in the required 
R → 1
limit. Thus, the average 
of a single expectation value of an operator or of a product of operators Ô_1 (such as e.g. those appearing in a 2-point function)
is unaffected by measurements, 

    [⟨Ô_1⟩_m⃗]=
    ∑_m⃗⟨0|   Ô_1 
    K̂^†_m⃗K̂_m⃗|0⟩ 
    =
    ⟨0|Ô_1 
    |0⟩,

where the last equality follows from the POVM condition,

Eq. <ref>.
Coming back to Eq. <ref>, if we replicate the Hilbert space R times, Eq. <ref> can be written as,

    [⟨Ô_1⟩_m⃗ ...
      ⟨Ô_N⟩_m⃗]
          =lim_R→ 1∑_m⃗R ⊗⟨0|𝒪_1^(1)𝒪_2^(2)…𝒪_N^(N) (K̂^†_m⃗K̂_m⃗)^⊗ R|0⟩^⊗ R
          
     =lim_R→ 1Tr(𝒪̂_1^(1)𝒪̂_2^(2)…𝒪̂_N^(N)(|0⟩⟨0|)^⊗ R∑_m⃗(K̂_m⃗^†K̂_m⃗)^⊗ R).

 Here,
 the trace `Tr' is now performed in the replicated Hilbert space, and superscripts on the operators indicate 
 which Hilbert space factor, in the R-fold tensor product Hilbert space, they act on.
 For the measurement protocol
 discussed in subsection <ref>,

after 
“softening" the measurement outcomes to take on continuous values, 
we can replace 𝐊̂_m⃗ in Eq. <ref> by 𝐊̂_t⃗ in Eq. <ref>, and also replace the sum ∑_m⃗ by the 
integral over t_i
as in
Eq. <ref>. Thus, we will make the following substitution in Eq. <ref>

    ∑_m⃗ (K̂_m⃗^†K̂_m⃗)^⊗ R→  [ 
    ∏_i∈ even (∫_-∞^+∞ dt_i P(t_i) ) ] (K̂_t⃗^†K̂_t⃗)^⊗ R.

Moreover, since Ê_i is an hermitian operator,
we have
K_t⃗^†= K_t⃗,
and using Eq. <ref>, we can 
write


    (K̂_t⃗^†K̂_t⃗)^⊗ R=1/(𝒩')^RLexp{2∑_i= even t_i(∑_a=1^RÊ_i^(a))}.

As discussed in Section <ref>, P(t_i) is a symmetric distribution under t_i→-t_i. 
Here {t_i}_i∈ 
even are independent random variables with joint distribution P̃({t_k})=∏_i
∈ 
even
P(t_i), and the first and second moments of the distribution are given by

    t_i =0, 
      t_i t_j
    = 2  Δ̃δ_i,j.

Here, Δ̃
    quantifies the strength of the measurements, and we assume that higher 
cumulants
of P(t_i) vanish; and
they will be shown to not change the physics at long distances in App. <ref>.
Then  using Eq. <ref> and Eq. <ref>
we obtain using the cumulant expansion
[We note that the normalization of the distribution P(t_i) is chosen such that it satisfies Eq. <ref> and hence it is not normalized as a probability distribution. However, multiplying and dividing P(t_i) by an appropriate overall trivial constant we can use the formula for cumulant expansion, which is valid for probability distributions.]

    ∫_-∞^∞   (∏_i
        =
        evend t_i P(t_i))(K̂_t⃗^†K̂_t⃗)^⊗ R∝
       ∝exp{Δ̃∑_i=even4(∑_a=1^RÊ_i^(a))^2}
       ∝exp{4Δ̃∑_i=even∑_a,b=1
    a≠ b^RÊ_i^(a)Ê_i^(b)}

 where in Eqs. <ref> and Eq. <ref>, we have dropped 
unimportant overall multiplicative constants and consequently
replaced the equality signs by proportionality signs, and in Eq. <ref> we have used Ê_i^2=1 (see Eq. <ref>).
Using Eqs. <ref>, <ref> and <ref>, we can 
then write the measurement averaged moments of expectation values as

    [⟨Ô_1⟩_m⃗ ...
      ⟨Ô_N⟩_m⃗]   ∝lim_R→ 1Tr(𝒪̂_1^(1)𝒪̂_2^(2)…𝒪̂_N^(N)×
    × (|0⟩   ⟨0|)^⊗ Rexp{4Δ̃∑_i=even∑_a,b=1
    a≠ b^RÊ_i^(a)Ê_i^(b)})


In the derivation of Eq. <ref>, we assumed that operators 𝒪̂_i commute with the Kraus operator K̂_m⃗ [For the O'Brien-Fendley chain, we will consider calculating measurement averaged moments of the correlation function for operators σ̂^z_i and Ê_j. As noted earlier, the operators Ê_j at even sites j commute with each other, and hence the operator Ê_j at an even site j also commutes with the Kraus operator K̂_m⃗ (see Eq. <ref>). 
Moreover, if we choose the operator σ̂^z_i to lie on an odd sites i, it commutes with Ê_j=1/√(2)(σ̂^z_j+1σ̂^z_j-σ̂^x_j) for all even sites j (see Fig. <ref>) and thus, it also commutes with the Kraus operator 𝐊̂_m⃗.
Therefore, the positions of operators for which we study the correlation functions in this paper can always be slightly “tuned" such that they commute with the Kraus operator 𝐊̂_m⃗].
We will close this section by discussing the case when this assumption is not satisfied.
See also the discussion at the end of App. <ref>.
Since
each operator 𝒪̂_i is either a `local'
operator
or a product of `local' operators, it will commute with most 
Kraus operators
K̂_j,m_j in
the product K̂_m⃗=∏_jK̂_j,m_j, and it might not commute with only a few K̂_j,m_j which have support on the same sites j as the operator 𝒪̂_i. 
We expect such local commutator terms of operator Ô_i to generically be subleading in scaling dimension [We note that in Ref. <cit.> they have made a similar observation in a related context.], such that the leading order long distance behavior of the measurement averaged moments of the ground state expectation value of 
operator
𝒪̂_i is still given by Eq. <ref>.

.8cm



§ FIELD THEORY REPRESENTATION AND MEASUREMENT RG FIXED POINT




 §.§ Field Theory Representation


In field theory language, the ground state density matrix of the O'Brien-Fendley chain at the
Ising
tricritical point can be written
as a path integral [Compare also analogous discussions for different systems in Refs. 
<cit.>, <cit.>
.] over the cut cylinder shown in Fig. <ref>,


    |0⟩⟨0|=      lim_β→∞e^-β H/Z=1/Z∫ D ϕ e^-S_*|ϕ(x,0^-)⟩⟨ϕ(x,0^+)|
    
    
    S_*=      ∫ d τ∫ d x {1/2(∂_xϕ)^2+1/2(∂_τϕ)^2+
    g^*_3ϕ^6},
    
    
    Dϕ=      ∏_τ=-β/2^+β/2∏_xdϕ(x,τ),


where
|0⟩ is the ground state of the tricritical Ising Hamiltonian, and 
S_* is the effective Landau-Ginzburg 
(-Zamolodchikov <cit.>)
fixed point action of the Ising tricritical point,
defined on the 2d space-(imaginary)time
geometry 
in Fig. <ref>. [
In the present case of two dimensions there is no meaning to a perturbative study of the Landau-Ginzburg action about the Gaussian theory, since the field ϕ is dimensionless by naive 
power-counting, and a non-perturbative tool is needed. This is provided in Ref. Zamolodchikov1986
where it is shown that non-perturbative field identifications following from the exact equations of motion are exactly those obtained from the corresponding unitary minimal model CFT.]
We can insert the path integral representation from Eq. <ref> into Eq. <ref>, and replace the local operators
(or products of local operators)
Ô_i with the corresponding continuum fields 𝒪^(a_i)_i(x,0^-)
in the respective replica copy “a_i".
Following
Eqs. <ref>,
<ref>, 
we 
can also replace the
measurement operator Ê^(a)_i in Eq. <ref> by the corresponding continuum energy scaling operator
ℰ^(a) in replica copy “a".
The field ℰ is expressed in terms of the Landau-Ginzburg field ϕ by

    ℰ(x,τ)= :ϕ^2   :(x,τ),

where `: :' indicates
standard `normal ordering' [
subtraction of the singular terms in the operator product expansion [OPE]]
of the field ϕ^2 <cit.>.
Thus, in
continuum 
language,
we obtain the following expression for the averages

    [⟨Ô_1⟩_m⃗ ...
      ⟨Ô_N⟩_m⃗]∝
       lim_R→ 1∫_ϕ^(a)(x,0^-)
     =ϕ^(a)(x,0^+)[∏_a=1^RD ϕ^(a)] 𝒪_1^(1)𝒪_2^(2)…𝒪_N^(N)
      e^- 𝕊
       where, -𝕊=
    ∑_a=1^R
    (-1)
    
    S_*^(a)
    +
    Δ∫_-∞^+∞ dx Φ(x)
    
       Φ(x) :=
     ∑_a,b=1 
     a≠ b^Rℰ^(a)(x,0) ℰ^(b)(x,0) and  Δ=(constant)×Δ̃.


Due to the trace in Eq. <ref>, the τ=0^- and τ=0^+ boundaries of the cut cylinder in Fig. <ref> will be glued, and the field configurations ϕ^(a)(x,0^-) and ϕ^(a)(x,0^+) are identified for all replica indices “a" as shown in Eq. <ref> [Important differences in the gluing of boundary field configurations could occur if the fields 𝒪_i^(a) cannot be expressed locally in terms of the Landau-Ginzburg field ϕ and its normal ordered higher powers. This issue is addressed in App. <ref>].A different perspective to verify the form of defect interaction appearing in Eq. <ref> and <ref> is to consider symmetries of the system, and in particular 
Kramers-Wannier duality.
Note that although our ground state |0⟩ is 
invariant under Kramers-Wannier duality,
an individual quantum trajectory will generally not be invariant under it.

However, since we average over all measurement outcomes,
we expect this symmetry to be restored in an average sense.
Thus, the K-W symmetry will appear as an 
average (“weak") symmetry
of the ensemble of quantum trajectories. 
This 
implies, in particular,
that although the total replica action (in IR) will be not invariant if we take ℰ^(a)→-ℰ^(a) in a single replica, the action will be invariant if we perform the transformation ℰ^(a)→-ℰ^(a) for all replica indices (a) simultaneously.
The 
most
RG relevant 
perturbation
supported on the τ=0 time-slice 
and
in the presence of
this 
average (“weak") symmetry
is of
the
form ℰ^(a)ℰ^(b) for replica indices a≠ b [Note that a term with equal replica indices of form ℰ^(a)ℰ^(a), can be evaluated
using point splitting and the operator product expansion <cit.> (OPE)

    ℰ^(a)×ℰ^(a)=I^(a)+ℰ'^(a)

where the operator
ℰ'=
:ℰℰ: 
=  :ϕ^4:, 
in every replica “a",
is irrelevant 
as an operator with support on the 1-dimensional τ=0 time slice at the Ising tricritical point.].
Finally, we must consider the sum of terms ℰ^(a)ℰ^(b) over all possible pairs of unequal replica indices for the action to be symmetric under permutation of replica indices.
This gives us Eq. <ref> with Φ(x) in Eq. <ref> back [Due to high scaling dimensions, terms with more than four ℰ^(a) fields and pairwise unequal replica indices are irrelevant under RG at the tricritical Ising 
point.
The term with exactly four ℰ^(a) fields (with pairwise unequal replica indices) is relevant at the tricritical Ising point and is less relevant than Φ(x) in Eq. <ref>.
Moreover, we argue in App. <ref> and <ref> that this term is expected to be irrelevant at the new IR 
fixed point. 
].





 §.§ Controlled Perturbative Renormalization Group Analysis 

At the
Ising tricritical point,
the scaling dimension of the field ℰ= :ϕ^2: is
X_ℰ =1/5
<cit.>.
Thus, the RG eigenvalue of the coupling constant Δ in Eq. <ref> is y_Δ=1-2X_ϵ=3/5>0,
implying that the perturbation is relevant.
To study the effect of this perturbation, we will use
a perturbative RG 
analysis,
controlled by a small parameter ϵ.
To obtain such a small parameter ϵ, 
we will consider
the following 
generalization 
of the  
action 
in Eq. <ref>,
where


    -𝕊=
    ∑_a=1^R
    (-1)
    S_*^(a)   
    +
    Δ∫_-∞^+∞ dx   Φ(x)
    
    Φ(x)=∑_a,b=1 
     a≠ b^Rℰ^(a)(x,   0)ℰ^(b)(x,0).

But now we consider, instead of Eq. <ref>, 
the more general fixed point described by the
action

    S_*=∫ d τ∫ d x {1/2(∂_xϕ)^2+1/2(∂_τϕ)^2+
    g^*_m-1ϕ^2(m-1)},

where

    ℰ=:ϕ^m-2:,

and where  m≥ 4 is an even integer.

Note that 
setting
m=4 in the above equations, we recover the problem at hand, i.e. the 
action 
and the field ℰ given by Eq. <ref> and <ref>, respectively.
For any  
integer m≥ 3 (even or odd), the
fixed point action in Eq. <ref> 
describes exactly <cit.>
the 
multi-critical points famously known as the m^th 
unitary  minimal model conformal field theories (CFTs)
of  central 
charge <cit.> 

    c(m)=1-6 m(m+1).

(The same comment as in
footnote <cit.> applies to this Landau-Ginzburg action.)
We note that for arbitrary integer values m ≥ 3, the operator ℰ in Eq. <ref> is no longer the `energy' field of the Ising multi-critical point described by the  Landau-Ginzburg action in 
 Eq. <ref> (which would be :ϕ^2:).
However, we will restrict ourselves to only even values of m [We will relax this restriction in Sect. <ref>, where we will consider an odd m minimal model, namely the Ising CFT.], and keep using the symbol ℰ for 
the field in Eq. <ref>
for the following reason.
For the central charges c(m), Eq. <ref>, with even integer values of m≥ 4, there is another critical model with the same central 
charge, in addition to that described by the action in Eq. <ref>.
This is the tricritical q-state Potts model, where the value of q is given <cit.>
by

    √(q) = 2 cosπ m,    m≥ 4    ( even).

When q=2, this is of course the tricritical Ising model, which is described by the Landau-Ginzburg action
in Eq. <ref>
above, but for other values of the number q of 
Potts states in Eq. <ref>, e.g. for q=3,
it is a
slighly different theory than the one in Eq. <ref>,
with the same central charge <cit.>.
This  will not be of relevance for the observables of interest to us, which 
turn out to be present in both theories (see also below). For example and of particular interest to us, when
m≥ 4 is even, the operator ℰ from Eq. <ref> is precisely the same operator as the energy (`thermal') operator in the 
tricritical q-state Potts model of the same central charge <cit.>. (In CFT language, that operator is the so-called Kac-Table primary operator φ_1,2 which has the scaling dimension listed in Eq. <ref> below.)
When m→∞, the value of q approaches q=4, describing the q=4 state
tricritical Potts model, which turns out to be the same  as the critical (ordinary) q=4-state Potts model at central charge c=1 <cit.>.
Moreover, for even m, all operators that appear when performing repeated operator product expansions of ℰ with itself are operators present simultaneously 
in both, the 
tricritical q-state Potts model and the Landau-Ginzburg multicritical point described by Eq. <ref> [As already mentioned, the energy operator of the tricritical q-state Potts model is the so-called Kac-Table operator φ_1,2, and under repeated OPEs with itself, it generates the set of Kac-Table operators φ_1,n, all of which are common to both critical systems. (This set of operators forms an operator algebra closed under the operator product expansion.)], 
and all correlation functions of an arbitrary number of ℰ operators are exactly the same in both systems. 
Since, as we will discuss shortly, we will be interested in computing the RG equation (beta function) for the coupling constant Δ in the generalized model 
Eq. <ref>
for even values of m, 
which is 
uniquely 
determined [in a given RG scheme] (to arbitrary loop order) by the set of  the correlation functions of an arbitrary number of ℰ operators (which, as just mentioned, are the same for both systems), we can use either the Landau-Ginzburg 
formulation of Eq. <ref> or  equivalently the tricritical q-state Potts model formulation, both yielding the same result for this  RG 
equation [See also the discussion in App. <ref>.].

Specifically, we will proceed as follows. We are interested in the properties of the
replica
field theory in Eq.
<ref> – <ref>
when m=4, describing the effects of the quantum mechanical measurements on the tricritical Ising ground state, as described in the previous sections. We will  study the generalization of this field theory to large even values of the parameter m which, as already mentioned, provides an expansion parameter ϵ that is small when m (even) is large. This is a pure field theory problem. 
We will find that for large even values of  m, the field theory in
 Eq. <ref>
 will exhibit a fixed point at a non-vanishing value 
 Δ_* of the coupling constant Δ, controlled by  the parameter ϵ=3/(m+1), small when the even integer m is large.
 At this fixed point, we compute universal  properties (including critical exponents) of a variety of observables
 perturbatively controlled by  the small parameter ϵ. This is the same logic as in the familiar Wilson-Fisher ϵ=
 4-d expansion in dimensions d smaller than 4.  In contrast, here we always remain in 2=(1+1)
 dimensions, but we  vary the central charge c(m) by varying the even integer m [This is an expansion in √(3 2(1-c(m)))
 about c(m)=1, which can  equivalently  be viewed (as discussed above) as an expansion in
 3 2π√(4-q) of the tricritical q state Potts model about the q=4 state Potts model.]. (This type of ϵ-expansion within conformal perturbation theory in two dimensions was first performed in Refs. <cit.>
 and subsequently used in many works.).
 This approach allows us to establish that at the 
 finite-Δ_* fixed point the system has an
 extremely rich universal scaling behavior (to be discussed in subsequent sections), which we can access in a controlled manner perturbatively in ϵ (in the sense of an ϵ-expansion).  
 Physically, this
 rich scaling behavior originates, as
 m → 4,  from the intrinsic 
 randomness resulting from the 
 indeterministic  outcomes of  the quantum mechanical measurements performed on the ground state of the Ising tricritical point.
We now proceed to discuss the RG equation for the coupling constant Δ in Eq. <ref>.
 For an arbitrary even integer 
 m≥ 4, 
 the scaling dimension of ℰ in Eq. <ref> (with action S_* in Eq. <ref>) is <cit.>,

    X_ℰ=1 2 - 3  2 (m+1).

Thus, the RG eigenvalue of the coupling constant Δ
is

    y_Δ= 
    1-X_Δ
    =1 - 2X_ℰ = 3 (m+1)def=ϵ

which  is greater than zero, implying that the perturbation is relevant and we will flow away from the
unperturbed fixed point at Δ=0 for any given
m.
To obtain the 1-loop RG equation for the coupling constant Δ, we will need the OPE of the operator
  Φ(x) (from Eq. <ref>
  with ℰ from Eq. <ref>) with itself (see for example 
Refs. JLCardy_1986RGOPE,LUDWIG198797,cardy_1996,LUDWIGWIESE).
 For any  conformal minimal model with even m≥ 4 in Eq. <ref>, the fusion rule [As already recalled in footnote <cit.>, the operator ℰ is the Kac-Table operator ℰ=φ_1,2] 
of ℰ=:ϕ^m-2: with itself is given by,

    ℰ×ℰ=I+ℰ'

where ℰ'=:ϕ^2m-4: is another scaling field in the m^th 
Landau-Ginzburg theory
and it is irrelevant on
the 1-dimensional τ=0 time slice,
for any m≥4 [The scaling dimension of ℰ' in 
the m^th 
minimal model is 2(m-1/m+1)>1 for any even m≥4 (see <cit.>)
].
In tricritical q-state Potts language where, as mentioned above, ℰ is the (leading)
energy  operator, ℰ' is simply the subleading energy  operator.
From Eq. <ref>,<ref>, one obtains the OPE

    Φ(x_1) Φ(x_2) ∼b  |x_1 - x_2|Φ(x_2) +… ,
     
           where
          
     b = 4 (R-2),

and
the 
ellipsis indicates  fields which are irrelevant when supported on the τ=0 time-slice
for m≥ 4 (which includes the m=4 Ising tricritical case,  Eq. <ref>), and can be ignored.
To 1-loop order, the RG equation is then given by 
<cit.>,
<cit.>,
<cit.>

    d Δ d ℓ
    = y_ΔΔ +  b Δ^2 +  O(Δ^3).

Thus, when the number of replicas is  R<2, there is a new fixed point at a non-vanishing positive value

    Δ_*=
    -1  b y_Δ +  O (y_Δ^2)
    =
    ϵ/4(2-R) +  O (ϵ^2) >0   
          ϵ=3/m+1.

We are interested in the limit R→ 1 relevant for measurements
(satisfying the Born rule)
<cit.>,
and
since Δ describes a second cumulant, we are interested in  a non-negative value Δ≥ 0.
Finally, we also note that  the RG analysis for the replica action 
    from Eq. <ref>
    (with action S_* in Eq. 
    <ref> and m≥ 4)
     has been 
     performed to two loop order [in a dimensional regularization  [by ϵ=3/(m+1)] RG scheme, with minimal subtraction of poles in ϵ]
in Ref. <cit.>, and from 
this analysis we 
obtain the following fixed point coupling up to second order in ϵ,

    Δ_*=
       ϵ/4(2-R)+ϵ^2/4(2-R)^2+  O(ϵ^3),

a result that will be used further below.
 Note that 
 the 1-loop results in Eq. <ref> and Eq. <ref> match 
 as they should because the  1-loop contribution b to  the
 RG equation  is independent of the used RG scheme.
 .8cm


§ CORRELATION FUNCTIONS

In this section, we will discuss measurement-averaged moments of various correlation functions in the ground state of the tricritical O'Brien-Fendley chain.
Following the logic of the preceding section, we will first express these measurement-averaged moments at the Ising tricritical point as correlation functions  of  corresponding fields in  the replica  Landau-Ginzburg field theory describing the  Ising tricritical point as in 
Eqs. <ref>,
<ref>, 
<ref>. 
Subsequently, we will consider the generalization of these correlation functions to the sequence of field theories with parameter m≥ 4 (even) in Eqs. <ref>, <ref>, <ref>,
which possess a small expansion parameter ϵ when m is large. We will identify the generalization of the fields of the Ising tricritical point to general values of (even) m, and calculate their correlation functions in the replica field theory at the RG fixed point controlled by ϵ (discussed in the preceding section). This provides, analogous to the ordinary Wilson-Fisher ϵ = (4-d)-expansion, a controlled expansion of critical exponents and other universal properties in an expansion in ϵ. Just as in the case of Wilson-Fisher expansion the  case of d=2 and d=3 dimensions is of  particular interest, of  particular interest to us is the case of m=4.

.5cm



 §.§ Measurement-averaged moments of the spin-spin correlation function
Lg

If we consider two lattice spin operators σ̂^z
at the Ising tricritical point
at
sites i and j (also see footnote <cit.>), 
we can use Eqs. <ref>, 
<ref>,
<ref>, 
<ref>
to 
express the measurement-averaged
N^th moment of their
ground state
correlation 
function in replica field theory language:

At long wavelengths and to leading order in scaling dimension, the lattice operator σ̂^z is represented <cit.> at the Ising tricritical point of the O'Brien-Fendley chain
by the spin field σ(x)=ϕ 
appearing in the action Eq. <ref>.
The N-th moment average ⟨σ̂^z_iσ̂^z_j⟩^N
is then given in continuum language by the following correlation function in the replica field theory for the Ising tricritical point,
discussed in 
Eqs. <ref>, 
<ref>,
<ref>, 
<ref>



    ⟨σ̂^z_iσ̂^z_j⟩^N∼⟨𝔖^{α_i}(x, 0) 𝔖^{α_i}(y, 0)⟩,

where
we have defined 𝔖^{α_i}(x, τ) as

    𝔖^{α_i}(x, τ)
    :=
     [ ∏_i=1^Nσ^(α_i)(x,τ) ],

and
1 ≤α_i≤ R are  pairwise distinct 
replica indices in the  R-replica theory.
As R→ 1,
the physics at long distances is determined by the 
new, measurement-dominated fixed point discussed in the previous section, and the
correlation function in Eq. <ref> will  asymptotically exhibit power law behavior,


    ⟨𝔖^{α_i}(x, 0) 𝔖^{α_i}(y, 0)⟩∝1/|x-y|^2X^(σ),R=1_N, 
    
           as  |x-y|→∞ .

Here
the power law exponent 
X^(σ),R=1_N
characterizes the scaling behavior of the measurement averaged N^th moment of the ⟨σ̂^z_iσ̂^z_j⟩ correlation function at the Ising tricritical point.We can evaluate the power law exponent 
X^(σ),R_N
in an expansion in ϵ=3/m+1 
at the new fixed point Δ_* 
discussed in the previous Section <ref>,
by computing the above correlation function 
with  the
generalized replica action in 
Eq. <ref> – <ref>.
To this end, we use the  spin field of the generalization of the Ising tricritical point to the tricritical q-state Potts model for
√(q) = 2 cosπ m with m≥ 4 even, which has scaling dimension <cit.> 

    X_σ=m^2-4/8m(m+1),
      (m≥ 4,   even),

and which we denote by  the same symbol σ(x)
as that used above in the tricritical Ising case. The tricritical Potts spin field is known [
As already mentioned, for even m≥ 4
the tricritical q-state Potts energy operator
ℰ
corresponds to the Kac-table primary field φ_1,2, 
while the leading tricritical Potts spin field σ
corresponds to the Kac-Table primary field φ_m/2, m/2 <cit.>. The OPE of these two fields reads φ_1,2×φ_m/2, m/2=
φ_m/2, m/2+1 + φ_m/2, m/2-1=
φ_m/2, m/2 + φ_m/2, m/2+2, where in the last equality we have used the symmetry of the Kac Table, h_r,s=
h_m-r, m+1-s, and φ_m/2, m/2+2 denotes the subleading tricritical Potts spin field σ'.] 
to have a natural OPE with the tricritical Potts energy operator,

    σ×ℰ = σ + σ',

where the subleading tricritical Potts spin field σ' has scaling dimension
X_σ'=9m^2-4/8m(m+1).
We note 
in passing that the
scaling dimensions of the
tricritical q-state Potts spin and subleading spin fields also
match those of the following fields 
in the Landau-Ginzburg formulation [The only difference for even m ≥ 6 is that in the Potts formulation there are two different spin and subleading spin fields, degenerate in scaling dimension. <cit.>
This doubling of spin fields in the Potts formulation will be of no relevance for us since we will only be interested in the two-point function of the tricritical q-state Potts spin fields.],

    σ(x)=:ϕ^m/2-1:(x)
         σ'(x)= :ϕ^3(m/2-1):(x) .

At m=4, this gives back the spin field and the subleading spin field  of the tricritical Ising CFT,
σ(x)=ϕ
and σ'(x)= :ϕ^3:.

Let us 
denote, for general even values of m,
the coefficient of the field σ in the OPE of ℰ and σ 
in the tricritical q-state Potts model
by c_σℰσ.
Then, 
the OPE of 𝔖^{α_i}
(recalling its definition in Eq. <ref>)
with Φ(x) is given by

    Φ(x) ·𝔖^{α_i}(y,0)
     ∼N(N-1)c^2_σℰσ/|x-y|^2X_ℰ  𝔖^{α_i}(x,0)+ …
 
where the ellipsis indicates 
fields 
which, at ϵ=0 (m=∞),
have scaling dimensions different from those of
𝔖^{α_i}(x,0).
Upon
using
the
OPE in 
Eq. <ref>, the 1-loop RG equation for 𝔖^{α_i}(x,0)
is given by


    d g_{α_i} d ℓ
    =(1-N   X_σ) g_{α_i} +   2N(N-1)c^2_σℰσΔ g_{α_i}+…,

where the scaling dimension  X_σ of the tricritical Potts spin field σ(x)  was 
recalled in Eq. <ref>,
and
we have defined g_{α_i} as the coupling constant for the term 
∫ dx 𝔖^{α_i}(x,0)
when added to the 
action.
The 
ellipsis in the above equation indicates
not only the higher order terms but also quadratic terms involving couplings other than g_{α_i} and Δ.
This yields the  decay exponent for the 
correlation function of the replicated product of tricritical Potts spins (i.e. 𝔖^{α_i}(x,0)) at the new fixed point Δ_* of the field theory
Eq. <ref> – <ref>
to first order in ϵ,

    ⟨𝔖^{α_i}(x,0) 𝔖^{α_i}(y,0)⟩∝1/|x-y|^2X^(σ),R_N,
    X^(σ),R_N=    
         NX_σ -   
        N(N-1)c^2_σℰσ/2(2-R) ϵ+
        𝒪(ϵ^2) .

The OPE coefficient c_σℰσ can also be expanded in powers of 
ϵ about ϵ=0 (corresponding to m=∞).
The m=∞ conformal minimal model corresponds to the
4-state
tricritical Potts model, 
and we can replace the 
OPE coefficient
c_σℰσ in the above equation by 
its value
in the 
 4-state
tricritical Potts model <cit.>; any 
ϵ-dependence of the OPE coefficient c_σℰσ will only yield contributions of order
𝒪(ϵ^2)
to Eq. <ref>.
The OPE coefficient
in the tricritical 4-state Potts model is 
equal <cit.>
to c_σℰσ=
1/√(2), yielding

    X^(σ),R_N=
        N/8[
    1-1 3(1+6(N-1)/(2-R) )  ϵ
    +𝒪 ( ϵ^2 )
    ].


In the limit R→ 1 the above 
expression 
provides, as m → 4,  an expansion  in ϵ=3/(m+1)
(in the sense of the ϵ-expansion) of
the scaling
dimension
of the measurement averaged N^th moment of  
the
⟨σ̂^z_iσ̂^z_j⟩ correlation 
function at the Ising tricritical point,

    X^(σ),R=1_N=
        N/8[
    1-1 3(1+6(N-1) )  ϵ
    +𝒪 ( ϵ^2 )
    ].


Note that this expression shows that for the first moment, N=1, the first order correction in ϵ
to the exponent X_σ  
(observed without measurements) vanishes, consistent with the
expected absence of corrections to X_σ arising from Born-rule measurements in the tricritical Ising case, m=4, to any order in ϵ due to the result in Eq. <ref>.



0.5cm



 §.§ Measurement averaged moments of the  energy-energy correlation function Lg

.2cm
Let us now consider, again at the Ising tricritical point, two energy operators Ê (given in Eq. <ref>) on links i+1/2 and j+1/2 of the lattice
(also see footnote <cit.>).
We can again use 
Eqs. <ref>, 
<ref>,
<ref>, 
<ref>
to 
express
the measurement averaged N^th moment of the correlation function ⟨Ê_i+1/2Ê_j
+1/2⟩
in field theory language.
In continuum language, as discussed in Section <ref>, the lattice operator Ê is given by the energy scaling field ℰ
at the Ising tricritical point.
Thus,
from Eq. <ref>, the measurement averaged N^th moment 
of ⟨Ê_iÊ_j
⟩ (as before, we have dropped the lattice offset +1/2 on the Ê operators for notational simplicity)
can be written as


    ⟨Ê_iÊ_j⟩^N∼⟨𝔈^{α_i}(x, 0) 𝔈^{α_i}(y, 0)⟩

where we have defined,

    𝔈^{α_i}(x, τ)
    :=
     [ ∏_i=1^Nℰ^(α_i)(x,τ) ]

and α_i are pairwise distinct replica indices in the R-replica field theory. Again, as R→ 1,
the physics at long distances is determined by the 
new, measurement-dominated fixed point discussed in the previous section, and the
correlation function in Eq. <ref>
will  asymptotically exhibit power law behavior,


    ⟨𝔈^{α_i}(x, 0) 𝔈^{α_i}(y, 0)⟩∝1/|x-y|^2X^( E),R=1_N, 
    
           as  |x-y|→∞ .

Here
the power law exponent 
X^( E),R=1_N
characterizes the scaling behavior of the measurement averaged N^th moment of the
⟨Ê_i
Ê_j
⟩
correlation function at the Ising tricritical point.

Analogous
to the discussion of the   moments of the
spin operator
in the preceding subsection, we can evaluate
the power law exponent 
X^( E),R_N
 in an expansion in ϵ=3/m+1 
 at the new fixed point Δ_* 
discussed in the previous Section <ref>,
by computing the above correlation function 
 with  the
generalized replica action in 
Eq. <ref> – <ref>.
Unlike the product 
𝔖^{α_i}
of replicated spin fields in Eq. <ref>, 
the product 𝔈^{α_i}
of replicated energy fields 
doesn't turn out  to be
<cit.>
a scaling operator at the new RG fixed point (even to 1-loop order).
The corresponding scaling operators at the new fixed point 
are instead
given by a linear
superposition of
𝔈^{β_i} with different possible sets of replica indices {β_1, ⋯, β_N}.
These scaling operators at the new fixed point transform in irreducible representations 
of the permutation group S_R of the R replicas
(as introduced in Ref. <cit.>).
Thus, the correlation function 
⟨𝔈^{α_i}(x, 0) 𝔈^{α_i}(y, 0)⟩
will be expressed as a sum of power laws
which, at large distances,
turn out to be dominated <cit.>
by
the leading (smallest)
scaling dimension in the sum. 
Details are provided in App. <ref>. In a theory with R replicas, we
obtain

    ⟨𝔈^{α_i}(x, 0) 𝔈^{α_i}(y, 0)⟩∝1/|x-y|^2X^(ℰ),R_N,     |x-y|→∞

where for the case of Born rule measurements (R→ 1),
 

    X^(ℰ),R=1_N=1=1/2-ϵ/2+𝒪(ϵ^3)
          X^(ℰ),R=1_N>1=N/2[1+ϵ-(3N-5)ϵ^2+𝒪(ϵ^3)].


Analogous to the case of the moments of
the spin operator, as m→ 4, the above expression for X^(ℰ),R=1_N provides an expansion
in ϵ=3/(m+1) (in the sense of the ϵ-expansion) of
the scaling dimension of the measurement averaged N^th moment of the ⟨Ê_i
Ê_j
⟩ 
correlation function at the Ising tricritical point. 

We note that for the first moment, N=1, 
the above scaling dimension
matches with the  scaling dimension of the energy operator 
X_ℰ (from 
      Eq. <ref>)
at the unperturbed fixed point up to second order in ϵ=3/(m+1).
  This is again consistent with  Eq. <ref> which implies that, in the tricritical Ising case, m=4, there should be no corrections 
      to X_ℰ
      at any order in ϵ  arising from measurements following the Born-rule.

We close this section by noting that it turns out 
<cit.>
that  the Nth moments of the 
correlation function of the subtracted energy operator describing the deviation from its expectation value in a fixed quantum trajectory, 

    δÊ_i
    := Ê_i -
    ⟨Ê_i⟩, 
    
          ⟨δÊ_i δÊ_j⟩=
     ⟨Ê_i Ê_j⟩ 
    -
    ⟨Ê_i⟩ ⟨Ê_j⟩,

decay with a single power law, and not with a sum of different power laws as the Nth moments listed in Eq. <ref>. That is, at the Ising tricritical point, the moments of the resulting connected correlation function of energy operators decay with a single power law,

    ⟨δÊ_i δÊ_j⟩^N
    = [ 
    ⟨Ê_iÊ_j⟩
    -
    ⟨Ê_i⟩⟨Ê_j⟩ ]^N∝
          ∝1  
        |x-y|^
            2X̃^( E),R=1_N,
      N ≥ 1,

where the expression for X̃^( E),R=1_N is that on the right hand side of
Eq. <ref>, however now valid  for all positive integers N, including N=1.
In the language of the replica field theory, this is written in the form

    ⟨𝔈_NNR(x, 0) 𝔈_NNR(y, 0)⟩∝1/|x-y|^2X̃^(ℰ),R=1_N,
        as  |x-y|→∞

where the field 𝔈_NMR(x, 0), transforming in a specific irreducible representation of the permutation group S_R of the replicas, is defined in Eq. <ref> of 
App. <ref>.





 §.§ Multifractality of Scaling Dimensions

We know 
from the POVM condition, Eq. <ref>,
that the measurement-averaged
first moment of 
correlation functions 
such as
⟨σ̂^z_iσ̂^z_j⟩  and ⟨Ê_iÊ_j⟩ 
exhibits
the same power law behavior as 
in the unmeasured ground state of the tricritical Ising Hamiltonian.
(This has also been verified within the ϵ expansion to the order we have evaluated  it - see 
Eqs. <ref>,
 <ref>  above.)
From Eqs. <ref> and  <ref>,
however, 
one sees
that for
each of the two operators σ̂^z_i and δÊ_i
we have obtained an infinite number of independent scaling dimensions which are associated 
with the measurement-averaged higher
integer moments of their correlation functions.
This is a signature of multifractality.
In a fixed quantum trajectory both correlation functions 
⟨σ̂^z_iσ̂^z_j⟩  and ⟨δÊ_iδÊ_j⟩ are  bounded from above and are non-negative
for sufficiently weak measurement strength [
Since the operators σ̂^z_i and Ê_i have eigenvalues ± 1,
in any quantum state
the correlation functions 
⟨σ̂^z_iσ̂^z_j⟩  and ⟨δÊ_iδÊ_j⟩ are
both bounded 
from above and below.
[E.g., the correlation function ⟨σ̂^z_iσ̂^z_j⟩ 
lies in the interval
[-1,1] 
(consider (σ̂^z_i-σ̂^z_j)^2), and analogously ⟨δÊ_iδÊ_j⟩=⟨Ê_iÊ_j⟩-⟨Ê_i⟩⟨Ê_j⟩ 
lies in the interval [-2,2]).]
If the given correlation function (either ⟨σ̂^z_iσ̂^z_j⟩ or ⟨δÊ_iδÊ_j⟩) in the 
critical ground state |0⟩ is oscillating between positive and negative values
with a period consisting  of a certain number of lattice sites
as we vary j, we can choose to look at a subset of sites j for which the correlation function is strictly positive.
Now, if we consider performing weak measurements (small λ) on the ground state |0⟩, a particular quantum trajectory will be given by |ψ_m⃗⟩=
K̂_m⃗|0⟩/
√(p_0(m⃗)
) (see Eq. <ref> and Eq. <ref>). The correlation function 
⟨σ̂^z_iσ̂^z_j⟩ (or ⟨δÊ_iδÊ_j⟩=⟨Ê_iÊ_j⟩-⟨Ê_i⟩⟨Ê_j⟩) in this quantum trajectory and for a given value of |j-i| will be an analytic function of the measurement strength λ. 
Since λ=0 corresponds to performing no measurements at all and given that the correlation function  ⟨σ̂^z_iσ̂^z_j⟩ (or ⟨δÊ_iδÊ_j⟩) was positive-valued in the ground state |0⟩, for sufficiently small values of λ the correlation function will also be positive in a given quantum trajectory obtained upon measurements. 
For sufficiently weak measurement strength λ, we can thus 
restrict ourselves to correlation functions ⟨σ̂^z_iσ̂^z_j⟩ and ⟨δÊ_iδÊ_j⟩ which are bounded from above and are  non-negative.].
This is in line with 
the  field theory calculation
which represents a controlled perturbative RG calculation in the 
(weak) strength of measurements.
Then, the Nth moments  
of these correlation functions
for integer values of N are known to determine the  entire probability distribution 
of these correlation functions
(and are analytic functions of N). 
(See e.g. Refs. LUDWIG1990infinitehierarchy,fellerintroduction,Witten2019,Boas1954EntireFunction,Titchmarsh1939.)
Thus, the exponents X^(σ),R=1_N
and X̃^( E),R=1_N, while initially defined for integer values of N, are in fact defined for 
real values of N (by analytic continuation).
Hence, to
each of the two operators
σ̂^z_i and δÊ_i is associated  a continuous spectrum of scaling dimensions, obtained by continuously varying N. 

Moreover, physically, while correlation functions represent random 
observables which are not self-averaging (as also  reflected by the non-linear dependence of
Eqs. <ref> and  <ref>
on the moment order N), their logarithm is self-averaging, and a cumulant expansion in the logarithm of the correlation function corresponds to a Taylor expansion of X^(σ),R=1_N
and X̃^( E),R=1_N in N about N=0 (see, e.g. Refs. DERRIDA-PhysRepts1984,LUDWIG1990infinitehierarchy,ZabaloGullansWilsonVasseurLudwigGopalakrishnanHusePixley,LiVasseurFisherLudwig,JianShapourianBauerLudwig).
This provides the typical scaling exponents, 

    X^(σ),R=1_typ=
    lim_N→ 0X^(σ),R=1_N/N,
    
          X̃^( E),R=1_typ=
    lim_N→ 0X̃^( E),R=1_N/N

where

    log⟨σ̂^z_iσ̂^z_j⟩=
    -2X^(σ),R=1_typlog|x-y|,
    
          log⟨δÊ_iδÊ_⟩=
        -2X̃^( E),R=1_typlog|x-y|.

Specifically,
Eq. <ref>,

    X^(σ),R=1_typ=
       1/8[1+5/3ϵ+𝒪(ϵ^2)],

and 
Eq. <ref>,

    X̃^(ℰ),R=1_typ
    =1/2[1+ϵ+5ϵ^2+𝒪(ϵ^3)],

provide, as m→ 4, an expansion in ϵ=3/(m+1) of the scaling exponents of,
respectively, the typical spin-spin and  the typical (connected) energy-energy correlation function with Born-rule measurements on the ground state of the tricritical Ising Hamiltonian.


 §.§ Logarithmic Correlation Functions

   Until now, we have used the replica field theory formalism with
   an arbitrary 
   number R of replicas and its R→1 limit to evaluate various quantities averaged with Born-rule probabilities (see Eq. <ref>).
   In taking the R → 1 limit, the scaling dimensions of two operators which are distinct when R ≠ 1 
   may become equal at R=1.
   Such a collision of scaling dimensions while taking replica limits can give rise to “logarithmic correlation functions” at the measurement-dominated fixed point. The corresponding
   logarithmic factors multiplying the power law decay of certain correlation functions at criticality are a hallmark of so-called logarithmic CFTs, which are  a class of non-unitary CFTs.
   (See e.g. Ref. 
   Gurarie1993,GurarieLudwig2005,VasseurJacobsenSaleur,CardyLogarithm1999,DavisCardy2000,CardyLogarithm2013).
We demonstrate in the present subsection that the indeterministic (random) nature of measurement outcomes performed on a critical ground state generates critical states that carry these hallmarks of logarithmic CFTs.
    In this section, we address these logarithmic CFT features of the measurement-dominated fixed point, and
    in particular, highlight correlation functions which contain multiplicative logarithms of distance on top of a power law decay. As discussed in Sect. <ref>, the scaling operators at the new fixed point Δ_*(ϵ) (Eq. <ref>) transform in irreducible representations of the symmetric group S_R and the correlation functions of such operators exhibit a pure power law decay at the new fixed point.
    Let us consider two operators 𝒪 and 𝒪̃ both transforming in irreducible representations of the symmetric group S_R s.t. the two correlation functions ⟨𝒪𝒪⟩ and ⟨𝒪̃𝒪̃⟩ are pure power law decaying at the new fixed point Δ_*(ϵ), 
    i.e.

    ⟨𝒪(x,0) 𝒪(y,0)⟩=A(R)/|x-y|^2X(R),
    ⟨𝒪̃(x,0) 𝒪̃(y,0)⟩=Ã(R)/|x-y|^2X̃(R).

Let us suppose that the correlators have colliding scaling dimensions in the replica limit R→ 1, 

    lim_R→ 1 (X̃(R)-X(R))=0    (but X̃(R)≠ X(R)  if  R≠1 ),

and that the 
amplitudes 
A(R) and Ã(R) of the correlators can be normalized s.t.

    lim_R→ 1 (Ã(R)+A(R))=𝒦,

where 𝒦 is a constant.

Then as R→ 1, we can write,

    Ã(R)      =-A(R)+
        𝒦
    X̃(R)      =X(R)+a(R-1)

where 
a≠ 0 
is also a
constant independent of R.

Then in the limit R→ 1,

    ⟨𝒪̃(x,0) 𝒪̃(y,0)⟩+ ⟨𝒪(x,0) 𝒪(y,0)⟩=
          =Ã(R)/|x-y|^2X̃(R)+A(R)/|x-y|^2X(R)
          =-A(R)+
       𝒦/|x-y|^2X(R)+2a(R-1)+A(R)/|x-y|^2X(R)
          =1/|x-y|^2X(R)[
         𝒦
         +2aA(R)(R-1)log |x-y|+
                                                           +𝒪((R-1))]


Now the last term in the above equation clearly vanishes as R→1 and therefore we are left with,

    lim_R→ 1  [⟨𝒪̃(x,0) 𝒪̃(y,0)⟩+ ⟨𝒪(x,0) 𝒪(y,0)⟩]=
    =lim_R→ 1𝒦+2aA(R)(R-1)log |x-y|/|x-y|^2X(R=1).

Therefore, in addition to Eq. <ref> and <ref>, if we have 

    lim_R→ 1A(R)(R-1)=finite

or equivalently,

    A(R)=𝒪(1/R-1)

we see that the following correlation function will be logarithmic at the new fixed point,

    lim_R→ 1  [⟨𝒪̃(x,0) 𝒪̃(y,0)⟩+ ⟨𝒪(x,0) 𝒪(y,0)⟩].

In App. <ref>, we identify two such operators 𝒪 and 𝒪̃ which transform irreducibly under the symmetric group S_R,

    𝒪=1/R-1𝔈_20R=1/R-1∑_a,b=1
    a≠ b^Rℰ^aℰ^b

and

    𝒪̃   =1/(R-1)(R-2)𝔈_22R
       =1/(R-1)(R-2)∑_a,b=1
    a≠ b; a,b≠ 1,2^R(ℰ^a-ℰ^1)(ℰ^b-ℰ^2).

Here the field 𝔈_NMR, transforms in a specific irreducible representation of the symmetric group S_R and is defined in Eq. <ref> of 
App. <ref>.
The normalization factors in front of 𝔈_20R and 𝔈_22R are chosen such that they satisfy  the criterion in Eqs. <ref> and <ref>.
(See App. <ref>.)

In App. <ref>, we show that the scaling dimensions of above two operators (which have unequal scaling dimensions at a generic R≠ 1) become equal to each other at R=1.
Then, following our analysis above, the correlator (see App. <ref> for its detailed form)

    lim_R→ 1  [⟨𝒪̃(y,0) 𝒪̃(x,0)   ⟩+ ⟨𝒪(y,0) 𝒪(x,0)⟩]=
       4⟨ℰ^1(y,0)ℰ^1(x,0)ℰ^2(y,0)ℰ^3(x,0)⟩
       -3⟨ℰ^1(y,0)ℰ^2(y,0)ℰ^3(x,0)ℰ^4(x,0)⟩

is logarithmic at all fixed points Δ_*(ϵ) parameterized by even m. In particular, as m→ 4, we obtain
the result that the following correlator averaged over 
measurements
with Born-rule probability should be logarithmic at the Ising tricritical 
point,

    4⟨Ê_iÊ_j⟩⟨Ê_i⟩⟨Ê_j⟩-3⟨Ê_i⟩^2⟨Ê_j⟩^2∝log|j-i|+𝒪(1)/|j-i|^2X^(ℰ),R=1_N=2,

where X^(ℰ),R=1_N=2 is given by Eq. <ref> 
and 𝒪(1) denotes a constant.

Finally, we note that in logarithmic CFTs the dilation operator of the scale transformations is not diagonalizable, and the logarithmic correlation functions are associated to Jordan cells of the dilation operator <cit.>.
In particular, the Jordan cell or the `logarithmic pair (C,D)' associated to the logarithmic correlation function in Eq. <ref> is formed by the following scaling operators in the R→1 replica limit,

    D=   𝒪+𝒪̃
    →     ℰ^1ℰ^2+∑_α≠ 1^Rℰ^αℰ^1+∑_α≠ 2^Rℰ^αℰ^2-∑_α,β=1
    α≠β^Rℰ^αℰ^β,

and 

    C=(X(R)-X̃(R))𝒪→-a×∑_α,β=1
    α≠β^Rℰ^αℰ^β

where the universal constant a 
is defined in Eq. <ref>.
The correlation functions of the operators C and D  are given by,

    ⟨ D(x,0)D(y,0)⟩   =Ã(R)/|x-y|^2X̃(R)+A(R)/|y-x|^2X(R),
    ⟨ C(x,0)D(y,0)⟩   =A(R)(X(R)-X̃(R))/|x-y|^2X(R),
    ⟨ C(x,0)C(y,0)⟩   =A(R)(X(R)-X̃(R))^2/|x-y|^2X(R),

and thus in the limit R→1 (see Eq. <ref>),

    ⟨ D(x,0)D(y,0)⟩   ⟶2ζ×(log|x-y|+𝒪(1))/|x-y|^2X^(ℰ),R=1_N=2,
    ⟨ C(x,0)D(y,0)⟩   ⟶-ζ/|x-y|^2X^(ℰ),R=1_N=2,
    ⟨ C(x,0)C(y,0)⟩   ⟶ 0

where ζ:=lim_R→1a(R-1)A(R).

.8cm


§ ENTANGLEMENT ENTROPIES


At long wavelengths, the n^th Rényi entanglement entropy 
of the ground state of a translationally invariant,
i.e. non-random
(1+1)d CFT
can be expressed as the logarithm of 
the correlation function of 
two n-twist fields
by considering n copies of the corresponding 2D CFT <cit.>. (See also Ref. HolzheyLarsenWilczek1994.)
Following our calculations of measurement averaged moments of correlation functions and, in particular, the measurement averaged logarithm of correlation functions 
in Sect. <ref>, 
we 
will now also
calculate the average of the logarithm of the twist field correlation functions 
(which are 
the Rényi entropies) using the controlled
perturbative RG expansion.
This will involve calculating the correlation function of multiple copies of twist fields
in the generalized replica action given in Eq. <ref> – <ref>. 
Since the twist fields are geometrical in nature, their generalization to higher m theories appearing in Eq. <ref> is natural.
This will allow us to evaluate the 
universal coefficient of the logarithm of subsystem size of the measurement averaged Rényi entanglement entropies at the tricritical Ising point in an expansion in ϵ=3/(m+1).
To calculate the n^th Rényi entropy, we will have to consider n copies of a state, corresponding to a given set of measurement outcomes, and `glue' the copies to form a n-sheeted Riemann surface (see App. <ref>).
Moreover, to be able to 
perform the average over measurement outcomes of the logarithm of the n-twist field correlation function,
we will have to introduce another replica index k. Overall we need to introduce R=nk+1 replica copies, where the additional copy comes due to the Born rule probability factor p_0(m)
(analogous to Eq. <ref>),
and the limit k → 0 or R → 1 will correspond to the Born-rule measurement averaged Rényi entropy(s) <cit.>.
The details of these calculations are given in  
 App. <ref> and we 
 obtain the following expression
 for the measurement averaged
n^th Rényi entropy S_n,A

    S_n,A=1/1-nd/dk|_k=0⟨∏_j=1^k(Τ^(j)_n(u,0)(Τ^(j)_n)^-1(v,0))⟩_Δ_* .

Here, Τ^(j)_n(u,0) denotes the twist field,  where
the superscript j 
specifies
which copy of 
the Riemann surface (out of k copies) the twist field corresponds to, and the subscript n signifies that we are dealing with twist fields for a 
n-sheeted
Riemann surface
(compare Fig. <ref>).
The subscript
Δ_* indicates that the correlation function is evaluated at the 
measurement-dominated
fixed point.
We now evaluate the scaling dimension of the replicated twist field,
occurring in the correlation function in Eq. <ref>,
in the replica field theory Eq. <ref>, <ref>, <ref>
to 1-loop order  in the ϵ expansion using the OPE.
The coefficient 
of the twist field 
in the OPE of the twist field
with the perturbation Φ(x)
is calculated in App. <ref> and is equal to kI_n, where I_n is 
defined as

    I_n=n/2π∫_0^∞ds 1-s^n-1/(1-s)(1+s^n)+
     𝒪 (ϵ ).

Then the 1-loop RG equation for the scaling dimension of the twist fields [ ∏_j^kΤ^(j)_n ]
can be obtained as usual, e.g., from the RG equation for  a coupling constant g_(n,k) for the term  ∫ dx [∏_j^kΤ^(j)_n] when added to the action. This yields

    dg_(n,k)/dl=(1-kd_n)g_n,k+2kΔ I_ng_n,k+…

where d_n:=c/12(n-1/n) is the scaling dimension of twist field  𝒯_n 

in the unperturbed CFT
<cit.>, and the 
 ellipsis represents terms  that will end up contributing only to corrections of second order in ϵ.
Then [We note that the above result for the scaling dimension of  the k^th moment 
of the n-twist field at the new Δ_*-fixed point is linear in k to 1-loop order.
However, in analogy with observations made in 
Ref. LiVasseurFisherLudwig in a related context,
we expect non-linearities in k to appear in higher orders in ϵ.
We plan to address the calculation of these non-linearities  in future work.]

    ⟨∏_j=1^k(Τ^(j)_n(u,0)(Τ^(j)_n)^-1(v,0))⟩_Δ_*∝(u-v/a)^-2d^*_n,k
       where,   d^*_n,k=kd_n-2kΔ_* I_n+𝒪((Δ_*)^2).

Here,
we have inserted 
a
short distance cutoff a 
to make the final result dimensionless.
The 
measurement-averaged
n^th Rényi entanglement entropy for an interval 
of length l=|u-v| is then given, using Eq. <ref>, by

    S_n,A      =c^(eff)_n/3lnl/a+𝒪(1)  where,
    
    c^(eff)_n      = 
    c(m) (1+1/n/2)-3 I_n/(n-1)ϵ+𝒪(ϵ^2),

where c(m) is the central charge of the unperturbed theory,
Eq. <ref>, 
while I_n is the integral in   Eq. <ref>.
To obtain the 
measurement-averaged
von Neumann entanglement entropy, we 
take the limit n→ 1 in the above equation. From Eq. <ref>,
one obtains

    dI_n/dn|_n=1=1/2π∫_0^∞ds ln(s)/s^2-1=π/8.

Thus, using Eqs. <ref>, <ref> and  <ref>, the 
measurement-averaged
von Neumann entropy is

    S_1,A=lim_n→1S_n,A
    =c^(eff)_n=1/3lnl/a+𝒪(1) ,

 where c^(eff)_n=1  at the
 measurement-dominated fixed point Δ_*
 is given by

    c^(eff)_n=1=
    c(m)
    -3π/8ϵ+𝒪(ϵ^2).

This provides, as m → 4, an expansion in ϵ=3/(m+1) of the “effective central charge” 
at the measurement-dominated fixed point  of the tricritical Ising ground state.
Note that, unlike 
in translationally invariant (i.e., unmeasured and thus non-random) CFTs
<cit.>, 
here the measurement averaged 
n^ th
Rényi entanglement entropies  S_n,A are not simply
a 1/2(1+1/n) 
multiple
of the von Neumann entropy. 
Rather, the universal coefficients of the logarithm
of subsystem size
are all independent 
of each other
for different measurement averaged Rényi entanglement entropies.
This feature is similar to having a hierarchy of 
independent (“multifractal”)  scaling dimensions 
for measurement-averaged moments of correlation functions of operators,
as discussed in Section <ref>.
Note that the n^th Rényi entropy being 
1/2(1+1/n) times the von Neumann entanglement entropy plays a central role in the calculation of entanglement spectrum shown in Ref. CalabreseLefevre2008 for the usual (unmeasured) 1d critical ground states.
Thus, we expect 
the entanglement spectrum of the present system under measurements 
to
exhibit qualitatively different universal features when compared to the (unmeasured) 1d critical ground states.
 If we have our system at finite temperature, the unmeasured state is given by Gibbs 
state ρ=e^-β H/Z, in contrast to the (pure) critical ground state.
Following
steps that parallel
the pure state 
calculation above (using twist fields), we find
that the measurement averaged n^th Rényi entanglement entropy can be written as 

    S_n,A=1/1-n(d/dk|_k=0𝒵(β)_A/𝒵(β)_∅)

where now,

    𝒵_𝒜(β)=   ∑_𝐦⃗Tr((K_𝐦⃗e^-β H K_𝐦⃗^†)^⊗ nk+1𝒮^k_n,A)
    𝒵_∅(β)=   ∑_𝐦⃗Tr((K_𝐦⃗e^-β HK_𝐦⃗^†)^⊗ nk+1).

(See Eq. <ref> for the definition of 𝒮^k_n,A). Then following the discussion in App. <ref>,
one 
verifies
that the average entanglement entropy is still given by Eq. <ref>, but
now the twist field correlation function
in this equation is calculated on a cylinder
of
finite circumference β instead of a plane. 
Since we are interested in evaluating this correlation function 
of twist-fields
at the 
new fixed point Δ_*, which is conformally invariant,
we can use the 
conformal transformation 

    w→ z=β/2πlog w

to map the twist field correlation on the plane (given in Eq. <ref>) to the twist field correlation function on the cylinder.
Thus, the measurement averaged n^th Renyi entanglement entropy for a region of length l≡|u-v| at finite (inverse) temperature β is given by

    S_n,A=   c^(eff)_n/3ln(β/π asinhπ l/β)+𝒪(1),
 
where the universal coefficient c^(eff)_n is given 
by Eqs. <ref>, <ref>.
At inverse  temperature β<<l,
this reduces to

    S_n,A/l ∼   c^(eff)_nπ/31/β.

If we take the subsytem size l to approach
the length L of the total system,
the universal coefficients c_n^eff 
will also appear in the  measurement averaged (not entanglement) Rényi entropies calculated for the full mixed state of the system at finite temperatures [Since the subsystem size approaches the system size L, the twist fields will be sitting at the ends of the one dimensional quantum system and such correlator will no longer be a pure power law, and will depend on the full operator content of the theory. However, the leading order piece proportional to the system size will not depend on the boundary condition and will be the same as in Eq. <ref>.]. 
At finite temperatures satisfying the condition β<<L, the measurement averaged Rényi entropy of the  mixed state of the system is
then 
given by

    R_n/L∼ c^(eff)_nπ/31/β+𝒪(1/β^2),

which is extensive in 
the total system size L.
We also note that, since the Rényi entropy of the full system is a self-averaging quantity,
it is thus represented by its average in Eq. <ref>.
This expression should be contrasted with
the Rényi-index n dependence of 

the 
extensive Rényi entropies
for 
the unmeasured, i.e. non-random
1d quantum critical 
system
at thermal equilibrium,
which is given by 
<cit.>,
<cit.>

    R_n/L=
    c(m) · [1 2  (1 + 1 n)]  π/31/β
        +𝒪(1/β^2),

where c(m), Eq. <ref>,
is the central charge of the corresponding 
unmeasured 2D CFT.
We note that due to the n-dependence of the universal coefficient c_n^eff
(from Eq. <ref>, <ref>)
in Eq. <ref>,
the leading order finite temperature behavior of the measurement averaged n^th Rényi entropy of the full mixed state of the system does not satisfy the simple relation in Eq. <ref>, which is valid for translationally invariant (unmeasured, non-random) CFTs.

.8cm


§ EFFECTIVE “GROUND-STATE DEGENERACY" LG

At measurement-induced phase transitions of deep  random quantum circuits with measurements in the bulk of the space-time of the circuit there exists a universal quantity known as the “effective central charge” which is defined in terms of  the replica limit R→ 1 of the derivative with respect to R of  the universal finite-size correction of the free energy of the circuit on a cylinder or strip of finite circumference or width. In a sense, it replaces the notion of the central charge  which is zero at R=1, in the non-unitary CFT describing the transition. The “effective central charge” has been shown in Ref. ZabaloGullansWilsonVasseurLudwigGopalakrishnanHusePixley,KumarKemalChakrabortyLudwigGopalakrishnanPixleyVasseur
to represent the universal finite-size scaling behavior of the Shannon-entropy of the measurement record of the circuit, the latter providing an
expression for 
the logarithm of the partition function in the language of the measurements performed on the circuit.
The “effective central charge” is   not equal to the universal coefficient of the logarithm of subsystem size of the  entanglement entropy at measurement-induced transitions in these deep random quantum circuits.

In  CFTs with boundary (or defect, before folding
<cit.>, <cit.>) there is a quantity referred to as  the “ground-state degeneracy g" or “zero-temperature entropy" S:= ln g which is a universal constant associated with any specific conformally invariant boundary RG fixed point. It plays a role for boundary (defect) CFTs analogous to the role played by the central charge in a bulk CFT. In particular, in unitary CFTs  it decreases upon boundary RG flows, a property often referred to as the “g-theorem” 
<cit.>,<cit.>.
The defect (boundary, after folding) piece 
ln Z_d 
(subscript d standing for “defect”)
of the logarithm of the partition function of a CFT on a
cylinder
of large length β and  
circumference
L≪β
has [With periodic boundary conditions in 
the spatial direction of size L,
we have to consider the CFT with a defect on a torus with radii β and L, where β denotes the inverse temperature (see Fig. <ref>). 
After folding the torus at τ=β/2 and at τ=0 (the location of the defect), we obtain a finite 
`double-sheeted' cylinder of circumference L and length β/2. 
The τ=0 boundary of this `doubled-sheeted'  cylinder is associated with the defect, while the boundary at τ=β/2 is `trivial' and it moves off to infinity in the limit β→∞ of interest, 
as we are interested in the ground state of the system.
Thus, the defect free-energy can be thought of as being associated 
with the boundary free-energy of this semi-infinite 
(`double-sheeted')  cylinder]
a non-universal contribution f_d per unit 
length 
L
of the defect, plus a universal constant, 
length-L
independent contribution S=ln g,

    ln Z_d= f_d·L
    + S,
      S=ln g.

In the 
defect (boundary, after folding) CFT of interest in the present paper the universal quantity S(R)= ln g(R) depends on the number R of replicas and must vanish in the R→ 1 limit due to the POVM condition Eq. <ref>.


In general we obtain for the type of measurement problems on a critical ground state discussed in the present paper, analogous to the logic used in Ref. ZabaloGullansWilsonVasseurLudwigGopalakrishnanHusePixley 
to obtain the Shannon entropy of a deep circuit with bulk measurements,
an expression for the partition function Z_R of our system from Eq. <ref> and Eq. <ref> upon  setting all operators to the identity,

    Z_R=1+r=
    ∑_m⃗
    p(m⃗)  [p(m⃗)]^r
    
           or:
          ln Z_R=1+r=
    
          
    =ln{∑_m⃗ [p(m⃗)
    + r p(m⃗) ln (p(m⃗)) +  O(r^2)
    ]
    }
           and:
          d  dr_|r=0ln Z_R=1+r=
    ∑_m⃗ p(m⃗) ln (p(m⃗))=
    
          
    = - S_Shannon({p(m⃗)}
    = f_d,eff·L  +    S_eff,

where

    S_Shannon({p(m⃗)}
    =
    -
    ∑_m⃗ p(m⃗) ln (p(m⃗))

is the Shannon entropy of the measurement record, while

    f_d,eff :=  ( d  dr_|r=0 f_d(R=1+r)  ) =
     non-universal
           and
          S_eff:= d  dr_|r=0 S(R)=
     d  dR_|R=1ln g(R)=
     
          =
       ( d  dR_|R=1 g(R)  )/g(R=1) := ln g_eff
      =  universal

The “effective boundary entropy” S_eff, the logarithm  of the “effective boundary degeneracy” g_eff, therefore characterizes the universal constant, i.e. system-size-L independent part of the Shannon entropy of the measurement record on the critical ground state.
The Shannon entropy thus expresses the Born-rule averaged defect free energy in terms of a quantity directly related to the measurements.
The “effective boundary entropy” S_eff
plays a role in our problem of measurements on the quantum critical ground states
analogous to the role played by the “effective central charge" in the deep  circuits with bulk measurements where, as already mentioned, the latter 
describes
the universal finite-size scaling information contained in the measurement record in the space-time bulk of the circuit. 


The value of g_0 at our  ultraviolet  
defect (boundary) fixed point Δ=0 is 
S_0=ln g_0=0
since there the space-time has no defect  at all and thus there is no 
defect (boundary)
contribution to the free energy of the space-time.
We have calculated the universal boundary entropy S(R)=ln g(R) at our new fixed point Δ_*, 
Eq. <ref>,
using our ϵ-expansion.
Following Ref. AffleckLudwig1991 and JengLudwig, we obtain

    g(R) = g_0 + δ g(R),   where g_0=1, 
          ln g(R) =  ln [1 + δ g(R)]
    = δ g(R) +  O (δ g(R) )^2,
    
          δ g(R)=
    -
     π^2/24R(R-1) (R-2)^2ϵ^3 +  O(ϵ^4).

We note that ln g(R), the  universal constant  contribution from the boundary to the logarithm of the  partition function vanishes as R→ 1 to the order in ϵ we are considering,  consistent with requirement from the POVM condition enforcing a partition function equal to unity in this limit.
The  “effective ground state degeneracy”
g_eff
of the 
defect (boundary)
fixed point Δ_* is thus found to lowest non-vanishing order in the ϵ expansion to be

    g_eff = 1 + δ g_eff,
    
          δ g_eff :=
     (d 
     dR)|_R=1δ g(R)
    = -π^2/24ϵ^3 +  O(ϵ^4).

Thus,
the  universal constant contribution to the Shannon entropy of the   measurement record on the tricritical Ising ground state is, owing to  Eq. <ref>, given by the following expansion

    [ S_Shannon({p(m⃗)})  ]_ universal  part
    =
    
          
    =-S_eff= - ln g_eff = π^2/24ϵ^3 +  O(ϵ^4),
 
as m→4.

We close this section by noting that the `boundary entropy' <cit.> has recently also been discussed in the different context  of decoherence in Refs. 
ZouSangHsieh,AshidaFurukawaOshikawa.
.8cm


§ MEASUREMENTS OF LG IN THE QUANTUM ISING MODEL


(a): So far we addressed in this paper the problem of 
measurements with the energy operator
Ê_j+1/2, 
Eq. <ref>,  performed on the ground state of the tricritical quantum Ising model (using the formulation by O'Brien and Fendley). We developed an ϵ=3/(m+1) expansion for the resulting rich and novel critical properties, where m ≥ 4 is an even integer characterizing a subset of minimal model  CFTs which can be represented by tricritial q-state Potts models (√(q)=2 cosπ m, Eq. <ref>). The tricritical Ising model itself, the aim of our study, corresponds to the smallest even value m=4.

(b):  In the present section we address the problem of 
measurements with the Pauli spin operator σ̂^z_i, performed on the ground state of the critical (not tricritical) quantum Ising model. We will again develop an ϵ=3/(m+1) expansion for corresponding resulting rich and novel critical properties of this system. Here, however, m≥ 3 is an odd integer that characterizes another (complementary) subset of minimal model CFTs, which can be represented as a subset of Ising multicritical points (namely those with m= odd). The Ising model itself, the aim of our study, corresponds to the smallest odd value m=3.

There is a common principle unifying problems (a) and (b), which allows us to apply the logic we have already developed for (a) also to (b) - with some modifications:

As described above, the continuum field  E describing the measurement operator Ê_j+1/2
Eq. <ref>, <ref>
in the 
O'Brien-Fendley model
is the energy operator 
of the tricritical Ising point.
This, as already mentioned above,
corresponds to the so-called Kac-Table primary field φ_1,2 of the minimal model CFT
[see,  e.g., Refs. BelavinPolyakovZamolodchikov1984, di1997conformal]
with m=4 describing the tricritical Ising model. Moreover, for arbitrary even values of m ≥ 4, the same
Kac-Table primary field φ_1,2 of the minimal model CFT corresponds to the energy operator ℰ in the corresponding
tricritical q-state Potts model that we use to define our ϵ expansion, i.e. ℰ=φ_1,2 for m≥ 4 even. At the same time, this operator is represented in the Landau-Ginzburg description with action Eq.
<ref>
by 

    tricritical  Ising:
           E = φ_1,2 = :ϕ^m-2:,
        when m≥ 4,   m=  even,

(see Eq. <ref>).


On the other hand, for the m=3 conformal minimal model, describing the critical Ising model, the continuum field representing the spin operator σ̂^z_i is also represented by the Kac-Table φ_1,2 operator at that value m=3. 
Moreover, for arbitrary odd values of m ≥ 3, the same
 Kac-Table primary field φ_1,2 of the minimal model CFT is 
 represented in the Landau-Ginzburg description with action Eq. <ref> again by φ_1,2 = :ϕ^m-2:, with the crucial difference that since now m= odd, this field  now changes sign under the Ising Z_2 symmetry ϕ→ -ϕ.
 We will use this field 
 as the generalization of the spin operator of the Ising model (m=3 minimal model CFT) to the minimal model CFT at odd m ≥ 3, and will denote it by the symbol
 
    critical  Ising:
          𝒮 := φ_1,2 = :ϕ^m-2:,
         when m≥ 3,  m=  odd.

The operator 𝒮 plays, for the ϵ=3/(m+1) expansion with odd m≥ 3 in the Ising case, exactly the same role that the operator ℰ played for the ϵ=3/(m+1) expansion with even m ≥ 4 in the tricritical Ising case that we have already described in this paper. 
Various
discussions and calculations discussed so-far for the tricritical Ising case can literally be taken over to the Ising case by simply replacing ℰ by 𝒮, and by replacing  “m≥ 4, m= even” by “m≥ 3, m= odd”.


However, before proceding with this,
we 
need to understand a subtlety of the Ising case, m=3. This will be done below after briefly reviewing again our ϵ expansion approach. 


Let us recapitulate that in  Sect. <ref> we
started with a quantum critical ground state in the universality class of the
Ising tricritical 
point (m=4^th minimal model) and considered the problem of performing measurements
of the  lattice energy operator Ê_j+1/2 in Eq. <ref> (which 
is odd
under Kramers-Wannier duality)
on it.
In the process of analyzing this 
problem within an ϵ expansion, 
we introduced 
the
generalized replica action in Eq. <ref>, with action S_*  corresponding to the m^th minimal model CFT (Eq. <ref>) and a field ℰ in this CFT.
We motivated the choice of the field ℰ in these higher minimal model CFTs by restricting
to only the even m minimal model CFTs
and using the tricritical q-state Potts formulation of the even  m minimal model
CFTs,
where the field ℰ in Eq. <ref> corresponds to the 
energy operator in the tricritical q-state Potts model, which for q=2 is the same as the tricritical Ising point.
This provided the basis for the ϵ
expansion, where the control parameter ϵ=
3/(m+1) is small when the even integer m≥ 4 is large.

Until now, for us the
generalized replica field theory
in 
Eqs. <ref> – <ref>
with even
m≥ 4 minimal model 
CFTs
only served as a tool to
perturbatively study the replica field theory for measurements at the Ising tricritical point (i.e. the replica field theory at  m=4) in 
the
small parameter ϵ=3/(m+1) via the ϵ expansion we developed  in Sect. <ref>
above.
However, as purely a defect field theory problem with replica action in Eqs. <ref> – <ref>, it is clear that the perturbative RG analysis 
presented
in Sect. <ref>
applies to any m^th minimal model with m≥ 4,
including also
the minimal model
CFTs
with odd m.
This is because the field ℰ, Eq. <ref>,
being the
energy operator
of the tricritical q-state Potts models, when m ≥ 4 is even, was only used to motivate the choice of field ℰ in the
minimal models with larger parameter m>4, and the field theory problem is perfectly well defined with just Eqs. <ref> – <ref> 
also for odd values of m, as far as the RG analysis is concerned [
Unlike the even m minimal model CFTs, there does not exist a tricritical q-state Potts model with the same central charge as the odd m minimal model CFTs.
However, this is immaterial to the RG analysis of the replica action in Eq. <ref> for the multicritical point given by an odd m minimal model CFT in Eq. <ref> and with the symbol ℰ replaced by 𝒮=:ϕ^m-2:.
We note that for the odd m minimal model CFTs the field 𝒮=:ϕ^m-2: is one of the spin fields of the multicritical point and not an energy field, and this distinction is also inconsequential to the RG analysis of the replica action presented in Sect. <ref>.  As already mentioned above, for both odd m and even m minimal model CFTs, the field :ϕ^m-2: is the so-called Kac-Table operator φ_1,2. See also footnote <cit.>
].
Excluding
Sect. <ref> on the (replicated) tricritical Potts spin correlation functions, the results from perturbative RG analysis for the entanglement entropies in Sect. <ref> and the (replicated) ℰ=:ϕ^m-2:  field correlation functions in Sect. <ref> also readily extend to all 
m≥ 4 (both even and odd)
minimal model CFTs
in Eq. <ref>. However, the physical meaning of this replica field theory in 
Eq. <ref>, and the corresponding ϵ=3/(m+1) expansion, is completely different in the case of m ≥ 3 with m= odd, as compared to the case of m≥ 4 with m= even, discussed in Sect. <ref>: The former case 
will serve to describe
the ϵ=3/(m+1) expansion for the problem of σ̂^z_i measurements on the ground state of the critical quantum Ising Hamiltonian. In this case, it is useful to rename the operator
ℰ, Eq. <ref> as 𝒮, Eq. <ref>, which is now odd under the Ising Z_2  symmetry. (In the Ising case,
m=3, 𝒮 is the continuum field representing the
Pauli σ̂^z_i operator that is measured in this case.) 
For general m≥ 3 with m=
 odd
the replica field theory action is the same as in Eqs. <ref> – <ref>, but with the field ℰ replaced by 𝒮, i.e. 

    -𝕊   =
    ∑_a=1^R
    (-1)
    S_*^(a)
    +
    Δ∫_-∞^+∞ dx   Φ(x)
    Φ(x)   =∑_a,b=1 
     a≠ b^R𝒮^(a)(x,0)𝒮^(b)(x,0),
    
    S_*   =∫ d τ∫ d x {1/2(∂_xϕ)^2+1/2(∂_τϕ)^2+
    g^*_m-1ϕ^2(m-1)},

with the notation 𝒮 as defined in Eq. <ref>.



For the m=3 minimal model, 
the lowest value of m= odd, describing the Ising critical point,  however, there is an 
additional subtlety that arises in the RG analysis of the replica action in Eqs. <ref> – <ref>.
For minimal models m≥ 4, the 
terms denoted by the ellipsis  
“…" in the OPE in Eq. <ref>
of the perturbation Φ (Eq. <ref>
and
Eq. <ref> for  m= even and odd, respectively), with itself contained only irrelevant fields localized 
on the τ=0 time-slice. Hence 
we obtained
Eq. <ref> at 1-loop order, which described the RG flow of Δ.
Precisely at the Ising critical point  m=3 however, for an arbitrary number of replicas R, the OPE in Eq. <ref>
of the perturbation Φ,
Eq. <ref>,
with itself 
contains a term
which is 
exactly
marginal. 
In particular, in the case of m=3, the following term 

    ∑_α𝔢^(α)  where   𝔢=:ϕ^2:=:ϕ^2m-4:|_m=3

 appears on the RHS of Eq. <ref>. Here,
 :ϕ^2m-4: is irrelevant on the τ=0  time-slice for minimal models with m≥ 4
 but it 
 is exactly marginal at m=3, i.e. 
  at
 the Ising critical point.
This term, however, 
turns out to come (see Eq. <ref> in App. <ref>) 
with a coefficient (R-1) in the OPE in Eq. <ref>, and hence it 
vanishes
in the limit 
R→1.
Moreover,
we show
in App. <ref>
that in the limit R→1 such a term 
cannot
be generated under the
RG in any order of the
coupling constant Δ of the perturbation Φ.
Since the  exactly  marginal term in Eq. <ref> cannot be produced under the RG in the R→ 1 limit at m=3, the Ising case [Recall from the discussion above that, when m>3, this term is replaced by an irrelevant operator on the time-slice which can be ignored.], 
the RG analysis performed in Sect. <ref>, <ref> and <ref> will also
provide an expansion in large odd  m (small ϵ= 3/(m+1))
for the generalized replica theory 
in Eqs. <ref> – <ref> all the way  down to
m=3, i.e. down to the Ising critical point.

Thus in the replica limit R → 1, the 1-loop RG equation derived in Eq. <ref> also applies to an expansion in ϵ and m= odd,  down to  
the Ising critical point.



To discuss this case in detail, let us consider the 1d quantum Ising model at its 
critical
(not tri-critical)
point described by the Hamiltonian in Eq. <ref>, which lies in the universality class of the m=3 minimal model CFT.
Let us consider performing (weak-) measurements with operator σ̂_i^z at all sites i 
on the ground state of the critical Ising model (Eq. <ref>).
Since (σ̂_i^z)^2=1 and σ̂_i^z at different sites commute with each other, one 
immediately verifies
that the details from sections <ref> and <ref>, 
where the 
O'Brien-Fendley chain at its Ising tricritical point was discussed,
generalize straightforwardly 
to the case of
measurements with σ̂_i^z
on the ground state of the critical quantum Ising model, where the measurement operator 
Ê_i
of the former is now replaced by σ̂_i^z.
In particular, the measurement averaged moments
of correlation functions for this measurement protocol are given by

    [⟨Ô_1⟩_m⃗ ...
      ⟨Ô_N⟩_m⃗]   ∝lim_R→ 1Tr(𝒪̂_1^(1)𝒪̂_2^(2)…𝒪̂_N^(N)×
    × (|0⟩⟨0|)^⊗ R   exp{4Δ̃∑_i=integer∑_a,b=1
    a≠ b^R(σ̂^z_i)^(a)(σ̂^z_i)^(b)}).

Here, the state |0⟩ 
now denotes
the ground state of the 
critical Ising Hamiltonian listed in Eq. <ref>.
In the above equation, in contrast to
the corresponding equation Eq. <ref>
of the tricritical point for the O'Brien-Fendley chain, we have a sum over all sites i and 
operators (σ̂^z_i)^(a).
This is because in 
the present
section we are performing measurements at all sites i with operator σ̂^z_i,
instead of with operator Ê_i 
for even
i.
Just as in the tricritical Ising case, Eqs. <ref>, <ref>,
we have gone over to a formulation using continuous (“softened”) measurement outcomes t_i with a symmetric distribution P(t_i), which we again for now first assume to be a zero-mean Gaussian Eq. <ref> (only second cumulant non-vanishing). 


In continuum language, one 
now sees that 
equation
Eq. <ref> above
reduces to
Eq. <ref>
but where  the action S_* in 
Eq. <ref>
is now
the 
Landau-Ginzburg(-Zamolodchikov) action of the Ising critical point, i.e.

    S_*=∫ d τ∫ d x {1/2(∂_xϕ)^2+1/2(∂_τϕ)^2+
    g^*_2ϕ^4},

as opposed to that of the Ising tricritical point in Eq. <ref>.
Also, in contrast to Eq. <ref>, the perturbation Φ(x) is now
given by

    Φ(x)  =∑_a,b=1 
     a≠ b^R𝔰^(a)(x,0) 𝔰^(b)(x,0),

where the field 𝔰(x,τ) is the the continuum field corresponding to the  lattice operator σ̂^z_i at the Ising critical point with
the scaling dimension 1/8 <cit.>.
As discussed at the start of this section, the field 𝔰 is given by the Kac's table field φ_1,2 at the Ising critical point, and thus can be obtained by taking m→3 limit of the field 𝒮 in Eq. <ref>. 
The special symbol 𝔰(x,τ) for the field 𝒮 in the Ising case m=3 (and only for m=3) is used to stress the additional subtlety arising in this case.
In particular, substituting m=3 in the generalized replica field theory in Eqs. <ref>, <ref>, 
and <ref>, we precisely recover the replica field theory for measurements performed with the operator σ̂^z_i at the Ising critical point.
We note that the role played by the  
average (“weak")
Kramers-Wannier symmetry mentioned in the last paragraph of Sect. <ref> is now played by the 
average (“weak")
Ising Z_2 symmetry. Thus, the perturbation
in Eq. <ref> represents the most RG relevant perturbation invariant under 
the (“average”) Ising Z_2 and replica permutation symmetries. Less relevant interaction terms which can be thought of as being associated with higher cumulants of the distribution P(t_i) are discussed in App. <ref> and are found to be irrelevant at the new fixed point Δ_*. This means that our result will also  be valid for 
weak
measurements where the distribution P(t_i) is a (normalized) sum of delta functions.
Since the replica field theory for measurements performed with the operator σ̂^z_i at the Ising critical point corresponds to the limit m→ 3 of the generalized 
replica theory in 

Eq. <ref> with odd m≥ 3,
we see,  
following the discussion on the analogy between replica field theories in Eqs. <ref> – <ref> for even m and replica field theories in Eqs. <ref> – <ref> for odd m, that the expansion in ϵ=3/(m+1)


    Δ_*=
       ϵ/4+ϵ^2/4+  O(ϵ^3)

(obtained by taking R→1 in Eq. <ref>) provides, 
as m → 3, the location (in coupling constant space) of the
measurement-dominated fixed point which
governs the IR physics 
of measurements with the spin-operator σ̂_i^z performed  at all sites  on the ground state of the critical quantum Ising model [In App. <ref>, we will show that the higher (>2) cumulants of P(t_i) (analogous to the case of tricritical Ising point) are inconsequential to the IR physics of measurement averaged quantities at Ising critical point. 
In particular, the higher cumulants generate terms where an even number (≥ 2) of pairwise unequal replica copies of spin field s(x,τ) interact with each other on the τ=0 time slice.
Out of these the 4-replica and 6-replica terms are relevant, the 8-replica term is marginal, and all the other higher replica terms are irrelevant at the m=3 minimal model CFT, i.e. the Ising critical point.
Moreover, the aforementioned relevant and marginal terms at the Ising critical point (m=3)
are irrelevant at fixed points described by the large-m minimal model CFTs.
Then following the standard reasoning used in the case of the ϕ^6 interaction at the Wilson-Fisher fixed point in d=4-ϵ dimensions, the relevant and marginal terms at the Ising critical point are expected to be irrelevant at the 
new
fixed point Δ_* even at m=3. This is discussed in more detail in App. <ref>.
Also see App. <ref> for a general argument for the irrelevance of higher cumulants (2k≥ 4) based on avoided level crossings.]. 
By reasoning completely parallel to the tricritical case discussed above,
 the measurement-averaged n^th Rényi entanglement entropy S_n,A for an interval of length l of the ground state of the critical Ising chain
 subjected to σ̂^z_i measurements at all sites is given by the ϵ-expansion from Eq. <ref>, 

    S_n,A=c^(eff)_n/3lnl/a+𝒪(1)  where,
       c^(eff)_n= 
    c(m)(1+1/n/2)-3 I_n/(n-1)ϵ+𝒪(ϵ^2)
       I_n=n/2π∫_0^∞ds 1-s^n-1/(1-s)(1+s^n)+
     𝒪 (ϵ ),
 
but now 
in the limit m→ 3 
with ϵ=3/(m+1).
Finally, since the field 𝔰(x,τ) is 
the m→3 limit of the 
field 𝒮 
defined 
in Eq. <ref> 
for general odd-m minimal model CFTs (which in turn forms the analogue of the field ℰ Eq. <ref> defined for even-m minimal model CFTs),
the measurement averaged N^th moment of the correlation function of the σ̂_i^z operator at the Ising critical point will be given by
m→ 3
limit of Eq. <ref>,
 

    X^(σ̂_ Is),R=1_N=1=1/2-ϵ/2+𝒪(ϵ^3)
       X^(σ̂_ Is),R=1_N>1=N/2[1+ϵ-(3N-5)ϵ^2+𝒪(ϵ^3)].



This shows that at the Ising critical point, the scaling dimensions of the measurement averaged moments of the σ̂_i^z correlation function 
exhibit
multifractal scaling
(see Sect. <ref>). 
In particular, 
at the Ising critical point with measurements, the typical
connected
correlation function of the σ̂_i^z operator will be given by the following power law exponent

    X̃^(σ̂_Is), R=1_typ=1/2[1+ϵ+5ϵ^2+𝒪(ϵ^3)]

as m→3, where the definition of
X̃^(σ̂_Is), R=1_typ
is completely analogous to the definition of
X̃^(ℰ), R=1_typ in the
tricritical Ising case. [I.e., X̃^(σ̂,Is), R=1_typ
is obtained from the moments of the subtracted Pauli spin operator describing the deviation from its expectation value in a fixed quantum trajectory, 
δσ̂^z_i
:= σ̂^z_i -
⟨σ̂^z_i⟩, and 
⟨δσ̂^z_iδσ̂^z_j⟩=
⟨σ̂^z_iσ̂^z_j⟩ 
-
⟨σ̂^z_i⟩⟨σ̂^z_j⟩.
]

.9cm


§ CONCLUSIONS AND DISCUSSION

We have demonstrated that performing weak measurements on relatively simple quantum critical ground states can give rise to critical states with highly complex and novel scaling behavior described by novel universality classes. 
We started our study with the critical ground state in the universality class of the
 Ising tricritical
point in the lattice formulation by O'Brien and Fendley, and subjected it to weak measurements with a lattice
operator which corresponds to the continuum energy operator at the Ising tricritical point.
The described weak measurements turn out to be a relevant perturbation at the 
 Ising tricritical point and the critical properties of the states obtained upon measurements are no longer dictated by the 
Ising tricrticial
point itself.
We showed that the critical behavior of the tricritical Ising ground state subjected to the described weak measurements is governed by a new, measurement-dominated fixed point, which occurs at a finite strength of measurements. 
We presented a controlled perturbative RG analysis, i.e. an ϵ expansion, to study the universal critical properties of this measurement-dominated fixed point and
we calculated a variety of universal quantities (described below) in this ϵ expansion.
We found the first manifestation of the novel scaling properties of the measurement-dominated fixed point in the scaling properties of the 
 measurement-averaged
moments of correlation functions. 
In particular, we showed that the measurement averaged Nth moments of both the spin and the energy correlation function at the tricritical Ising point decay with independent power-law exponents for each N.
Thus, there exists an infinite number of independent scaling exponents associated with each correlation function.
 Moreover, noninteger moments N of the correlation functions  
also exhibit
scaling behavior, resulting in a continuous spectrum of scaling exponents for each operator, spin and energy.
Each continuous spectrum of scaling exponents is related to 
a
universal scaling form of the probability distribution of the given correlation function in states obtained upon measurements, and we determined the typical scaling behavior of the spin and the connected energy correlation function.
We also
demonstrated the presence of
logarithmic CFT features 
at
the measurement-dominated fixed point, 
in particular
the presence of logarithmic correlation functions.
We showed that, unlike in usual (unitary) CFTs
where all correlation functions are power law decaying, 
measurement-averaged correlations functions may possess
a multiplicative logarithm of distance on top of a power law decay. Such logarithmic correlation functions are associated with the non-diagonalizability of the dilation operator, and we also identified the `logarithmic pair' of scaling operators that span the 2× 2 Jordan cell of the dilation operator corresponding to the obtained logarithmic correlation function.Another novel feature of the finite measurement strength fixed point was found in the universal coefficients 1/3c_n^(eff) of the logarithm of subsystem size in the measurement averaged nth Rényi entanglement entropies. 
We found that similar to the infinite hierarchy of scaling exponents in the case of moments of the correlation functions, the universal coefficients c_n^(eff) associated with the nth measurement averaged Rényi entanglement entropies are also independent of each other for different n.
This is 
in contrast
to the unmeasured 1d quantum critical ground states 
 (and all unitary CFTs)
where the universal coefficients of the logarithm of subsystem size for all nth Rényi entropies are 
all related
solely to a single number,
the central charge of the corresponding 2D CFT.
We showed that c_n^(eff) also appears in the coefficient of the leading order finite temperature correction to the measurement averaged extensive nth Rényi entropy of the full (thermal) mixed 
Gibbs state
of the system.The problem of performing weak measurements on a 1d quantum critical ground state can be formulated as a field theory problem with a one dimensional defect at the zero-time slice of the corresponding (replicated) (1+1)d CFT.
We showed, generally, that for a given 1d quantum critical ground state, the universal “Affleck-Ludwig" effective boundary entropy associated with this defect (boundary, after folding) appears as a constant, system size independent piece in the Shannon entropy of the measurement record on the ground state. 
In the case of the tricritical Ising ground state subjected to weak measurements with the energy operator, we calculated this universal contribution to the Shannon entropy to leading order in
the ϵ expansion.
We note that the role of the effective boundary entropy in the case of a 1d critical ground state subjected to measurements is analogous to that of the `effective central charge' 
at the measurement-induced transition of a deep quantum circuit, 
where the latter characterizes the finite size 
dependence of
the Shannon entropy of the measurement record on the bulk of the deep quantum circuit at the measurement-induced transition.
Finally, we also studied the ground state of the quantum critical Ising model subjected to weak measurements with the spin operator σ̂_i^z.
By appropriately generalizing the controlled perturbative RG analysis, i.e. the ϵ expansion, developed in the case of the tricritical Ising point, we demonstrated that the critical behavior of the critical Ising ground state subjected to weak measurements with the σ̂_i^z operator is also governed by another measurement-dominated fixed point, which occurs
at a finite strength of measurements.
At the Ising critical point, we determined the power-law exponents of the measurement averaged moments of the σ̂_i^z correlation function to two-loop order in the ϵ expansion and 
found these exponents to be independent of each other.
We also obtained the power law exponent of the typical connected correlation function of the σ̂_i^z at the Ising critical point in the ϵ expansion.
Lastly, we also calculated, to leading order in the ϵ expansion, the universal coefficients 1/3c_n^(eff) of the logarithm of subsystem size in the measurement averaged nth Rényi entanglement entropies at the Ising critical point.
Again, analogous to the case of the tricritical Ising point, the universal coefficients c_n^(eff) at the Ising critical point are also found to be independent of each other for different values of n.



1.5cm


One of us (AWWL) thanks Sam Garratt for an inspiring discussion on Ref. GarrattWeinsteinAltman2022 in Fall 2022, as well as  especially Romain Vasseur and Chao-Ming Jian for collaboration on several previous works in the related area of measurement induced phase transitions.


.8cm



 

§ 
HIGHER 
CUMULANTS
AND A COMMENT ON `NON-LOCAL' FIELDS





 §.§ Higher Cumulants: Ising tricritical point


We begin by discussing the Ising tricritical point.
In Eq. <ref>, we averaged over measurement outcomes by assuming that only the second cumulant of the distribution P(t_i) is non-zero.
In this Appendix, we will provide justification for why the higher even cumulants of P(t_i) [The distribution P(t_i) is taken to be an even function of t_i to satisfy Eq. <ref>.] cannot change the
critical behavior of the system at long distances.
A 
non-vanishing (2n)^th cumulant (Δ̃_2n) of P(t_i) will give rise to the following term

    Δ̃_2n∑_i=even4(∑_a=1^RÊ_i^(a))^2n

in the 
exponential
on the RHS of Eq. <ref>.
The above expression can be 
simplified, using ( Ê_i^(a))^2=1, which shows that it corresponds to
a superposition of terms of the following form (summed over all even i)

    ∑_a_j_1,a_j_2,…,a_j_2k^all indices are
    pairwise distinctÊ_i^(a_j_1)Ê_i^(a_j_2)⋯Ê_i^(a_j_2k),

where k is an integer less than n.
In continuum language, we can replace each Ê^(a_j_l)_i in the above expression with the corresponding continuum field ℰ^(a_j_l)(x,0) in the replica copy “a_j_l".

Going over to
continuum language, 
the above expression 
thus reads

    ∑_a_j_1,a_j_2,…,a_j_2k^all indices are
    pairwise distinct(ℰ^(a_j_1))(ℰ^(a_j_2))⋯(ℰ^(a_j_2k)).


In principle, in each of the 
parentheses
in the above expression we can have a contribution from the 
subleading energy field ℰ” 
which (just like the leading energy field ℰ) is also odd under K-W duality 
and can appear in the continuum representation of the lattice operator 
[compare Eqs. <ref>,<ref>].
However, ℰ” (scaling dimension =3) is highly irrelevant as a field with support on the 1-dimensional time slice τ=0, and we can drop it.
Coming back to Eq. <ref>, since the scaling dimension of the field ℰ
at the Ising tricritical point is 1/5, for all k>2 the term appearing in Eq. <ref> is irrelevant as a field with support on the 1-dimensional time slice τ=0, and again we can ignore it.
The only relevant term coming from the higher cumulants
k≥ 2
(apart from the k=1 term already appearing in Eq. <ref>) is
the k=2 term,

    ∑_a,b,c,d^all indices are
    pairwise distinctℰ^(a)ℰ^(b)ℰ^(c)ℰ^(d),

which arises
from all cumulants higher than or equal to the four.
This term has scaling dimension 4/5 < 1
(while being less relevant than 
Φ(x) in Eq. <ref>.)

The discussion above of scaling dimensions of the operators in
Eq. <ref>
arising from higher cumulants
was referring to the  Ising tricritical point where  the measurement strength is Δ=0. However, since we are in fact
interested in the measurement-dominated fixed point at which Δ_*≠0, which we control within  the ϵ=3/(m+1)-expansion, we are really concerned with the relevance/irrelevance of these operators at the 
Δ_* ≠0 fixed point. Now, all “higher-cumulant” (k≥ 2) operators
in
Eq. <ref>
are highly irrelevant at the 
Δ_* ≠0 fixed point when ϵ=3/(m+1) is small, i.e. when m is large:
At ϵ=0 (where 1/m=0)
they have scaling dimensions =2k × (1/2)>1, i.e. are irrelevant (by integers)
on the one-dimensional time-slice when k≥ 2, and for small ϵ those scaling dimensions only change  by small amounts (of order ϵ, ϵ^2, ... etc.) when going to the finite-Δ_* fixed point of order ϵ. 
More specifically, one can show explicitly <cit.>
that 
the operators  in 
Eq. <ref>
become even more irrelevant at the Δ_* ≠0  fixed point within the 
1-loop
epsilon expansion, as compared to their dimensions 
=2k × (1/2) at the Δ=0 fixed point.
(I.e. the order ϵ 
shifts  
of
their scaling dimensions away from their already highly irrelevant ϵ=0 values are all positive.)
This is a familiar feature of the epsilon expansion, well  known
already from that of ϕ^4 Landau-Ginzburg theory in d=4-ϵ dimensions
where, although the ϕ^6 perturbation is relevant at the Gaussian fixed point when d<3 (while being irrelevant when d>3), it is
irrelevant at the Wilson-Fisher fixed point of physical interest for  all dimensions d ≥ 2.
Analogously, in the epsilon expansion from
Sect. <ref> of interest in this paper, while
the  operator in Eq. <ref> associated with the fourth cumulant k=2 is, at the unperturbed fixed point Δ=0 (analogous to the Gaussian fixed
point in ϕ^4 Landau-Ginzburg theory),
relevant  when m=4 and irrelevant for all even values  m >4,
it is analogously expected to be irrelevant at the  Δ_*≠ 0 fixed point of interest, 
Eq. <ref>, for all even values 
m ≥ 4, i.e. including m → 4. (A general argument for the irrelevance of all higher cumulants (k≥ 2) based on avoided level crossings is presented in 
App. <ref>.)

Hence, we do not expect the higher 
cumulants
of  the distribution P(t_i) to change the long distance behavior of the system. This implies in particular that in  the case weak measurements with discrete measurement outcomes 
(Eqs. <ref>, <ref>, <ref>), where P(t_i) is a (normalized) sum of delta functions [compare the discussion below Eq. <ref>] and thus contains even cumulants higher than the second, the same critical behavior results as in the case of a zero-mean Gaussian distribution P(t_i).





 §.§ Higher  Cumulants: Ising Critical Point






Following the above discussion for the tricritical Ising case, we will now  
provide a justification for why the higher even cumulants of P(t_i) are not expected to change the critical long-wavelength properties
in the case of measurements with the σ̂^z_i 
operator on the ground state of the critical quantum Ising model.
Analogous to Eq. <ref>, the higher cumulants for the σ̂^z_i measurements will give rise to terms of form

    ∑_a_j_1,a_j_2,…,a_j_2k^all indices are
    pairwise distinct(σ̂^z_i)^(a_j_1)(σ̂^z_i)^(a_j_2)⋯(σ̂^z_i)^(a_j_2k),

which will appear in the 
exponentional 
in Eq. <ref>.
In continuum language, we can replace each (σ̂^z_i)^(a_j_l)  operator
in the above equation by the continuum field 𝔰^(a_j_l)(x,τ).  Since
the scaling dimension of 𝔰^(a_j_l)(x,τ) is 1/8, all the terms with k>4 appearing in the above equation are irrelevant at the
(unmeasured)
Ising critical point.
Thus, at the Ising critical point, the
4-replica and 6-replica terms (corresponding to k=2 and k=3 in 
Eq. <ref>) are relevant, 
while the 8-replica term (corresponding to k=4) is marginal
(while these terms are all less relevant than the perturbation Φ(x) in 
Eq. <ref>
corresponding to k=1).

In analogy with the tricritical Ising case in the preceding subsection, the discussion above of the scaling dimensions of the operators in Eq. <ref> arising from higher cumulants
was referring to the Ising critical point where the measurement strength is Δ=0. However, again, as we are interested in the measurement-dominated fixed point at which Δ_* ≠ 0, which we control within the ϵ = 3/(m+1)-expansion [where now m= odd], we are really interested in the relevance/irrelevance
of these operators at this 
new
fixed point. Again, all “higher-cumulant” operators (k≥ 2) in Eq. <ref> are highly irrelevant at the
Δ_* ≠0 fixed point when ϵ = 3/(m+1) is small [For odd m>3 minimal models, we denote the generalization of the field 𝔰 by the symbol 𝒮 defined in Eq. <ref>], i.e. when m= odd is large: At ϵ=0 (1/m=0) they have again  scaling dimensions =2k × (1/2)>1, i.e. are again irrelevant (by integers) on the one-dimensional
time-slice when k ≥ 2. And again, for small ϵ those scaling dimensions
only change by small amounts when going  to the finite-Δ_* fixed point of order ϵ. Again,  specifically, within the 
1-loop
epsilon expansion  <cit.> these operators become more irrelevant as compared to their already irrelevant scaling dimensions
=2 k × (1/2) at Δ=0. Again, in analogy with the discussion of the ϕ^6 term in the d=4-ϵ expansion of ϕ^4 Landau-Ginzburg theory, the operators in Eq. <ref> with k=2, 3, 4 are expected to be irrelevant at the Δ_* ≠0 fixed point of interest for all odd values of m ≥ 3, i.e. including m=3. 
(For a general argument for the irrelevance of all higher cumulants (k≥ 2) based on avoided level crossings  we refer again to  
App. <ref>.)

Hence, again,
we do not expect the higher 
cumulants
of  the distribution P(t_i) to change the long distance behavior of the system. 
This  implies again in particular  that in  the case of  weak measurements
with discrete measurement outcomes 
(Eqs. <ref>, <ref>, <ref>),
where P(t_i) is a (normalized) sum of delta functions (compare discussion below Eq. <ref>) and thus contains even cumulants higher than the second, the same critical behavior results as in the case of a zero-mean Gaussian distribution P(t_i). 







 §.§ Locality of Observables



We will close this section by discussing the significance 
of `locality' of
fields
𝒪_i
associated with lattice operators 𝒪̂_i and
which appear
in Eq. <ref>.
Note that in deriving 
Eq. <ref>,
we assumed that the operators 𝒪̂_i in their continuum representation correspond to `local' fields 𝒪_i, i.e. 𝒪_i can be expressed in terms of local combinations
involving
the Landau-Ginzburg field ϕ(x,τ) and its normal ordered higher powers.
Important differences in the gluing of field configurations {ϕ^(a)(x,0^-)}_j=1^R and {ϕ^(a)(x,0^+)}_j=1^R (appearing in Fig. <ref> and Eq. <ref>) could occur if the fields 𝒪_i are non-local.
An example of this is seen in the case of measurements performed on the Tomonaga-Luttinger liquids (TLLs), studied in Ref. <cit.>, when calculating the correlations functions of phase e^iθ(x). The phase θ(x) is termed as a `non-local' field in the bosonic theory of the field ϕ(x) [∂_xϕ(x) is proportional to the density of the TLLs] as they satisfy the following equal time commutator

    [ϕ(x),θ(x')]=iπ H(x-x')= 
          iπ    x≥ x' 
    
        0      x<x'
       .

In the calculation of phase correlation functions, as noted in Ref. <cit.>, the two field configurations  {ϕ^(a)(x,0^-)}_j=1^R and {ϕ^(a)(x,0^+)}_j=1^Rdiffer with each other on an interval of values of position x, and are not `identified/glued' on this interval.
In this work, we have only considered correlation functions (and their moments) of lattice operators 𝒪̂_i which correspond to `local' fields in continuum, i.e. they can be expressed as local combinations of Landau-Ginzburg field ϕ and normal ordered higher powers of ϕ.






§ IRREDUCIBLE REPRESENTATIONS OF THE SYMMETRY GROUP

Unlike the unperturbed critical theory (Δ=0), the operators ∏_i=1^Nℰ^(α_i)(x,0)
 (1≤α_i≤ R 
 in a theory
 with R replica)
 are no longer scaling operators at the new fixed point (Δ=Δ_*). 
 Rather, as discussed in Ref. <cit.> and <cit.>, the scaling operators at the new fixed point are formed out of linear superposition of operators ∏_i=1^Nℰ^(α_i)(x,0) for different choices of replica indices {α_i}, and they transform in irreducible representations of the symmetric group S_R. Following Ref. <cit.>, the 
 corresponding 
 scaling operators at the new fixed point are given by 
 
    𝔈_NMR=
    
    ∑_α_i≠α_j
    1≤α_i≤
    R-M(ℰ^(α_1)-ℰ^(R))…(ℰ^(α_M)-ℰ^(R-M+1))ℰ^(α_M+1)…ℰ^(α_N)
    
    (0≤ M≤ N).

The scaling dimensions of the above operators are 
calculated to two-loop order in Ref. <cit.> in a dimensional regularization [by ϵ=3/(m+1)] RG scheme, with minimal subtraction of poles in ϵ. From their analysis, the scaling dimension of the operator in Eq. <ref> is given by


    X^(ℰ),R_NM=NX_ℰ-γ(Δ_*) … (X_ℰ=1/2-3/2(m+1))
       Δ^*=ϵ/4(2-R)+ϵ^2/4(2-R)^2+  O(ϵ^3)  … (ϵ=3/m+1)
       γ(Δ)=2b̃_NMRΔ-8(N(R-N)+(N-1)b̃_NMR)Δ^2
        + O(Δ^3)
       b̃_NMR=2((N -M)R-N^2 +M(M -1))

In table <ref>, 
we list the scaling dimensions X^(ℰ),R_NM for R=0,1 and
N=1,2,3.
Out of the N scaling dimensions, corresponding to different values of M in Eq. <ref>, the smallest one dictates the 
power law behavior
of ⟨ℰ(x,0)ℰ(y,0)⟩^N when |x-y|→∞. 

 Setting R=1 and minimizing the scaling dimension X^(ℰ),R_NM in Eq. <ref> over possible values of M 
 ,
we obtain Eqs. <ref> and <ref>. 


As an aside, we note that in Ref. <cit.> they were in interested in the limit R→0 (which corresponds to quenched disorder), and in this limit the smallest scaling dimension for a fixed N in Eq. <ref> is given by,

    X^(ℰ),R=0_N=N/2(1-ϵ^2/4(3N-4)+𝒪(ϵ^2)).

0.6cm


 §.§ Colliding Scaling Dimensions in Replica Limit Lg

As discussed in Sect. <ref>, scaling dimensions of operators with unequal scaling dimensions at generic replica number R≠1 can become equal to each other at R=1.
To see this collision of scaling dimensions in replica limit R→1, we consider two operators 𝔈_20R and 𝔈_22R from Eq. <ref>.
We note that the correlation function of the 𝔈_20R operator is given by,

    ⟨𝔈_20R(r,0)𝔈_20R(0,0)⟩ = 2R(R-1)×
       (⟨ℰ^1(r)ℰ^1(0)ℰ^2(r)ℰ^2(0)⟩+
                         +(R-2)(R-3)/2⟨ℰ^1(r)ℰ^2(r)ℰ^3(0)ℰ^4(0)⟩
                                      +2(R-2)⟨ℰ^1(r)ℰ^1(0)ℰ^2(r)ℰ^3(0)⟩),

while for 𝔈_22R operator,

    ⟨𝔈_22R(r,0)𝔈_22R(0,0)⟩ = (R-3)(R-2)^2(R-1)×
       (⟨ℰ^1(r)ℰ^1(0)ℰ^2(r)ℰ^2(0)⟩+ ⟨ℰ^1(r)ℰ^2(r)ℰ^3(0)ℰ^4(0)⟩
                                                    -2⟨ℰ^1(r)ℰ^1(0)ℰ^2(r)ℰ^3(0)⟩).

Ignoring the overall R dependent constants, clearly, the expressions in parentheses in Eqs. <ref> and <ref> are identical to each other in R→1 limit.
Thus, the two operators 𝔈_20R and 𝔈_22R have colliding scaling dimensions in the replica limit R→1,
i.e. the scaling dimensions of the two operators are equal to each other in the limit R→1 at the new fixed point Δ_*(ϵ) (ϵ=3/(m+1)) for all even values of m. 
(This can also be verified using the ϵ-expansion
for the scaling dimensions of the two operators using Eq. <ref>.)
Moreover, with the given normalization for operator 𝒪 (Eq. <ref>) and operator 𝒪̃ (Eq. <ref>), it can be easily verified that the criterion in  Eq. <ref> is satisfied by the amplitudes of correlators  ⟨𝒪(r) 𝒪(0)⟩ and ⟨𝒪̃(r) 𝒪̃(0)⟩.
Finally, since the correlation functions in the parentheses of Eqs. <ref> and <ref> are physical correlators, we expect to get a finite answer for them in the R→1 
limit, and thus operators 𝒪 and 𝒪̃ also satisfy the criterion in Eq. <ref>.
As discussed in Sect. <ref>, such colliding of scaling dimensions give rise to logarithmic correlation functions at the new fixed point.
.8cm


§ DETAILS OF ENTANGLEMENT ENTROPY CALCULATION

Given a set of measurement outcomes 
m⃗=
{m_j}, the state obtained after measurements 
is

    |Ψ_{m_j}⟩=K̂_𝐦⃗|0⟩/√(⟨0|(K̂_𝐦⃗)^†K̂_𝐦⃗|0⟩).

The n^th Rényi entanglement entropy of a spatial region A=[u,v] in this state is given by

    S_n,A(|Ψ_{m_j}⟩)=1/1-nln{Tr_A [ρ_A (|Ψ_{m_j}⟩)]
    ^n},

where the reduced density matrix is

    ρ_A(|Ψ_{m_j}⟩)=   Tr_A̅(|Ψ_{m_j}⟩⟨Ψ_{m_j}|)
    =   Tr_A̅(K_𝐦⃗|0⟩⟨0|(K_𝐦⃗)^†)/Tr(K_𝐦⃗|0⟩⟨0|(K_𝐦⃗)^†)

and A̅ is complement of spatial region A. Then


    S_n,A(|Ψ_{m_j}⟩)=      1/1-n{ln(Tr_A(Tr_A̅(K_𝐦⃗|0⟩⟨0|(K_𝐦⃗)^†))^n)
        -ln(Tr(K_𝐦⃗|0⟩⟨0|(K_𝐦⃗)^†))^n}
    
        =      1/1-n{lnTr(𝒮_n,A(K_𝐦⃗|0⟩⟨0|(K_𝐦⃗)^†)^⊗ n)
        -lnTr((K_𝐦⃗|0⟩⟨0|(K_𝐦⃗)^†)^⊗ n)}
    
        =      1/1-nlim_k→ 01/k×{Tr(𝒮^k_n,A(K_𝐦⃗|0⟩⟨0|(K_𝐦⃗)^†)^⊗ nk)-Tr((K_𝐦⃗|0⟩⟨0|(K_𝐦⃗)^†)^⊗ nk)}.

Here the permutation operator 𝒮_n,A 
is defined <cit.> as

    𝒮_n,A=xΠ1.2χ_g_xand  g_x=
       (1,2,…,n )    x ∈ A
    identity=e     x ∈A̅
    ,

where g_x labels the permutation on site x, and 1.2χ_g_x =∑_[i]|i_g_x(1)i_g_x(2)… i_g_x(n)⟩⟨i_1i_2… i_n| is its representation on the replicated on-site Hilbert space. 
Since the operator (K_𝐦⃗)^⊗ nk=K_𝐦⃗⊗ K_𝐦⃗⋯⊗ K_𝐦⃗
commutes with the permutation operator 𝒮_n,A, 
using cyclicity of trace we can write Eq. <ref> as

    S_n,A(|Ψ_{m_j}⟩)=1/1-nlim_k→ 01/k×{Tr(𝒮^k_n,A(|0⟩⟨0|)^⊗ nk(K̂_m⃗^†K̂_m⃗)^⊗ nk)-Tr((|0⟩⟨0|)^⊗ nk(K̂_m⃗^†K̂_m⃗)^⊗ nk)}


Then the average of the n^th Rényi entropy over the measurement outcomes with Born rule is given by,

    S_n,A=   ∑_𝐦⃗p_0(𝐦⃗)S_n,A(|Ψ_{m_j}⟩)
    =   lim_k→ 01/(1-n)k∑_𝐦⃗p_0(𝐦⃗){Tr(𝒮^k_n,A(|0⟩⟨0|)^⊗ nk(K̂_m⃗^†K̂_m⃗)^⊗ nk)-Tr((|0⟩⟨0|)^⊗ nk(K̂_m⃗^†K̂_m⃗)^⊗ nk)}.

Since p_0(𝐦⃗)=Tr(K_𝐦⃗|0⟩⟨0|(K_𝐦⃗)^†)=Tr(|0⟩⟨0|K_𝐦⃗^†K_𝐦⃗)
we obtain

S_n,A= 1/1-nlim_k→ 01 k [ 𝒵_A -𝒵_∅ ],
where


    𝒵_𝒜=∑_𝐦⃗Tr(𝒮^k_n,A(|0⟩⟨0|)^⊗ nk+1(K̂_m⃗^†K̂_m⃗)^⊗ nk+1)  and  𝒵_∅=∑_𝐦⃗Tr((|0⟩⟨0|)^⊗ nk+1(K̂_m⃗^†K̂_m⃗)^⊗ nk+1).

Owing to the POVM condition
Eq. <ref>,
lim_k → 0𝒵_∅=1,
we can write the measurement averaged n^th Rényi entropy as

    S_n,A=1/1-nlim_k→ 01/k(𝒵_A/𝒵_∅-1)=1/1-n(d/dk|_k=0𝒵_A/𝒵_∅).

Using Eq. <ref>, 
<ref>,
<ref> and following the arguments in 
the derivation of Eq. <ref>, 𝒵_A can be written as


    𝒵_𝒜=∑_𝐦⃗Tr(𝒮^k_n,A(|0⟩⟨0|)^⊗ nk+1(K̂_m⃗^†K̂_m⃗)^⊗ nk+1)∝
          ∝∫∏_a=1^nk+1Dϕ^(a) e^-∑_a=1^nk+1S_*^(a)+Δ∫ dx Φ(x) Tr(𝒮^k_n,A|{ϕ^(a)(x,0^+)}⟩⟨{ϕ^(a)(x,0^-)}|)

where,

    Φ(x)=∑_a,b=1 
     a≠ b^Rℰ^(a)(x,0) ℰ^(b)(x,0)  and |{ϕ^(a)(x,0^±)}⟩=⊗_a=1^R|{ϕ^(a)(x,0^±)}⟩


Upon making use of the definition Eq. <ref>,
the factor Tr (𝒮^k_n,A|{ϕ^(a)(x,0^+)}⟩⟨{ϕ^(a)(x,0^-)}|) in Eq. <ref> does the job gluing the nk+1 replicas into k n-sheeted Riemann surfaces, as illustrated in Fig. <ref>:
Each of these k n-sheeted Riemann surfaces contain n replicas which are glued in the spatial region A along the τ=0 equal-time slice, and there is one additional replica representing a plane that remains unglued.
Thus we conclude that the  
ratio 𝒵_A/𝒵_∅ of partition functions equals the correlation function of two 
twist fields describing these  k  n-sheeted Riemann surfaces. Thus we can express the measurement-averaged Rényi entropies from
Eq. <ref> in terms of the twist fields as

    S_n,A=1/1-nd/dk|_k=0⟨∏_j=1^k(Τ^(j)_n(u,0)(Τ^(j)_n)^-1(v,0))⟩_Δ_*,

where
the superscript j on the twist fields 
denotes
the
Riemann surface (out of k) to which the twist field corresponds, and the subscript n 
indicates 
that we are dealing with twist fields for a n-sheeted Riemann surface. 
(Τ^(j)_n)^-1 denotes the twist field conjugate (“inverse”) to 
Τ^(j)_n.
.8cm



§ OPE COEFFICIENT OF TWO TWIST FIELDS INTO  LG




In order to compute the scaling dimension of the twist field at the new fixed point to 1-loop order in the small parameter ϵ= 3/(m+1),
we will need the OPE  coefficient 
with which the perturbation Φ(x) 
(from Eqs. <ref>, 
<ref>) appears
in the OPE of the twist fields.
This is equivalent to finding the following three point
correlation function
in the unperturbed replica theory (with action in Eq. <ref>)

    ⟨∏_j^k(Τ^(j)_n(u,0)(Τ^(j)_n)^-1(v,0))Φ(x)⟩=
    ∑_a,b=1 
     a≠ b^nk+1⟨∏_j^k(Τ^(j)_n(u,0)(Τ^(j)_n)^-1(v,0))ℰ^(a)(x,0)ℰ^(b)(x,0)⟩.

We know from Ref. <cit.> that

    ⟨ℰ^(a)(x,0)ℰ^(b)(x,0) ⟩_(ℛ_n)^k+1=
    ⟨∏_j=1^k(Τ^(j)_n(u,0)(Τ^(j)_n)^-1(v,0))ℰ^(a)(x,o)ℰ^(b)(x,0)⟩/⟨∏_j^k(Τ^(j)_n(u,0)(Τ^(j)_n)^-1(v,0))⟩.

The 
LHS
of the above equation calculates the correlator for the two 
specified 
fields
ℰ^(a) and ℰ^(b)
in the geometry shown in Fig. <ref>, which involves k
copies of a n-sheeted Riemann 
surface
and 
one  plane, denoted by the subscript (ℛ_n)^k+1 on the correlator.
On the LHS of the above equation, the index a is to be thought of as equal to a combined index (i,α). 
The index i here indicates either a Riemann surface out of the k 
copies of the n-sheeted Riemann 
surface (when i∈{1, ..., k})
or it indicates the plane 
(when i=k+1). When the index i corresponds to a Riemann surface, the index α denotes the Riemann sheet of that n-sheeted Riemann surface on which the field is located [When the index i corresponds to the plane, there is no ambiguity of the Riemann sheet 
to
which the field belongs and α can be taken to be zero.].
On the other hand, the correlators on the 
RHS
of the above equation are evaluated in the  nk+1 independent replicas of the m^th minimal model and the 
labels a and b indicate
the replica copy of the theory.Now to evaluate
the correlator ⟨ℰ^(a)(x,0)ℰ^(b)(x,0) ⟩_(ℛ_n)^k+1, we can use a conformal transformation to map each
of the k copies of the n-sheeted Riemann surface
to 
a plane. 
In particular, we note that since ℰ is a 
(Virasoro) primary field, its 
expectation value on the plane vanishes [
All primary fields of a CFT  are by convention subtracted so that their expectation values in the infinite plane vanish identically.].
So ⟨ℰ^(a)(x,0)ℰ^(b)(x,0) ⟩_(ℛ_n)^k+1 is zero unless both ℰ^(a)(x,0) and ℰ^(b)(x,0) lie on the same n-sheeted Riemann 
surface, i.e. a=(j,α) and b=(j,β) for the same Riemann surface j [Here, the symbol β should not be confused with the inverse-temperature.]
. 
The correlator ⟨ℰ^(α)(x,0)ℰ^(β)(x,0) ⟩_ℛ_n 
for a single n-sheeted Riemann surface ℛ_n
can be calculated using the following conformal transformation 

    z=f(w)=(w-u/w-v)^1/n.

In particular, if w corresponds to a point (x,0) 
on
the α^th sheet in the n-sheeted Riemann surface,

    z=f(α,(x,0)_(x,0) on  the
    α^th sheet)=(x-u/x-v)^1/ne^2πα/niwith α∈{1,2,…,n}.

Since ℰ is a 
(Virasoro) primary
field,

    ⟨ℰ^(α)(x,0)ℰ^(β)(x,0) ⟩_ℛ_n=
    |dw_1/dz_1|^-X_ℰ|dw_2/dz_2|^-X_ℰ⟨ℰ^(α)(z_1)ℰ^(β)(z_2) ⟩_plane,

where w_1 
denotes position
(x,0) in the α^th sheet and w_2  
denotes position
(x,0) in the β^th sheet. Also,

    dw_1/dz_1=n(x-v)(x-u)/u-v(x-v/x-u)^1/ne^-2πα/ni,
    dw_2/dz_2=n(x-v)(x-u)/u-v(x-v/x-u)^1/ne^-2πβ/ni,
    and |z_1-z_2|=2|(x-u/x-v)^1/nsin(π(α-β)/n)|.

Then using Eq. <ref>
as well as ⟨ℰ^(α)(z_1)ℰ^(β)(z_2) ⟩_plane= 1/  ( |z_1-z_2|^2X_ℰ), we obtain

    ⟨ℰ^(α)(x,0)   ℰ^(β) (x,0)⟩_ℛ_n=
       = 
        |(u-v)/2n(x-v)(x-u)sin(π(α-β)/n)|^2X_ℰ
       =⟨ℰ^(a)(x,0)ℰ^(b)(x,0)⟩_(ℛ_n)^k+1,

where in the last equality we recall that a=(j,α) and b=(j,β).
The last equality in the above equation follows
because, as already mentioned above, we are interested in the case when both the fields ℰ^(a) and ℰ^(b) lie on the same Riemann surface ℛ_n, since otherwise the correlator is zero.
Finally, from Eq. <ref>, <ref> and 

⟨𝒯_n(u,0)(𝒯_n)^-1(v,0)⟩=
1/ |u-v|^2d_n,
we obtain
for the desired three point function

    ⟨∏_j^k(Τ^(j)_n(u,0)   (Τ^(j)_n)^-1(v,0))Φ(x)⟩=
       = C_n,k/|u-v|^2kd_n-2X_ℰ|x-u|^2X_ℰ|x-v|^2X_ℰ,
    
    with  C_n,k=k   ∑_α,β=1 
    α≠β^n1/|2nsin(π(α-β)/n)|^2X_ℰ.

Thus, 
 the required OPE coefficient is

    C_n,k=k∑_α,β=1 
    α≠β^n1/|2nsin(π(α-β)/n)|^2X_ℰ=k/2∑_α=1^n-11/(sin(πα/n))^2X_ℰ.

Moreover, as 
2X_ℰ=
1-ϵ,
with ϵ = 3/(m+1), the above OPE coefficient can be expanded in powers of ϵ as

    C_n,k=k/2∑_α=1^n-11/sin(πα/n)+𝒪(ϵ).

To obtain the von Neumann entanglement entropy, we also want to be able to
analytically continue
the n-dependence in the above expression to n → 1.
Thus, we want an expression for the above OPE coefficient 
which is an analytic function of n at n=0, and
which is valid for all real numbers n and which reduces to Eq. <ref> when n is a natural number (≥ 2).
Following Ref. <cit.>, we can write 
1/sin(π x)
in the form
(see also Ref. <cit.>)

    1/sin(π x)=1/π∫_0^∞dt t^x-1/1+tx∈(0,1)   implying
    1/sin(πα/n )=1/π∫_0^∞dt t^α-n/n/1+t=n/π∫_0^∞ds s^α-1/1+s^n.

From Eq. <ref>, the OPE coefficient then can be written as C_n,k=kI_n, where I_n is defined as

    I_ndef=n/2π∫_0^∞ds 1-s^n-1/(1-s)(1+s^n)+
        𝒪(ϵ).

We make use of Eq. <ref> in Sect.
<ref>
 below Eq. <ref>.
.8cm


§ HIGHER LOOP ORDERS IN THE  RG AND THE ISING CRITICAL POINT

We noted in Section <ref> that when m=3, i.e. in the case of 
the
Ising critical point, there is an additional subtlety associated with the RG analysis of the
replica action in Eq. <ref> – <ref>. 
This is due to the 
term in Eq. <ref>, which appears in 
the
OPE
in Eq. <ref> 
for a generic number R of replicas,
and although it is irrelevant for minimal models m≥4, it becomes marginal at m=3. 
We noted in Section <ref> 
that, as shown in Eq. <ref> below,
the coefficient of
this
marginal term in Eq. <ref> comes with a factor of (R-1) in the OPE in Eq. <ref>. 
Thus, in the
 R→1 limit, this term 
is not generated by the RG to second  (1-loop) order  in the coupling constant Δ of the perturbation Φ.
In this appendix, we will show that the term in Eq. <ref> cannot be generated
by the RG
to any (higher-loop) order in
the coupling Δ in the replica limit R → 1, relevant for Born-rule measurements.
We will  provide two different arguments, (i) and (ii).

(i): In the first argument we use the fact that 
all terms that could possibly
be generated  under the RG at arbitrary order in the coupling
Δ
can be obtained by analyzing the multiple OPE 

    Φ(x)×Φ(x)×⋯×Φ(x)_n  #  of  Φ(x)

where Φ from Eq. <ref> is the perturbation given by,

    Φ=∑_a,b=1 
     a≠ b^R𝔰^(a)𝔰^(b).

We use the well-known fact (see e.g. Refs. JLCardy_1986RGOPE, LUDWIG198797, cardy_1996, LUDWIGWIESE) which states that the only operators that can be generated under the RG to any order are the operators that appear in the multiple OPE of the perturbation in Eq. <ref>, and 
in multiple OPEs of the operators that appear 
in Eq. <ref>. –  A brief review of this can be 
found,  if desired, in App. <ref>. –
By analyzing these OPEs, we will show below that the marginal operator, Eq. <ref>, does not appear in any of these multiple OPEs in the limit R → 1, and thus cannot be generated in this limit in any order.

We begin by discussing
the possible operators 
that can occur in a multiple OPE of the operator Φ in Eq. <ref>.
The CFT describing the Ising critical point has three primary fields: I, 𝔰 and 𝔢
(identity, spin, and energy). The OPEs between these primary fields are 
given [
 In the above OPEs, we have 
 not written 
 the explicit coefficients 
which accompany
 the fields in the OPE and which depend on position of the fields. 
(For Ising critical point, this is same as the fusion rules of the theory.).] 
by,

    𝔰×𝔰=I+𝔢,
          𝔰×𝔢=𝔰,
          𝔢×𝔢=I,
    
        I×𝔰=𝔰,         I×𝔢=𝔢,    I× I=I

For the purpose of the following discussion, we only care 
about whether or not
a field appears in the OPE of 
two  given
fields, and its exact coefficient  is immaterial. 

Let us now first  consider the OPE in Eq. <ref> for n=2. Using the above OPE relations in each replica copy, we obtain

    Φ×Φ=4(R-2)Φ+4a_1(R-1)∑_a=1^R𝔢^(a)+
          +a_2
    ∑_a≠ b, a≠ c,
    b≠ c𝔢^(a)𝔰^(b)𝔰^(c)+a_3
    ∑_a,b,c,d^all indices are
    pairwise distinct𝔰^(a)𝔰^(b)𝔰^(c)𝔰^(d)

where a_1, a_2 and a_3 are R independent numerical constants. 
We see that the marginal term ∑_a=1^R𝔢^(a) comes with a prefactor (R-1), which vanishes in the
R→1 limit.
For n=3, the OPE in Eq. <ref> can be obtained by contracting 
the RHSs of Eqs. <ref> and <ref>. 
In particular, Φ on 
the
RHS of Eq. <ref> can be contracted with Φ in Eq. <ref>, and the 
term ∑_a=1^R𝔢^(a) will be again be produced with a (R-1) prefactor. Moreover, the term 
∑_a≠ b, a≠ c,
b≠ c𝔢^(a)𝔰^(b)𝔰^(c)
in Eq. <ref> can contract with Φ=∑_a≠ b𝔰^(a)𝔰^(b) to give,

    (∑_a≠ b, a≠ c,
    b≠ c𝔢^(a)𝔰^(b)𝔰^(c))×Φ=
        R-12∑_a𝔢^(a)+other terms

where 
again 
the marginal term ∑_a𝔢^(a) comes with a factor which vanishes in the limit 

R→1 of interest. Finally, we note that when contracted with Φ
the last term in Eq. <ref> cannot produce the marginal term [Note that the term 𝔰^(a_1)𝔰^(a_2)𝔰^(a_3)𝔰^(a_4) when contracted with 𝔰^(b_1)𝔰^(b_2) gives terms of the following form: 𝔰^(c_1)𝔰^(c_2), 𝔰^(c_1)𝔰^(c_2)𝔢^(c_3) and 𝔰^(c_1)𝔰^(c_2)𝔢^(c_3)𝔢^(c_4). Thus, when contracted with Φ, the last term on the RHS of the OPE in Eq. <ref> cannot produce the marginal operator ∑_a 𝔢^(a)].
Thus, one can conclude that as R→1, the marginal term ∑_a𝔢^(a) does also  not occur in the OPE in Eq. <ref> for n=3.We now present an induction argument for the absence of the 
operator
∑_a𝔢^(a) in the OPE of Eq. <ref> for
any 
n
number of operators Φ, and for the set of operators appearing in this OPE.
To this end, let
us assume that the 
operator
∑_a𝔢^(a) 
does not appear
in the R→1 limit in the OPE 
of 
n operators Φ.
Let us also assume that the
most general replica term that can occur in the OPE of
n
operators
Φ is

    ∑_a_1,a_2,…,a_2k,
    b_1,b_2,…,b_k^all indices are
    pairwise distinct𝔰^(a_1)𝔰^(a_2)⋯      𝔰^(a_2k)𝔢^(b_1)𝔢^(b_2)⋯𝔢^(b_l), 
          where 2k+l≤ 2n.

For
n=2 this corresponds to the terms appearing in 
Eq. <ref>, and this is our first step in the induction.
Now to 
obtain
the OPE of
(n+1) operators  Φ in Eq. <ref>, we have to contract all the terms 
that appear
in 
Eq. <ref> above
with Φ=∑_a≠ b𝔰^(a)𝔰^(b).
Since only the terms in the same replica copy can be contracted with each other,
out of all the sub-terms shown in Eq. <ref> which appear in OPEs of 
n operators Φ, 
only the following terms can be contracted with another Φ to get the marginal ∑_a𝔢^(a) term:

    ∑_a_i≠ a_j𝔰^(a_i)𝔰^(a_j)=Φ,
    ∑_a_i≠ a_j, a_i≠ b_k,
    a_j≠ b_k𝔰^(a_i)𝔰^(a_j)𝔢^(b_k).

From Eq. <ref> and Eq. <ref> we see that both of these
sub-terms
when contracted with Φ produce the marginal ∑_a𝔢^(a), and 
that the corresponding coefficient vanishes in the
R→1 limit in both 
cases.
Moreover, all terms that can appear in the OPE of Eq. <ref> with the perturbation Φ are again of the form of Eq. <ref> with n
replaced by n+1. This completes the induction argument.
Thus, in summary, we have proven so far that 

(a) the only operators that can appear in the multiple OPE of
n 
operators Φ are the operators of the form appearing in Eq. <ref>, and 

(b) of those the marginal operator, having k=0 and l=1, appears with a combinatorical coefficient that vanishes in the limit R→ 1.

Finally, since the operators appearing in Eq. <ref> can all be generated in the OPE in Eq. <ref>, and thus could be generated by the RG
(with combinatorical coefficients that we have not determined), we would 
have finished demonstrating that the marginal operator cannot be generated in the RG to any order, if we could show that the marginal operator cannot appear in the limit R→ 1 in the OPE of an arbitrary number of operators of 
the type listed in Eq. <ref>.  We will
now show that this is indeed the case. 

First, we observe that it is sufficient to show that this is the case for only two
such operators, because by definition the operators appearing in Eq. <ref> form a closed set of 
operators under the OPE [I.e., they form a closed Operator Algebra.]. 
Namely, when we consider an arbitrary number of successive OPEs of operators of the form of
Eq. <ref>, the marginal operator would not be generated  in this multiple OPE
if it was not generated in any of the individual successive OPEs in this limit (which involves only two operators).
On the other hand, we can show as follows that in
the OPE of two operators from Eq. <ref>
the marginal operator can only appear with a combinatorical coefficient that vanishes in the limit R → 1:

Note that only the terms in the same replica copy can contract under
the
OPE. Moreover, the marginal operator 𝔢 is only produced either when two 𝔰 fields are fused (see Eq. <ref>) or when the 𝔢 field fuses with the identity (see Eq. <ref>). Therefore, the general term which appears in Eq. <ref> can produce the marginal operator ∑_a 𝔢^(a) 
only (i) when it contracts with 
itself (i.e., both operators have the same values of k and l), i.e. with

    ∑_a_1,a_2,…,a_2k,
    b_1,b_2,…,b_k^all indices are
    pairwise distinct𝔰^(a_1)𝔰^(a_2)⋯      𝔰^(a_2k)𝔢^(b_1)𝔢^(b_2)⋯𝔢^(b_l),

or (ii)  when it contracts with another operator of the form in
Eq. <ref> with the same value of k but with l replaced by l+1, namely with

    ∑_a_1,a_2,…,a_2k,
    b_1,b_2,…,b_k^all indices are
    pairwise distinct𝔰^(a_1)𝔰^(a_2)⋯      𝔰^(a_2k)𝔢^(b_1)𝔢^(b_2)⋯𝔢^(b_l)𝔢^(b_l+1).

Let us first consider the OPE of the term in Eq. <ref> itself, i.e. with the term in Eq. <ref>.
Since we are interested in the coefficient of the marginal operator ∑_a𝔢^(a), we can consider two 
identical 𝔰^(a) 
fields, one in each of the two identical operators from 
Eq. <ref>
we are considering the OPE of, 
and contract 
these
two fields to produce the field 𝔢^(a), while the rest of the fields
in these two operators,
which includes both 𝔰^(a_i) and 𝔢^(b_i), should contract to produce the identity. 
Since a_i,b_i≠ a, the number of choices for the replica indices of the remaining 𝔰^(a_i) and 𝔢^(b_i) fields is given by R-12k+l-1. 
Thus, the marginal operator ∑_a𝔢^(a) appears with  a prefactor R-12k+l-1 in the OPE of the general term in Eq. <ref> with 
itself, and this prefactor thus vanishes in the limit R → 1.
When k=1 and l=0, this statement is the same as ∑_a𝔢^(a) appearing with a prefactor of (R-1) as shown in Eq. <ref>.
Analogously, one 
sees
that in the OPE of Eq. <ref> with Eq. <ref> 
the marginal operator ∑_a𝔢^(a) appears with  a prefactor 
R-12k+l, thus also vanishing in the limit R → 1.
When k=1 and l=0 in Eqs. <ref> and <ref>, this statement implies that the marginal operator  appears with a prefactor of R-12 in the OPE of Eq. <ref> and Eq. <ref>, which was verified in Eq. <ref>.
Thus, we see that whenever the marginal operator is produced under the OPE of two (same or different) general operators of  the type shown in Eq. <ref>, it always comes with a prefactor which vanishes in R→ 1 limit. This concludes our proof.
Thus, to summarize our argument (i),  we conclude that 
in the limit R→1, the 
operator
∑_a𝔢^(a) cannot be generated under the RG in any order in perturbation theory with Δ. 

One can check that 
this result also holds 
if we include higher replica terms 
arising from higher cumulants  discussed in App. <ref>, and the 
proof of this statement
proceeds analogous to the above discussion.
Finally, we note that
besides 
Φ, 
the exactly marginal operator and the higher replica terms discussed in App. <ref>,
all other 
terms 
that could be generated under the RG are of the form 
of those in Eq. <ref>, involving a mixture of 𝔰 and 𝔢 (the term in Eq. <ref> being the simplest example), and
are all irrelevant under the RG, as terms with support on the τ=0 time-slice.
.1cm
(ii): We will now give another argument 
which is perhaps more physical, for why 
the operator
∑_a𝔢^(a) cannot be generated under 
the RG in the limit R → 1 of relevance to 
Born-rule measurements. 
The operator
∑_a𝔢^(a), if
it were to be generated 
at any order in RG, can be handled non-perturbatively by using the exact solution due to Bariev <cit.>, and  McCoy and Perk <cit.>.
Using the exact 
solution one sees that when the Ising critical point CFT is perturbed with 
the exactly marginal operator 𝔢(x,τ) supported on the one-dimensional time-slice
and 
in the absence of any other perturbation, 
the power law exponent of
the 𝔰(x,τ) two-point correlation function along the time-slice (defect line) should change continuously with the 
coupling
strength of the
marginal operator supported on the defect line. 
In our replica field theory, in addition to
a possible 
perturbation ∑_a𝔢^a 
generated under the RG, we will also have the defect perturbation Φ itself, from  
Eq. <ref>.
Ignoring
higher cumulants, which
cannot change the low energy details (see App. <ref>), 
the same
replica field theory
would also arise
when we consider performing measurements with σ̂_i^z on the 
state 

    |ψ⟩=exp{κ∑_iσ̂^z_iσ̂^z_i+1}|0⟩     (κ≠0)

where |0⟩ is the ground state at the Ising critical point and the operator σ̂^z_iσ̂^z_i+1 
represents
the continuum field 𝔢 at the Ising critical point.
If we insert in Eq. <ref> in place of the 
state |0⟩ the state 
|ψ⟩ from Eq. <ref>, we 
see that upon going
over to the continuum 
formulation, we will 
obtain
the discussed replica theory with both, the marginal ∑_a𝔢^a 
as well as the Φ(x) interaction
added along the one-dimensional time-slice. 
Thus, if the marginal ∑_a𝔢^a term 
were to be generated under the  RG in the replica field theory for σ̂_i^z measurements
performed on the
critical ground state |0⟩, we get the same replica theory as that for σ̂_i^z measurements 
performed on the state |ψ⟩.
This is a contradiction because Eq. <ref> tells us that the measurement averaged correlation function of
the σ̂_i^z operator, which 
represents
the field 𝔰(x,τ), should be the same as that in the unmeasured state, be it |0⟩ or |ψ⟩.
Since the σ̂_i^z correlation function has different power law behavior in states |0⟩ and |ψ⟩, they cannot be described by the same replica field theory at any energy scale.
Thus, in the replica field theory for the  Ising critical ground state |0⟩ under Born-rule σ̂_i^z measurements, the marginal ∑_a𝔢^a term cannot be generated at any order in
the RG.
.8cm


§ BRIEF REVIEW - RG EQUATIONS FROM THE OPERATOR PRODUCT EXPANSION (OPE)




In general one is interested in computing expectation values of O, representing an operator or a product of operators in the perturbed theory such as in Eq. <ref>,

    ⟨ O⟩_Δ_0
    =Z_*
    Z_Δ_0 ⟨ O e^+Δ_0 ∫ dx Φ(x)⟩_*,

where expectation values ⟨…⟩_* are taken in the unperturbed (i.e. critical) CFT  (in the present case the Ising CFT, compare e.g.
Eq. <ref>).
Here Z_*= Z_Δ_0=0 is the partition function of the unperturbed CFT,  and Z_Δ_0 is the fully interacting partition function
obtained from Eq. <ref> by letting O→ 1.
The RG equations for all operators generated in perturbation theory to any order in Δ_0 is obtained by expanding the exponential on the right hand side of Eq. <ref>,

    ⟨…  1 ⟩_*+
    ⟨… (Δ_0 ∫_x_1Φ(x_1)
     )⟩_* +
    
          +
    ⟨…  (Δ_0^2 2!∫_x_1∫_x_2Φ(x_1) Φ(x_2)
     )⟩_*+ 
    
          +
    ⟨…  (Δ_0^3 3!∫_x_1∫_x_2∫_x_3Φ(x_1) Φ(x_2)
    Φ(x_3)
     )⟩_*+ 
    …

[We note that in the above formula (and also in the subsequent formulae in this appendix) the symbol ∫_x is a short hand for ∫d^2x/a^1-X_𝒜, i.e. in addition to the integral over coordinate x of the integrand field Φ_𝒜(x), the measure of the integral is normalized with a factor of short distance cutoff a raised to an appropriate power involving scaling dimension X_𝒜 of the field Φ_𝒜 so that the corresponding coupling constant (like Δ_0 when Φ_𝒜=Φ) is dimensionless.].
The following discussion is independent of the operator(s) O, indicated by the ellipses,  present in the expectation 
value [There is an analogous procedure to handle the RG equation of operators present in the expectation value, but this is not elaborated on here.].
In a general 
term in Eq. <ref>
we use the OPE which
expands the product of 
n
operators Φ into a complete set of operators Φ^ A located at, say, the position 
x_n of the last operator,

    Φ(x_1) …Φ(x_n-1) Φ(x_n)
    =
    
          
    =∑_ AC_ A [(x_1-x_n), (x_2-x_n), ..., (x_n-1-x_n)]  Φ^ A(x_n).

Most of the possible operators Φ^ A that appear
are irrelevant, and we will mostly be interested in relevant or marginal ones.
The integrals 
I_n-1 over the n-1 
relative coordinates appearing in the OPE coefficient C_ A are performed against a  suitable “cutoff function”  which restricts the absolute values of all relative coordinates within the range between a short-distance cutoff a and a long-distance cutoff L. (There are many options for the “cutoff function”, and our discussion and result will not depend on this choice.)
Inserting these integrals into Eq. <ref>
the latter reads

    ⟨…  1 ⟩_*+
    ⟨… (Δ_0 ∫_xΦ(x)
     )⟩_* +
    
          +
    ⟨…  (Δ_0^2 2!∑_ AI^ A_1∫_xΦ^ A(x)
     )⟩_*+ 
    
          +
    ⟨…  (Δ_0^3 3!∑_ AI_2^ A∫_xΦ^ A(x)
     )⟩_*+ 
    … =
    
          =
    ⟨… ( 1+
    Δ∫_x Φ(x) +∑_Φ^ A≠Φλ_ A∫_xΦ^ A(x) + ... )
    ⟩_*
          =
    ⟨…
    e^Δ∫_x Φ(x) +∑_Φ^ A≠Φλ_ A∫_xΦ^ A(x)⟩_*

where 
we have re-exponentiated in the last line of Eq. <ref> (using standard logic) and we defined

    Δ(Δ_0, a  L)=
    Δ_0 [1
    + Δ_0 2!I_1(a L)
    + Δ_0^2 3!I_2(a L) + 
    … ],
        
           and
          λ^ A(Δ_0, a  L)
    =
     [ Δ_0^2  2!I^ A_1(a L)
    + Δ_0^3 3!I^ A_2(a L) + … ], 
      
           for  Φ^ A≠Φ.

Here we used the abbreviation

    I_k(a L) :=
    I^ A_k(a L),    when  Φ^ A=Φ, 
      k=1, 2, ...

The dependence of the RG equations on Δ, at any order, is then obtained
for both of the couplings Δ and λ^ A in the standard manner:
The RG equation for Δ reads 
 
    d Δ(ℓ) dℓ=
     y_Δ·Δ(ℓ)  
     +(a∂∂ a)_|Δ_0 Δ(Δ_0, a  L)=
    
          
    =y_Δ·Δ(ℓ)  
     + b_2   Δ^2(ℓ)
     + b_3  Δ^3(ℓ) + …,

 where ℓ=ln(L/a) and Δ(ℓ=0)=Δ_0 is kept fixed,
 while  Eq. <ref> is used to re-express the right hand side order-by-order in terms of 
 Δ(Δ_0, a  L)=
 Δ(ℓ). Here, y_Δ  is the RG eigenvalue of the coupling Δ in the unperturbed CFT, i.e.
 y_Δ=
 1- X_Δ= 3/(m+1);  compare Eq. <ref>,
 but now with m=odd. The terms of up to order Δ^2(ℓ) (1-loop order) are those listed in Eq. <ref>.

 The RG equations for λ^ A read
 
    d λ^ A(ℓ) dℓ=
     y_ A·λ^ A(ℓ)  
     +(a∂∂ a)_|Δ_0 λ^ A(Δ_0, a  L) =
    
          
    =
     y_ A·λ^ A(ℓ) 
    + b^ A_2   Δ^2(ℓ)
     + b^ A_3  Δ^3(ℓ) + …,
    
           for Φ^ A≠Φ,

 where λ^ A(ℓ=0)=0, and
again Eq. <ref> is used to re-express the right hand side order-by-order in terms of 
 Δ(Δ_0, a  L)=
 Δ(ℓ).
 Here,
 y_ A are the RG eigenvalues of the couplings λ^ A in the unperturbed CFT.
 
At this stage of the discussion only powers of Δ(ℓ) appear on the right hand side of both RG equations Eqs. <ref>, <ref>, while we see from the latter equation that in general non-vanishing couplings λ^ A of operators Φ^ A≠Φ
are generated. These will then appear in the argument of the exponential of the last line of
Eq. <ref>. The key point then is the following: 
Upon the RG coarse-graining process
these thereby generated couplings will generate additional terms in the RG equations. These  
additional terms
can easily be understood and incorporated into our existing discussion due to the fact that the set of all thereby generated operators Φ^ A with couplings λ^ A form the set of operators
listed in Eq. <ref> which is  closed under the  Operator Product
Expansion (OPE). This means that all the terms generated by the RG from the couplings
λ^ A of operators Φ^ A≠Φ can be understood by simply generalizing Eq. <ref> to multiple OPEs of the operators appearing in Eq. <ref>, namely to

    Φ^ A_1(x_1)
    …Φ^ A_n-1(x_n-1) Φ^ A_n(x_n)
    =
       
    =∑_ AC_ A^ A_1,  ...,  A_n-1,  A_n [(x_1-x_n), (x_2-x_n), ..., (x_n-1-x_n)] Φ^ A(x_n).

Employing the same logic that led to the   RG equations Eqs. <ref>, <ref> now leads to the same RG equations but with arbitrary powers of the coupling constants λ^ A  appearing on the right hand side of these equations.
The key result of this analysis is that the only couplings λ^ A that can be generated under the RG are those of operators Φ^ A that can appear on the right hand side of the multiple OPE in Eq. <ref>. Such  contributions would be represented by a term  of order λ^ A_1…λ^ A_n-1 λ^ A_n 
on the right hand side of the RG  equation for λ^ A.
However we show in argument (i) 
of App. <ref> 
that all  the OPE coefficients  in 
Eq. <ref>,
involving on the left hand side operators
appearing in Eq. <ref>,
vanish in the limit R → 1, when  Φ^ A and λ^ A appearing on the right hand side corresponds to the exactly marginal operator
in 
Eq. <ref>
and its corresponding coupling constant. This then implies that the exactly marginal operator cannot be generated under the RG to any order in perturbation theory in the coupling Δ in the limit R → 1.
.8cm


§ IRRELEVANCE OF HIGHER CUMULANTS FROM AVOIDED LEVEL CROSSINGS


The  “higher-moment operators” in Eq. <ref>
and
in Eq. <ref>  whose scaling dimensions 
at the Δ_* ≠0 fixed point are of interest
in App. <ref> and App. <ref>, respectively, are conformal boundary operators: In the standard manner, these operators with support on the
one-dimensional τ=0 time-slice  in space-time can be viewed upon folding  the space-time
along 
this time-slice 
<cit.>,
<cit.>
as operators with support on the boundary, the real axis, of the tensor product of two identical copies of the  (non-random but replicated) bulk CFT, located in the upper half complex plane. The fixed point Δ_* ≠0
describes a scale- and conformally invariant boundary condition  B_Δ_* on the  two copies of the bulk CFT in the upper half plane, which determines all the universal properties of interest to us in this paper.
After conformal mapping from the upper half complex plane to
the interior of an infinitely long strip of finite width L with identical boundary conditions B_Δ_* on both sides, the spectrum of the Hamiltonian Ĥ_Δ_*, generating translations along the strip, is universally related  by finite-size scaling 
<cit.>,<cit.>
and the operator-state correspondence to the scaling dimensions of the set of all operators with support on the boundary B_Δ_*, of interest to us here.

Here we will discuss, 
for each value of m or equivalently of ϵ=3/(m+1),
the evolution (“spectral flow”) of a particular set of energy levels of the Hamiltonian Ĥ_Δ(ℓ) where the coupling constant 
Δ=Δ(ℓ)
(here ℓ=ln(L/a)), flows under the RG from the RG-unstable zero coupling fixed point
Δ=0 in the ultraviolet to the RG-stable finite-coupling
fixed point 
Δ_* = Δ_*(ϵ) in the infrared.
For each value of m 
(or equivalently
ϵ) this
describes an RG flow between two conformally invariant boundary conditions.
In the intermediate regime of length scales ℓ away from the two fixed points, the corresponding boundary condition B_Δ(ℓ) will not be scale- nor conformally invariant, and the entire spectrum of the corresponding
Hamiltonian Ĥ_Δ(ℓ)  will undergo an evolution, i.e. a spectral flow with the length scale ℓ. Because the operator coupling to Δ is invariant under the group S_R of permutations of the R replicas, we can classify all eigenstates of Ĥ_Δ(ℓ) according to irreducible representations of the permutation group S_R.
For each value of m we consider the spectrum of this Hamiltonian in each symmetry sector separately. Since the operators  in Eq. <ref>
and
in Eq. <ref>  of interest to us are singlets under permutations, we restrict attention to the  S_R-singlet sector of the spectrum of Ĥ_Δ(ℓ). We know that this Hamiltonian is, due to the large conformal symmetry,  integrable at the ultraviolet (Δ=0)  as well as the infrared fixed point (Δ_*≠0). However, at all intermediate scales ℓ  away from the two fixed points this Hamiltonian is not expected to be integrable since the operator Φ coupling to Δ 
(and thus setting the boundary condition B_Δ(ℓ) which determines the spectrum of Ĥ_Δ(ℓ)) is not expected to conserve a macroscropic number of the conformal conservation laws present at the two fixed points.
Given that the Hamiltonian Ĥ_Δ(ℓ) is not integrable at intermediate scales, the evolution of its spectrum as a function of scale ℓ=ln(L/a) in the S_R-singlet sector is expected to exhibit avoided level crossings. 
Now for each value of m (or ϵ),
as discussed in App. <ref> and App. <ref>, 
the scaling dimensions of the operators in  Eq. <ref>
and
in Eq. <ref> at the ultraviolet fixed point Δ=0
are equal to 
2k × X_ϵ= 2k × X_φ_1,2
when m= 
even, and 
2k × X_ϵ= 2k × X_φ_1,2 
when 
m= odd, respectively, and thus are strictly ordered in both cases. Here 
X_ϵ=X_φ_1,2
is given by  Eq. <ref> for both cases,  m= even and  m= odd (see footnote <cit.>). 
Note that in the limit m →∞ (ϵ→ 0), these dimensions become
2k × (1/2) =k since 
X_ϵ=X_φ_1,2 
→ 1/2.
In either case, all these operators with k>1  thus 
have, for any value of m (even or odd), scaling dimensions larger than the respective perturbation Φ (which corresponds to k=1) 
at the ultraviolet fixed point Δ=0.
For each case, i.e. for any even and for any odd value of m corresponding to 
Eq. <ref>
and
Eq. <ref> respectively, we expect,
given the avoided level 
crossings,
as we increase the scale ℓ=ln(L/a) to  run the RG via finite-size scaling 
from
the ultraviolet to the infrared fixed point 
Δ_*(ϵ),
that the relative ordering of these scaling dimensions for different values of k is preserved. In particular, we expect the scaling dimensions of all these operators with k>1 to remain larger than the scaling dimension of the operator which has the  smallest scaling dimension at the ultraviolet fixed point Δ=0, which is the one with k=1, corresponding to the perturbation Φ. But since we know that the perturbation Φ must be irrelevant (i.e. must have scaling dimension >1) at the infrared fixed point (as given by the slope of the corresponding RG beta function for Δ), we conclude that
the scaling dimensions of all the operators with k>1 will also  need to 
be
certainly larger than unity, due to avoided level crossings. 
This provides an argument supporting the irrelevance at the infrared fixed point of 
the operators in Eq. <ref>
and
in Eq. <ref> 
arising from all higher cumulants, both in the  tricritical Ising (App. <ref>)  
as well as in the Ising (App. <ref>) case.We close by noting that avoided level crossings of scaling dimensions of bulk
operators in RG flows between two 
(bulk) 2D RG fixed points, arising from perturbations breaking the integrability  of the ultraviolet CFT, have been observed
explicitly via the Truncated Conformal Space approach
<cit.>, 
e.g. see Ref. <cit.> Sect. IV.C, Figs. 7, 8.
Flows of boundary scaling dimensions in RG flows between two different boundary fixed points of the same bulk CFT have also been studied using the Truncated Conformal Space approach, see e.g. 
Ref. <cit.>;
the latter particular investigation is of less direct relevance for us since in this study only an integrable boundary perturbation is discussed, but it demonstrates the ability to study RG flows between boundary fixed points effectively within the Truncated Conformal Space approach. Finally, all spectra numerically obtained from 
K. G. Wilson's numerical renormalization group approach to the Kondo- and other quantum impurity problems
(see e.g. Ref.
Wilson1975,PangCox1991)
precisely observe <cit.>
related spectra of boundary RG flows between different fixed point boundary conditions on a fixed bulk CFT, exhibiting avoided crossings in a given symmetry sector. ]