Emergent Ashkin-Teller criticality in a constrained boson model

Referee: In my opinion, the authors make a reasonable case that their numerical results are consistent with the Ashkin-Teller universality class. However, I agree with Report 1 by Anonymous (Referee 2) on 2025-2-21 that the comment at the end of section 5 is very strange. My understanding is that certain boundary conditions may project out the sector of the ground state from the Hilbert space. Then one sees an effective central charge as defined, e.g., in Ref. [45]. However, this does not mean that the true central charge does indeed change. After all, a change of boundary conditions should not change the bulk physics, at least not in a system with short-range interactions. If the emergent Z2 symmetry is supposed to hinge on these boundary conditions, this is actually worrisome. In any case, this point needs to be clarified before the manuscript can be published.\

Response: We have now clarified this point to the best of our abilities. We have also responded regarding this to the other referee. Below, we present that response. \

a) We request the referee to note that we are working for a system with even number of plaquettes $N_p$ and odd number of bosons $N=N_p+1$ (a sector of the subsystem symmetry of the model as discussed in the work). This system of bosons has open boundary condition. In the boson language, for $\lambda>0$ and $\lambda \gg w$, it is easy to see that for this $N$, the ground state manifold consists of Fock states involving a single lone boson (other $N_p$ bosons are arranged along transverse diagonal of the plaquette; see Fig 1b where $N_p=6$ and $N=7$). \

b) Since different positions of the lone boson gives rise to degenerate Fock state, this leads to a $O(L)$ degenerate ground state manifold. This degeneracy is lifted by quantum fluctuations due to $H_0$ leading to a gapless boson ground state with two-fold degeneracy as shown in our work. Such a gapless state only occurs in this fixed $N$ sector for open boundary condition and is protected by the subsystem symmetry of the model. We request the referee to note that an analogous description for periodic boundary condition will requite $N= N_p-1$. \

c) In contrast for $\lambda <0, |\lambda|\gg w$, the ground state is gapped and unique (for both $N$ and for both boundary conditions). Thus the transition is between a gapped and a gapless ground state for a fixed $N$ and for the specified boundary condition; this is in contrast to the Ising model where it occurs between two gapped ground states. Although this is not a proof, the presence of a different ground state at $\lambda \gg w$ indicates the possibility that the transition may have different universality class. \

d) When we map this model to the spin language, i.e., a PXP model with additional three spin interaction (Eqs. 15 and 16 of the draft), this point becomes slightly more clear. We note the following. \

i) The ground state manifold of the PXP chain depends crucially on the boundary condition for $\lambda \gg w$. For periodic boundary condition, the ground state manifold (for even number of sites) contains two Fock states given by $|1,0,1,0, .... ,1,0\rangle$ and $|0,1,0,1,... ,0,1\rangle$, where $1$ indicates an up-spin and $0$ a down-spin. Thus the choice of any one of these breaks a $Z_2$ symmetry. \

ii) In contrast, for the open boundary condition, the ground state manifold has several other states. These additional states corresponds to states with a single domain wall. For example, one such Fock state in the manifold is given by $|1,0,0,1....0,1 \rangle$. Note that such domain wall states are forbidden in chains with periodic boundary condition due to the constraint of not having two up-spins next to each other (here these up-spins are on last and first sites of the chain which are adjacent if one imposes periodic boundary condition). The the ground state manifold has ${\rm O}(L)$ states; it therefore changes due to the presence of the boundary condition and, as our analysis indicates, leads to a gapless phase instead of a gapped one. This shows that the boundary condition in a constrained system can change the nature of the bulk phase. We have checked this by computing the structure factor for a $N$ site spin chain $S_q= \langle \psi_q^{\ast} \psi_q \rangle$ where $ \psi_q = (1/N) \sum_j n_j e^{i 2 \pi j/q}$ and $n_j =(1+\sigma_j^z)/2$ counts the number of up-spins on each site. For a PXP chain with periodic boundary condition $S_{q=2}$ saturates to $1/4$ in the thermodynamic limit as suggested by $1/N$ extrapolation; however in the presence of constraint and open boundary condition, a similar analysis (using $N \le 26$) shows that $S_q|_{N \to \infty} \sim 0.034$. This confirms the different nature of the ground state at large positive $\lambda/w =14$ for periodic and open boundary conditions for the present case. \

e)We hope that the above discussion makes it clear that in a fixed boson sector $N= N_p+1$, the nature of the ground state at large $\lambda$ depends on the boundary condition in the present model. This dependence arises due to fixing of $N$; we note that in this model, one can work with a fixed $N$ (for a given $N_p$) due to the presence of subsystem symmetry. One can see similar physics for periodic boundary condition; but this requires choosing the sector $N= N_p-1$ as discussed. The nature of the bulk ground state for this model depends on chosen on the boundary condition for a fixed $N$; this dependence can be tied to the presence of subsystem symmetry of the bosonic model as discussed. Since a quantum critical point separating any two phases depends on the nature of these phases, it is possible that its universality class also depends on the boundary condition. This is not a definitive proof, but is a scenario which our numerics seem to suggest. This situation is to be contrasted to the standard cases where usually the chosen particle number sector is not crucial to determining the property of the bulk ground state of the model. \ \

2: Beyond this, I regret to say that I do not have the impression of a finished manuscript even if it already went through several rounds of revision, specifically: A) The Landau-Ginzburg section 4.2 is very short and its role did not become completely clear to me. Also, I do not think that a Landau-Ginzburg theory will yield the correct critical exponents, but the universality class of the transition is the main focus of the present work. \

Response: We would request the referee to note that we have not attempted to derive scaling exponents from the Landau-Ginzburg theory anywhere in the manuscript precisely because of the reason mentioned by the referee. We have used it only to understand the nature and origin of the additional emergent symmetry at the critical point. It is well-known that since the Landau free-energy functional need to respect all microscopic symmetries of theory; often this restriction leads to additional emergent symmetries near the critical point. This type of analysis has long been carried out in the literature, for example, in Refs 44 and 45 of the present work. We have now put in a discussion regarding this in this section. \

B) The role of section 3.1 ("Numerical results") is not quite clear either. In particular, this comprises two references to entanglement entropy on less than one page, but no data is shown. I think that the authors should either show the data (if they consider it important) or remove these remarks.\

Response: This section discusses the entire set of numerical analysis of the critical point that we have carried out. The data for entanglement entropy was shown in Figs. 7(c) for $\alpha=1$ and Fig 9(c) and (d) in the limit of large $\alpha$. We have now moved the discussion entanglement entropy calculation in the main text and referred to these figures in appropriate places. We thank the referee for pointing this out. \

C) Appendix A makes the impression of an excuse to cite Refs. [50,51] by the group itself. By contrast, references for the well-known definition of entanglement entropy are missing. I recommend to cut this down to the essence, move the remainder into an appropriate place of the main text, and add references for the definition of the entanglement entropy $S_E$.\

Response: We put Refs 50 and 51 in the Appendix since they have details of the methods used. Since the details of the methods used are already published, we thought it would be fair to point it to the interested readers. Following suggestion of the referee, we have now put in more, earlier, refs on the subject and moved this section in the main body of the paper. \

3) Referee: There are further issues of a more typographic nature that I list below among "Requested changes".

Requested changes

Major points: 1) Clarify the discussion of the central charge $c$ in relation to boundary conditions at the end of section 5. \

2) Consider moving the Landau-Ginzburg section 4.2 to an appendix, add remarks on its role, and remove it from the abstract (after all, this cannot be a main point of this work). \

3) Clarify the role of section 3.1 ("Numerical results"). Furthermore, either show the results for the entanglement entropy or remove the related remarks from section 3.1. \

4) Shorten appendix A, move the result into the main text, and add references for the definition of the entanglement entropy $S_E$ \

We have now addressed all these points. \

1) We have modified the discussion and explained it. \

2) We have not moved the section on LG functional to the appendix since we believe that its an important part of the paper, being our only analytical, albeit qualitative, means to understanding the reason for the additional emergent $Z_2$ symmetry. We have, however, expanded this section and added clarification of several points as requested by the referee. \

3) We have now pointed out the role of Sec. 3.1 explicitly and added a paragraph the computation of entanglement entropy.\

4) We have removed App A and moved the discussion to section 3.1. \ \

4) Referee: Minor points (typographical errors etc.): \

5) Third paragraph of section 1: "disordered Ising system" "disordered Ising systems". \ 6) Third line of section 2: "have been discussed" "has been reviewed". After all, Ref. [34] is a review, the original publications are older than that. \ 7) Fig. 1 appears to be a low-quality and thus fuzzy image. Ideally, it would be converted to a vector graphics; at the very least, resolution should be increased. \ 8) Eq. (1) should terminate with a full stop ("."). \ 9) First line of section 3: "two limit" "two limits". \ 10) Eq. (7) should terminate with a full stop ("."), not a comma (","). \ 11) Figs. 5, 7, 6 seem to be cited in this order and thus should be reordered according to their appearance in the text.\ 12) Fourth line of section 4: In my opinion "It is well known" needs nevertheless to be substantiated by references.\ 13) Third line of caption of Fig. 1: I think an article "the" is missing on front of "Ising model".\ 14) I believe that Fig. 7(c) corresponds to $\alpha=1$, but that should be stated in the figure caption. \ 15) Eq. (15) should terminate with a full stop ("."). \ 16) Last line of page 13: "One of the source" "One of the sources". \ 17) First paragraph on page 14: "loan" "lone"? \ 18) First line of second paragraph of on page 14: "finite-sized analysis" "finite-size analysis".\ 19) Second line of page 15: insert "standard" before "density-matrix-renormalization group". After all, this is followed be a statement that there are variants that do the job.\ 20) Second line of second paragraph of on page 15: "a ultracold" "an ultracold". \ 21) Line below Eq. (21): "rho" "$\rho$". \ 22) Fix issues with the bibliography: [7,14,16-18,20,24,33,38-40,42,45,48-51] Lower-casing of names in titles ("Ashkin", "Elitzur", "Floquet", "Ising", "Mott", "Rydberg", "Teller", "Wolff"). \ 23) "q" "Q" and "u(1)" "U(1)". \ 24) "n=2" "N=2". \ 25) Break the title such that at least it does not spill beyond the boundary of the page.\

Response: We have now corrected these to the best of our abilities. Please note that we have put in a new version of Fig. 1 in the present version.

1) Referee: I appreciate the efforts made by the authors to improve the paper, but I am still not convinced that the claim of the Ashkin-Teller universality class is supported well enough. I believe that the operator content can be studied numerically in spite of the unclear relation to the microscopic operators mentioned by the author. \

Response: As we have explained before, it is unclear to us how one does this for the following reasons.

First, the operator content analysis of the CFT, other than that of the stress-energy tensor (which is similar to analysis of the energy spectrum), requires an identification of the primary operators of the CFT in terms of the microscopic variables. In our case since the additional symmetry at the critical point is emergent, we do not have a clear identification. For example, the existence of a local order parameter $O(r)$ in terms of the microscopic spin variables is unclear; if we could have identified this operator, we could have numerically checked, for example, if $\langle [{\rm Re}O(r)^2] [{\rm Re}O^2(0)]\rangle$ falls as $r^{-\eta_2}$ where $\eta_2=1-1/(2\nu)$ varies continuously across the critical line. This would have ensured the presence/absence of Ashkin-Teller criticality. However, this seems to be difficult for the reason mentioned above. We shall be happy to have a suggestion if the referee can point us towards a way to carry out this or similar analysis.

Second, regarding spectroscopic analysis, we have a) looked into the low-energy spectrum and obtained $z$ and $\nu$ from it. Our analysis shows a variable $\nu$ across the critical line. b) We have also shown that $\Delta_{20}$ and $\Delta_{30}$ shows scaling behavior similar to $\Delta_{10}$ (albeit with stronger finite-size corrections) and that this behavior is completely different from that for an Ising critical point (see Fig 6). Within the limitation of finite size in our numerics, we do not find any other numerical analysis that we could carry out. We do not think we can have other, different, information from analysis of spectrum of the model (or stress energy of the CFT).

Third, we also measured entanglement, which do not require knowledge of the relation of the primary CFT operators with the microscopic operators of the spin model. These measurement suggest a central charge which is different from $1/2$; we agree that these results have strong finite-size effect (in contrast to $\nu$). \

2) Referee: I am also confused with the comment "a change in boundary condition may change c (central charge)" at the end of Section 5. I did check Ref. [45], and if you define the "effective central charge" as in Ref. [45] it would be true, but perhapsit is rather a misnomer. As a general principle, the boundary conditions should not alter the bulk universality class which is characterized by the central charge. If the author somehow implies that Ref. [45] is related to the claim of the (bulk) Ashkin-Teller universality class, I am afraid that they are mistaken. It is also not clear to me how the discussion in Sec. 4.2 is related to the specific model in question.\

Response: We apologize for this confusing statement. We have now rewritten this part of the draft to make things clear. The gist of that discussion is as follows: \

a) We request the referee to note that we are working for a system with even number of plaquettes $N_p$ and odd number of bosons $N=N_p+1$ (a sector of the subsystem symmetry of the model as discussed in the work). This system of bosons has open boundary condition. In the boson language, for $\lambda>0$ and $\lambda \gg w$, it is easy to see that for this $N$, the ground state manifold consists of Fock states involving a single lone boson (other $N_p$ bosons are arranged along transverse diagonal of the plaquette; see Fig 1b where $N_p=6$ and $N=7$). \

b) Since different positions of the lone boson gives rise to degenerate Fock state, this leads to a $O(L)$ degenerate ground state manifold. This degeneracy is lifted by quantum fluctuations due to $H_0$ leading to a gapless boson ground state with two-fold degeneracy as shown in our work. Such a gapless state only occurs in this fixed $N$ sector for open boundary condition and is protected by the subsystem symmetry of the model. We request the referee to note that an analogous description for periodic boundary condition will requite $N= N_p-1$. \

c) In contrast for $\lambda <0, |\lambda|\gg w$, the ground state is gapped and unique (for both $N$ and for both boundary conditions). Thus the transition is between a gapped and a gapless ground state for a fixed $N$ and for the specified boundary condition; this is in contrast to the Ising model where it occurs between two gapped ground states. Although this is not a proof, the presence of a different ground state at $\lambda \gg w$ indicates the possibility that the transition may have different universality class. \

d) When we map this model to the spin language, i.e., a PXP model with additional three spin interaction (Eqs. 15 and 16 of the draft), this point becomes slightly more clear. We note the following. \

i) The ground state manifold of the PXP chain depends crucially on the boundary condition for $\lambda \gg w$. For periodic boundary condition, the ground state manifold (for even number of sites) contains two Fock states given by $|1,0,1,0, .... ,1,0\rangle$ and $|0,1,0,1,... ,0,1\rangle$, where $1$ indicates an up-spin and $0$ a down-spin. Thus the choice of any one of these breaks a $Z_2$ symmetry. \

ii) In contrast, for the open boundary condition, the ground state manifold has several other states. These additional states corresponds to states with a single domain wall. For example, one such Fock state in the manifold is given by $|1,0,0,1....0,1 \rangle$. Note that such domain wall states are forbidden in chains with periodic boundary condition due to the constraint of not having two up-spins next to each other (here these up-spins are on last and first sites of the chain which are adjacent if one imposes periodic boundary condition). The the ground state manifold has ${\rm O}(L)$ states; it therefore changes due to the presence of the boundary condition and, as our analysis indicates, leads to a gapless phase instead of a gapped one. This shows that the boundary condition in a constrained system can change the nature of the bulk phase. We have checked this by computing the structure factor for a $N$ site spin chain $S_q= \langle \psi_q^{\ast} \psi_q \rangle$ where $ \psi_q = (1/N) \sum_j n_j e^{i 2 \pi j/q}$ and $n_j =(1+\sigma_j^z)/2$ counts the number of up-spins on each site. For a PXP chain with periodic boundary condition $S_{q=2}$ saturates to $1/4$ in the thermodynamic limit as suggested by $1/N$ extrapolation; however in the presence of constraint and open boundary condition, a similar analysis (using $N \le 26$) shows that $S_q|_{N \to \infty} \sim 0.034$. This confirms the different nature of the ground state at large positive $\lambda/w =14$ for periodic and open boundary conditions for the present case. \

e)We hope that the above discussion makes it clear that in a fixed boson sector $N= N_p+1$, the nature of the ground state at large $\lambda$ depends on the boundary condition in the present model. This dependence arises due to fixing of $N$; we note that in this model, one can work with a fixed $N$ (for a given $N_p$) due to the presence of subsystem symmetry. One can see similar physics for periodic boundary condition; but this requires choosing the sector $N= N_p-1$ as discussed. The nature of the bulk ground state for this model depends on chosen on the boundary condition for a fixed $N$; this dependence can be tied to the presence of subsystem symmetry of the bosonic model as discussed. Since a quantum critical point separating any two phases depends on the nature of these phases, it is possible that its universality class also depends on the boundary condition. This is not a definitive proof, but is a scenario which our numerics seem to suggest. This situation is to be contrasted to the standard cases where usually the chosen particle number sector is not crucial to determining the property of the bulk ground state of the model. \ \

3) Referee: On the other hand, as I mentioned, this paper does contain some interesting observations. I would suggest two options: 1) Consult/collaborate with someone with a strong background of CFT to perform a more thorough comparison with CFT. 2) Revise the paper to objectively describe theoretically established statements and numerical observations. The authors may speculate/conjecture the Ashkin-Teller universality class in the Discussion section, but I think it better to remove the Ashkin-Teller from the title etc.\

Response: We have now rephrased the text/title etc using option 2. We hope that this version will be more suitable for publication. Regarding option 1) we did consult a few cft experts; however, we were told that the knowledge of the relation between the microscopic variables and the primary operators of the CFT is necessary to carry the analysis suggested by the referee.