Emergent Ashkin-Teller criticality in a constrained boson model

1) This manuscript contains results on the phase structure of a constrained one-dimensional bosonic model with subsystem symmetries. The main unusual feature that makes this system interesting is that it exhibits criticality with an extra emergent Z2 symmetry that changes the universality class of the transition. The authors support these claims with numerics, and field theoretic arguments for the extra emergent Z2 symmetry at the critical point.

The paper makes a nice observation on the unusual nature of the critical point of this hard-core bosonic model, which is certainly interesting. However, I think there are some related issues would be good to address or explore in the manuscript.

Response: We thank the referee for a careful reading of the manuscript. Below, we respond to their comments

1. Referring to this system as having a subsystem symmetry" is a bit misleading because that term usually refers to models in two or higher dimensions that have a symmetry operator that lives in a smaller-dimensional manifold (e.g., one-dimensional manifold of a two-dimensional system). Since the authors focus on an effectively one-dimensional model, they are really studying a model with some kind of Local Integrals of Motion (LIOMs, or also sometimes called a gauge symmetry"). Models with such LIOMs have been studied a lot in the literature in various contexts, e.g., in disorder-free localization. It might be good to use this terminology instead of subsystem symmetry" since it would then naturally lead to questions such as: Are these LIOMs responsible for the unconventional criticality? Does this phenomenology extend to other systems with LIOMs? It would also help if the authors then mention in the abstract that they are essentially studying one-dimensional models.

Response: Our reference to subsystem symmetry comes from the 2D version of this model discussed in SciPost Phys. 14, 146 (2023) where the role of subsystem symmetry has been discussed. In this work, where we consider the bosons with a slightly modified Hamiltonian on a two-leg ladder (or diamond chain lattice as pointed out by Ref 2), the subsystem symmetry of the 2D model, gets translated into conservation of bosons number in any vertically or horizontally connected sites ( paragraph 1 of Pg 4 and Fig 1). It still involves three sites and is not a local (single site) or global (conservation of total boson number) symmetry in the strict sense. When we map the model to a spin chain in Sec 5, the same conservation leads to the constraint condition of not having neighboring up spins. We have now discussed this in detail in the discussion section of the paper.

1. The analysis of Sec.~3.2 seems to be a fairly standard Schrieffer-Wolff" perturbation theory, but the authors do not make any reference to earlier works doing a similar analysis (e.g., Annals of Physics, Volume 326, Issue 10, October 2011, Pages 2793-2826). If there is any difference from the standard technique, it would be good to address it more clearly.

Response: This is a standard Schrieffer-Wolfe transformation which has been used since this transformation came out in 1966 in Phys. Rev 149, 491 (1966). It is now textbook material; we have now added relevant citations. We cited Ref 37 earlier in this context since such transformation was carried out there in the context of a similar bosonic theory.

1. The importance of α>0 in this analysis can be elucidated more clearly. For example, when α=−1, this model seems to map onto the quantum dimer model studied in Eq.~(3.1) of Phys. Rev. B 63, 224401 (2001), which has its own set of transitions, and also features an interesting Rokhsar-Kivelson" type of critical point. Since the model for any α<0 also has the same subsystem symmetry" or LIOMs that the authors discuss, does any of the physics of α>0 also occur for some α<0 ? It would be good if the authors also point out this connection to the dimer models, which are also related to the PXP-type models discussed in Sec.~5.

Response: We have now added a discussion regarding the importance of \alpha>0 in the model in Sec 2 of the paper. We thank the referee for pointing this out.

1. While the authors provide a clear intuitive reasoning for the existence of a gap in the large negative λ regime, they do not clearly explain the origin of the gapless spectrum in the large positive λ regime. Perhaps they can expand the discussion of this regime, and particularly make it clear what the broken Z2 symmetry is, and why that leads to gaplessness?

Response: Due to the presence of subsystem symmetry and due to our choice of 2N plaquettes and a reference state having 2N+1 bosons, it is mandatory that there exists a lone boson in the ground state manifold of states for the system at large positive \lambda. These states can be labelled by the position of this lone boson; consequently there are O(N) such states, as explained in Sec 3. These states are degenerate for $w=0$; our Schrieffer-Wolfe analysis shows that this degeneracy is lifted at finite $w$ leading to a two-fold degenerate gapless band (the energy dispersion showing this given by Eq 8; for the PBC its two-fold degenerate since it is invariant under k\to \pi+k, where $k$ is the momentum while for OBC the degeneracy comes from reflection symmetry as discussed in the last paragraph of Sec 3). The ground state breaks the Z_2 corresponding to this two-fold degeneracy. The virtual processes which allow for hopping of this loan boson for finite $w$ connecting such N states are shown in Figs 3 and 4. Such hopping leads to the gapless nature of the low-energy excitations can be seen from both Eq 8 and also exact numerics.

We think that these points are quite clearly mentioned already in Sec 3.

Response to Ref 2

1) In this paper, the authors study a constrained boson model with a subsystem symmetry on a "diamond chain" lattice. While a quantum phase transition of Ising universality class is expected, the authors claim that the actual transition belongs to the Ashkin-Teller universality class.

The model and various consequences of the subsystem symmetry is certainly interesting. However, I feel that the nature of the ground-state phase diagram and the quantum phase transition is not clarified enough. The main claim that the transition belongs to the Ashkin-Teller universality class is also not sufficiently justified in my view.

Response: We thank the referee for their comment. We respond to all their comments below.

1) First, in the λ≫w region, the authors find "gapless" phase with a spontaneously broken Z2 symmetry. I see the authors' argument why the system is "gapless" (in a certain sense) in the limit λ≫w. However, this seems to be attributed to a motion of a single defect, and does not seem to lead to gapless excitations with non-vanishing density of states in the thermodynamic limit. In other words, is the "gapless" phase described by a CFT? I suspect it corresponds to c=0 (if you study the low-temperature limit of the specific heat, for example). If the "gapless" nature discussed by the authors just comes from the motion of a single defect, it sounds more like an artifact and does not constrain the bulk behavior in the thermodynamic limit. In any case, the authors should clarify what "gapless" means for this phase.

Response: We would like to mention a few points here:

a) The gapless phase here has z=2 at low momentum as is evident from the dispersion given in the draft (E_k \sim \cos 2k). It is therefore not Lorentz invariant (gapless phase with z=2) and hence not conformal invariant since the latter require both Lorentz and scale invariance. This gapless phase is therefore more like a line of quantum Lifshitz points. We have now mentioned this point in Sec 3.

b) The specific heat of the model can be computed and it scales as \sqrt[T] for d=1 and z=2.

c) The gapless phase does not occur due to a single Fock state ( where the lone boson is taken as a defect); there are O(N) such states and all of them are low-energy eigenstates of H_1 (Eq 2) for $w=0$ and $\lambda>0$. This manifold of states is degenerate at w=0; their degeneracy is lifted by H_0 ( see Eq 2 and Sec 3.2 of the draft). Since the number of states in the manifold is O(N), it survives the thermodynamic limit. If the gaplessness occured due to a single defect state, it would not have survived the thermodynamic limit, but that's not the case here.

2) The claim that the phase transition belongs to the Ashkin-Teller universality class also looks weak to me. Apparently it is based on the numerical estimate of the central charge --- it looks significantly larger than the Ising value of 1/2. However, it does not also show that the central charge is the Ashkin-Teller value of 1. As the authors claim, maybe it is consistent with the hypothesis that the universality class is Ashkin-Teller. But I do not feel it is strong enough to be regarded as evidence. Just as a possibility, the presence of the moving defect might be "contaminating" the numerical central charge data, and the actual central charge could be Ising.

Response: We have already discussed this point in the Discussion section of our work. We agree that a concrete statement about Ashkin-Teller requires study on systems with larger L. However, we would like to point out the following:

a) Numerical data ( Fig 5) is clearly consistent with z=1 and \nu >1. This was our first indication that the transition is different from Ising (which has \nu=1$. Whereas the central charge data shows a lot of fluctuations, we find that estimation of \nu is reliable and has very little system size induced fluctuation. This does not seem to point towards an ising transition.

b) From Fig 7a, we find that \nu is a function of the parameter \alpha while $z$ is not. This seems to indicate ( without having to estimate c yet) weak universality; this is not something one finds in an Ising transition.

c) Finally, even within fluctuation c stays closer to unity than 1/2 (Fig 7d). Also relatively larger system size induced fluctuations for numerical estimation of central charge (compared to \nu or z) is common in the literature.

These three facts, and not the central charge data alone, led to conjecture that the universality is in the Ashkin-Teller universality class. As we mentioned, numerical studies on larger L are needed to settle this issue, but the above points (along with the effective field theory argument) seem to point towards a possibility of Ashkin-Teller universality.

3) I am not claiming this is true, but from what I find in the paper I cannot rule out such a possibility. Ashkin-Teller universality class is not just characterized by the central charge 1. The operator content is also well-understood. In particular, there must be the twist (or spin) operators with the conformal dimension (1/16,1/16) in the spectrum independently of the compactification radius ("Luttinger parameter"). There are also vertex operators whose conformal dimensions depend on the radius. I suggest the authors to study the operator content from the spectrum, in addition to the central charge. Finally, the phase diagram of the Ashkin-Teller model has been studied in many papers, and we can understand the phase diagram in terms of the perturbations to the Ashkin-Teller CFT. In order to establish the claim, I think the authors should make such an analysis for the present model.

Reply: This is certainly true. There are several papers in the literature where operator contents of the Ashkin Teller model has been studied. However the present case has important distinction with those models. For us the second Z_2 symmetry is emergent and we do not know of any microscopic local operator which we can associate with this symmetry. In other words, in the present model, the operators corresponding to the Ashkin-Teller model are not related in any simple way (that we could figure out) to the microscopic bosons due to the emergent nature of the symmetry. This prevents us from studying their conformal dimensions. For the same reason, it is difficult to understand relevant perturbations in terms of the microscopic bosons which will serve this purpose.

4) Some minor (non-essential) comments: - I think the authors' lattice is sometimes called as "diamond chain lattice" in quantum spin systems community. Perhaps they can mention this.

Response: We have mentioned this in the modified version

5) The original form of Eq. (11) (although not exactly Eq. (11) for open boundary conditions) was derived by Holzhey, Larsen, and Wilczek in Nucl.Phys. B424 443 (1994). Perhaps it is fair to cite it along with Calabrese-Cardy.

Response: We have cited this work in the current version. We thank the referee for pointing it out.

We hope that the above points will convince the referee regarding points made in the paper.

We thank the referee for their positive recommendation regarding our work.