# Quantum criticality in many-body parafermion chains

### Submission summary

 As Contributors: Eddy Ardonne · Ville Lahtinen Arxiv Link: https://arxiv.org/abs/1709.04259v2 (pdf) Date accepted: 2021-05-20 Date submitted: 2021-05-18 08:38 Submitted by: Ardonne, Eddy Submitted to: SciPost Physics Core Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We construct local generalizations of 3-state Potts models with exotic critical points. We analytically show that these are described by non-diagonal modular invariant partition functions of products of $Z_3$ parafermion or $u(1)_6$ conformal field theories (CFTs). These correspond either to non-trivial permutation invariants or block diagonal invariants, that one can understand in terms of anyon condensation. In terms of lattice parafermion operators, the constructed models correspond to parafermion chains with many-body terms. Our construction is based on how the partition function of a CFT depends on symmetry sectors and boundary conditions. This enables to write the partition function corresponding to one modular invariant as a linear combination of another over different sectors and boundary conditions, which translates to a general recipe how to write down a microscopic model, tuned to criticality. We show that the scheme can also be extended to construct critical generalizations of $k$-state clock type models.

Published as SciPost Phys. Core 4, 014 (2021)

Dear editor and referee,

We would like to start by mentioning that due to various reasons, the resubmission of our manuscript did not materialise. We assumed that it wasn't possible anymore to resubmit, but we were contacted, and told that resubmission was still possible. We of course fully understand if the referee and/or editor has changed her/his/their opinion about the manuscript.

We are pleased to see that the points mentioned by the referee under 'Strength' agree with what we think of the main points of the paper. We agree with the referee about the weaknesses of the paper. The results are more straightforward than profound, but we think, and the referee seems to agree, still interesting. It is also true that there are no physical applications, and having such applications would make the paper more interesting. Though not an excuse, our paper is not alone in this respect. There is not much we can do about these two weaknesses. We tried to improve the paper with respect to the second weakness mentioned by the referee, in that the paper does not properly discuss the relation with, in particular, the literature on the orbifold construction in CFT and extended chiral algebras. We added a discussion about this to the manuscript.

Below, we go through the list of requested changes in more detail.

Best regards,

Ville Lahtinen
Teresia Månsson
Eddy Ardonne

### List of changes

Requested changes
1- As mentioned above, they need to at least briefly discuss extended chiral algebras, and the relation to the pure Virasoro characters to extended characters. They should also be precise the relation of their construction to the orbifold construction.

Answer. We agree that the paper benefits from having a discussion on extended chiral algebras and the relation with the orbifold construction. One can either do this rather extensively, or concisely. We chose the latter, and briefly discuss these matters in section 2.

2- At the beginning of section 2, they omit some key words. The partition function of every 2d CFT ON THE TORUS must be modular invariant.

Answer. This is of course true, we added this, and the implication for the current setting, namely the spectrum of periodic one dimensional chains is given in terms of a modular invariant partition function.

3- Again at the beginning of section 2, for many properties of a CFT (e.g. the specific heat), it DOES NOT MATTER IF YOU ARE ON THE TORUS, and so the distinction between different modular invariants is not important. And as I said above, the literature (especially in the '80s) is filled with CFT papers discussing other modular invariants. The authors don't mention these, and don't mention any physical properties that are different (other than the excited state spectrum).

Answer. It is of course true that for many (bulk) properties, the boundary conditions do not matter, and the referee mentions the most important one, the specific heat, which is determined by the central charge. In general, however, finite temperature properties, do depend on the details of the (at least low-energy part of the) spectrum. These properties deserve more attention in the context of quantum chains. In addition, properties related to dynamics also depend on the full spectrum. We added a short discussion on this, and added references to some '80s papers, discussing different modular invariants.

4- on p.7, they say condensation makes less fields. That's not necessarily true, since in condensations, you double other fields.

Answer. Agreed, but we don't know any example where the number of fields increases. We added the word `generically'.

5- when discussing the Z_2 x Z_2 case, they need to mention GInsparg.

6- they should mention that their added terms to the Hamiltonian are conventionally called "twisted" boundary conditions, and are widely studied.

Answer. Well, the terms we add do not quite correspond to conventional twisted boundary conditions. For our construction, it is really important that the twist actually depends on the symmetry sector, which is not the case in the, indeed widely studied, twisted boundary conditions. We added a sentence towards the end of sec 2.2 to discuss this.

7- at the beginning of sec 3.1, I don't know what "on-site" Z_k symmetry means. If this is on-site, what is off-site?

Answer. We dropped the "on-site" here.

8- at the beginning of sec 4.2, they mention that the Bethe ansatz solution gives a factor of 2 in the relative Fermi velocities between the two critical points in Potts. That presumably means if you take the Hamiltonian limit of the same classical 3-state Potts model, this relation holds. That's maybe interesting, but why is this relevant here? They're not studying the classical model.

Answer. Up to this point, we were taking products of the same critical points, which meant that the velocity simply amounted to an overall scale of the energies. Here, however, we combine two different critical points, each with their own velocity. In order for the construction to work, we should rescale the on of critical points, such that both have the same velocity. This is the reason for the factor of 2 in the second term of the hamiltonian in eq. (34). Without this factor of 2, the spectrum would not be described by su(2)_3, but some deformed version thereof.

9- they should mention that the "other" critical point for 3-state corresponds to the anti-ferro Potts model. The fact that this is c=1 goes back at least to Saleur, Nucl.Phys. B360 (1991) 219

10- for general k, they shouldn't say they're generalizing the Potts chain. The conventional definition of the Potts chain is that it has S_k permutation symmetry, and it is not critical for k>4. The points they describe are usually called the Z_k parafermion critical points, and this is what their modular invariants are related to.

Answer. We tried to put things in a broader perspective, but we should of course not go against conventional nomenclature, so we updated the text to reflect this.

11- in the conclusion, they mention getting SU(2)_3 by combining ferro and anti-ferro Potts chains, and speculate this may be true for other k. That seems unlikely, unless the antiferromagnetic point for Z_k parafermions is always c=1. (If that's true, the authors should note it -- I don't know offhand the answer myself).

Answer. It is indeed true that the antiferromagnetic point for the Z_k parafermions is always c=1 (Albertini). In the mean time, it was shown that for k=5,7, the critical point is u(1)_2k, as required. We included both results in the conclusions.