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Quantum criticality in many-body parafermion chains
by Ville Lahtinen, Teresia Mansson, Eddy Ardonne
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Submission summary
| Authors (as registered SciPost users): | Eddy Ardonne · Ville Lahtinen |
| Submission information | |
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| Preprint Link: | http://arxiv.org/abs/1709.04259v1 (pdf) |
| Date submitted: | 2017-10-11 02:00 |
| Submitted by: | Ardonne, Eddy |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| 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 Potts models.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2017-11-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1709.04259v1, delivered 2017-11-15, doi: 10.21468/SciPost.Report.277
Strengths
1- Thorough examination of modular invariants in Z_3 invariant systems
2- Many explicit formulas make analysis clear
3- Explicit lattice models realising this behaviour
Weaknesses
1- None of it very profound, but rather an incremental extension of previous knowledge
2- Don't explain the relation to the existing CFT literature, especially that on orbifolds
3- No interesting physical applications (potential or actual)
Report
The authors have constructed modular invariants in conformal field theory by taking copies of three-state Potts model (with a brief generalisation to Z_k parafermions), and then either ``condensing anyons'' 0r utilising permutation invariants. They explain their construction clearly, and work out numerous of examples. They show nicely how to relate the various partition functions by taking linear combinations. They also show how to realise the models as quantum ``spin'' chain Hamiltonians (although not as 2d classical models).
I think the paper is worth of publishing in SciPost. It's good to keep developing methods of conformal field theory, especially those influenced by work in other fields, such as the study of anyons. However, I don't think the results have much physical significance; they don't really explain how their allegedly exotic critical points differ in any substantive way from just taking copies of Potts models. They don't provide any examples of physical behaviour being different as a result (I am aware that there is, but they don't discuss it).
Instead, they say trite things like "Usually, when talking about a critical point being described a given CFT, one refers to the diagonal invariant". That's simply not true (perhaps the authors' friends do, but that shouldn't be extrapolated to the rest of the world). The literature certainly gives lie to that fact. As I believe the authors know (they certainly ought to), when we discusses an extended chiral algebra (such as supersymmetry), often the non-diagonal modular invariant becomes diagonal. Literally hundreds of papers were written in the CFT world about these algebras, and it spawned an entire field of mathematicas (fusion categories etc, which the authors barely mention). To give one of many examples, I note that the authors don't even refer to Ginsparg's famous paper on c=1 theories, where he very clearly explains how to do an orbifold to construct various modular invariants from each other, in exactly the same fashion as the authors describe here. Again, literally hundreds of papers were written on orbifolds, so to say that people tend to always talk about diagonal modular invariants.
Thus the authors need to clarify the relation of their results to the existing CFT literature. In particular, they need to explain how it relates to the orbifold construction; I suspect many of their procedures are completely equivalent.
In general, the authors make a number of small comments like this that I believe give the wrong impression. I list them below.
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.
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.
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).
4- on p.7, they say condensation makes less fields. That's not necessarily true, since in condensations, you double other fields.
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.
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?
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.
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.
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).