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Critical properties of quantum three- and four-state Potts models with boundaries polarized along the transverse field

by Natalia Chepiga

Submission summary

As Contributors: Natalia Chepiga
Preprint link: scipost_202109_00009v1
Date submitted: 2021-09-06 16:59
Submitted by: Chepiga, Natalia
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational


By computing the low-lying energy excitation spectra with the density matrix renormalization group algorithm we confirm the boundary conformal field theory predictions for the three-state Potts minimal model in 1+1D with boundaries polarized in the direction of the transverse field. We further show that the transverse-polarized boundary conditions lead to scale-invariant conformal towers of states at the critical point of the quantum four-state Potts model - a special symmetric case of the Ashkin-Teller model. Finally, we phenomeno- logically establish the duality between fixed and free, and between transverse-polarized and three-state-mixed boundary conditions at the four-state Potts critical point.

Current status:
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Submission scipost_202109_00009v1 on 6 September 2021

Reports on this Submission

Anonymous Report 1 on 2021-10-21 (Invited Report)


1- Extended numerical calculation for non diagonal boundary condition quantum chain


1- It is seems the author ignore many results in the literature. Due to that he ignores exact results, and report some results (not precise)


The author provide a numerical investigation of the conformal dimensions
happening in some special boundary conditions of the the 3- and 4-state Potts
model. He uses an extension of the Density Matrix Rennormalization Group,
proposed previously by the author previously.

The main result of the paper is the verification that the predicted operator
content predicted by Afleck, Oshikawa and Saleur, in 1998 (the reference
(21) of the paper) is correct.

I have some objections for this paper.
a) Several studies of the operator content of the Potts models were done
in the last 20-25 years, that the author do not compare. Many of the
results were obtained by exploring the fact that the model is a
representation of the Temperley-Lieb algebra ( see for example the papers
of alcaraz, Batchelor,Rittenberg, on the above period).

b) Due to the point (a) several results were derived exploring the equivalence
of energy levels of the the XXZ quantum chain and the Potts (or the
Ashkin-teller). Since the XXZ can be solved by the Bethe ansatz (for some
boundaries), the low-lying levels can be estimated for lattices quantum
chains of quite large chains, or even analytically.

c) Due to the point b) for example, (see eq. 5.11 of Ann.Phys. 182 (1988)
the exact result for the sound velocity
of the 3-state model is \sqrt{3}/2 = 0.866035..., that is different from
the result assumed in the paper (0.857), actually the same problem already
appear in a previous publication of the author (ref.15). In the 4-state
Potts model the author should also know that the exact value is \pi/4=0.78539816

d) The point c) wonders me since using only lattices up to size 14, the
authors of J.Phys.A 19,107 (1985), obtain for the sound velocity of the
3-state Potts model a value .858-0.867 (not lattice 100 like the present
paper), this is an indication that the extrapolations may not be considered prop
erly in the present paper.

e) The author also present some considerations about a model (eq. 10) that
he claims to be the Ashkin-Teller model. The quantum Ashkin-Teller has a long
story, and the operator content on several boundaries are already known. The
author would help if write the Ashkin-Teller (also the Potts modes) in terms
of the standard Z(N) operator, satisfying the Z(N) exchange algebra, and
also in terms of two coupled Ising chains. In this formulation will
be more clear what are the mixed boundary conditions.

In summary, my overall vision of the present paper is that the author did
a lot of numerical work in a problem that is already known a lot of results,
without going in the details of the literature, and re deriving proximate
results whose numerically exact results are known.
Due to the above facts I believe this paper will bring confusion in the
literature and I recommend its rejection.

  • validity: low
  • significance: good
  • originality: low
  • clarity: ok
  • formatting: acceptable
  • grammar: -

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