# Self-dual $S_3$-invariant quantum chains

### Submission summary

 As Contributors: Paul Fendley Arxiv Link: https://arxiv.org/abs/1912.09464v3 (pdf) Date accepted: 2020-12-10 Date submitted: 2020-11-25 12:17 Submitted by: Fendley, Paul Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We investigate the self-dual three-state quantum chain with nearest-neighbor interactions and $S_3$, time-reversal, and parity symmetries. We find a rich phase diagram including gapped phases with order-disorder coexistence, integrable critical points with U(1) symmetry, and ferromagnetic and antiferromagnetic critical regions described by three-state Potts and free-boson conformal field theories respectively. We also find an unusual critical phase which appears to be described by combining two conformal field theories with distinct "Fermi velocities". The order-disorder coexistence phase has an emergent fractional supersymmetry, and we find lattice analogs of its generators.

Published as SciPost Phys. 9, 088 (2020)

We thank the referees for their detailed comments on the paper. We have implemented as many of their suggestions as we could, and so we hope the paper is now ready for publishing in SciPost.

### List of changes

We listed almost all of the changes in the reply to the referees' reports. In addition, we made a few more minor pedagogical improvements and corrected a few typos.

### Submission & Refereeing History

Resubmission 1912.09464v3 on 25 November 2020
Submission 1912.09464v2 on 17 September 2020

## Reports on this Submission

### Report 1 by Dirk Schuricht on 2020-11-28 (Invited Report)

• Cite as: Dirk Schuricht, Report on arXiv:1912.09464v3, delivered 2020-11-28, doi: 10.21468/SciPost.Report.2239

### Report

The authors have revised the manuscript and thereby increased its readability considerably. As far as I see they have addressed all the comments by the referees (except for one) and made good improvements accordingly. Thus in principle I support publication, but would like to ask again my previous question: At the end of Sec. 3 the authors state “the Fermi velocity vTCI in Figure 4 quite clearly is vanishing”, but to me the ratio of velocities in Fig. 4 stays finite everywhere. So how should I understand this? Maybe the authors still want to comment on this.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Paul Fendley  on 2020-11-28

(in reply to Report 1 by Dirk Schuricht on 2020-11-28)

Sorry for being slightly imprecise. We mean that it is quite clearly heading toward zero as $\theta$ is increased, and if extrapolated vanishes at approximately the value ($\sim .87\pi$) where the incommensurate phase begins. Although for reasons indicated in our first reply, doing numerics near or in the incommensurate phase is too difficult for us, we think the vanishing is a pretty reasonable inference from the figure. It also fits in with our other studies of the incommensurate phase, so I hope we can be forgiven for our slight imprecision.