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Self-dual $S_3$-invariant quantum chains
by Edward O'Brien, Paul Fendley
This is not the current version.
|As Contributors:||Paul Fendley|
|Arxiv Link:||https://arxiv.org/abs/1912.09464v2 (pdf)|
|Date submitted:||2020-09-17 13:24|
|Submitted by:||Fendley, Paul|
|Submitted to:||SciPost Physics|
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.
Submission & Refereeing History
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Reports on this Submission
Report 2 by Eran Sela on 2020-10-18 Invited Report
In this work the authors provide an extensive study of a 1D model motivated by self-duality in the 3-state quantum Potts model. In fact, following their earlier paper Arxiv:1908.02767 entitled 'The “not-A”, RSPT and Potts phases in an S3-invariant chain', they combine this Hamiltonian with a second Hamiltonian that has the same symmetry, and study a rich family of interesting phases and phase transitions. The paper is nicely written, with a broad introduction, and a wide exposition to CFT methods, algebraic methods, and comparison to numerical input. I recommend the paper for publication after the authors have considered the optional remarks/suggestions below.
1. The c=3/2 critical theory, which the authors suggest to decompose into two CFTs with different velocities, reminds me the paper M. Sitte, A. Rosch, J.S. Meyer, K.A. Matveev, M. Garst
Phys.Rev.Lett. 102 (2009) 176404. While that model has a different symmetry, it has an emergent Lorentz symmetry. Is it possible that at very low energies the two velocities flow to equal values?
2. As a follow up, in Fig. 4 the ratio of two velocities is plotted. At what value of $L$ this is taken? (what does it mean $L \to \infty$?) It could be useful to actually show the $L$ dependence.
3. From figure 5 the authors claim that one can see oscillations in the spin-spin correlation away from the point $\lambda_1=\lambda_P/3$. First of all, at what value of $L$ is this calculation done? Now, can the authors extract numerically an actual period from a FT analysis of the data? Or is it just noise?
If this statement is correct and the oscillations are real, it will be useful to show how the wave number behaves as function of $\lambda_1$.
4. This is an optional suggestion: The separation of the c=3/2 theory into 4/5+ 7/10 sectors could be further tested by looking at the symmetry resoled entanglement (Phys. Rev. Lett. 120, 200602 (2018)). One could check how each symmetry sector scales with $L$. Since only one of the two theories carries the S_3 charge, one could fit to Eq.15 in Phys. Rev. Lett. 120, 200602 (2018), and demonstrate scaling with the c=4/5 theory.
Report 1 by Dirk Schuricht on 2020-10-8 Invited Report
The authors study the phase diagram of S_3 invariant spin chains. They find several gapped and gapless phases, locate the phase boundaries and characterise the phases using both analytical and numerical techniques. Particularly interesting is the supersymmetric c=3/2 phase they establish in part of the phase diagram. The results are very interesting and certainly should be published.
However, before publication the authors should revise the presentation of their results. In general I find the discussions quite short, with lots of references to other works and few numerical results. What I am lacking is a more detailed explanation as to how the authors reach their conclusions, and maybe also some background informations to make the presentation easier to access. More specifically I would like to suggest to address the following points:
1. The phase diagram (Fig. 1) is hard to read. Maybe a better presentation can be found by adding lines as guide to the eye (see, eg, Fig. 1 in PRB 53, 3304 (1996) for an example). Also the numerical values of the locations of the phase transitions could be included to arrive at a figure containing all main results.
2. At the beginning of Sec. 2.1 the authors state “No self-dual operator obeying the symmetries of H(θ) exists in the Potts CFT, with the least irrelevant such operator having dimension 14/5” but somehow the 2nd part of the sentence seems to contradict the first.
3. Figs. 2 are way too small. Also the plotted ratio R_1^1 is not defined. In the corresponding discussion around (10), what does the statement “are the dimensions of operator creating the states labelled by a and b respectively” mean, ie, which states are precisely considered in the specific cases shown in Fig. 2 and what are the corresponding operators? Also, how does one arrive at the predicted values of 17/12 and 10/3 given the stated scaling dimensions?
4. How are the “ratios” plotted in Fig. 4 defined? Do I understand correctly that the state E_a, here labelled by the momentum k, is varied?
5. In Sec. 3 the discussion of the possible marginal perturbations is very short and mostly refers to App. A. When looking at Tab. 1 I see three operators with eigenvalues D,D’=+1 and scaling dimension 2. However, there are other operators with D,D’=+1 and smaller scaling dimensions, so why do they not appear as perturbations since, as far as I understand, they should be self-dual as well?
6. When discussing Eq. (12) the authors state “One of these operators couples the two theories.” but how do the authors reach this conclusion? Similarly they later 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?
7. Finally, in Sec. 3 the authors suspect the existence of an incommensurate region, but some numerical data supporting this would be helpful.
8. In the caption of Fig. 5 the authors state “The incommensurability is readily apparent in the oscillations“ but an argument why oscillations are related to incommensurability would be helpful (after all there are many systems with oscillations that have other origins).
9. The authors suspect an incommensurate phase between θ ≈ −0.70π and θ ≈ −0.73π, but again some supporting data would be nice.