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Emergent XY* transition driven by symmetry fractionalization and anyon condensation

by Michael Schuler, Louis-Paul Henry, Yuan-Ming Lu, Andreas M. Läuchli

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Authors (as registered SciPost users): Andreas Läuchli · Michael Schuler
Submission information
Preprint Link: scipost_202205_00017v1  (pdf)
Date submitted: 2022-05-18 15:11
Submitted by: Schuler, Michael
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


Anyons in a topologically ordered phase can carry fractional quantum numbers with respect to the symmetry group of the considered system, one example being the fractional charge of the quasiparticles and quasiholes in the fractional quantum Hall effect. When such symmetry-fractionalized anyons condense, the resulting phase must spontaneously break the symmetry and display a local order parameter. In this paper, we study the phase diagram and anyon condensation transitions of a $\mathbb{Z}_2$ topological order perturbed by Ising interactions in the Toric Code. The interplay between the global (``onsite'') Ising ($\mathbb{Z}_2$) symmetry and the lattice space group symmetries results in a non-trivial symmetry fractionalization class for the anyons, and is shown to lead to two characteristically different confined, symmetry-broken phases. To understand the anyon condensation transitions, we use the recently introduced critical torus energy spectrum technique to identify a line of emergent 2+1D XY* transitions ending at a fine-tuned (Ising$^2$)* critical point. We provide numerical evidence for the occurrence of two symmetry breaking patterns predicted by the specific symmetry fractionalization class of the condensed anyons in the explored phase diagram. In combination with large-scale quantum Monte Carlo simulations we measure unusually large critical exponents $\eta$ for the scaling of the correlation function at the continuous anyon condensation transitions, and we further identify lines of (weakly) first order transitions in the phase diagram. As an important additional result, we discuss the phase diagram of a resulting 2+1D Ashkin-Teller model, where we demonstrate that torus spectroscopy is capable of identifying emergent XY/O(2) critical behaviour, thereby solving some longstanding open questions in the domain of the 3D Ashkin-Teller models. To establish the generality of our results, we propose a field theoretical description capturing the transition from a $\mathbb{Z}_2$ topological order to either $\mathbb{Z}_2$ symmetry broken phase, which is in excellent agreement with the numerical results.

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Reports on this Submission

Anonymous Report 2 on 2022-8-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202205_00017v1, delivered 2022-08-06, doi: 10.21468/SciPost.Report.5506


This paper studies the confinement transition (condensation transition of the “m” anyon) in a toric code model with additional Ising interactions. The interest of the model lies in the nontrivial symmetry properties of the m anyon. As a result of these, the confined regime splits into two phases with different broken symmetries (either broken Ising symmetry or broken lattice symmetry). By “ungauging” the model is mapped to an Ashkin-Teller-like model, and much of the analysis relates to this magnetic model.

I am happy to recommend publication of this high-quality article in Scipost with minimal changes. The authors apply a range of techniques to the model and all of them appear to be carried out with a high level of professionalism. The phase transitions studied are “Landau*” transitions (i.e. transitions that are related to Landau critical points by a simple gauging procedure) so in that sense they are a type of transition that is broadly understood. Nevertheless, the present example has various subtleties and the analysis is worthwhile and interesting. The work also sheds light on the Ashkin-Teller model, as it is pointed out that a known phase transition there is due to the change of sign of the 4-fold anisotropy perturbation to the XY model. Finally, the work is useful as a demonstration of a numerical tool, the “Critical Torus Energy Spectrum” (CTES), that has not been used all that widely so far. Comments:

- Eqs 1, 2. The phase diagram that is explored has two important parameters. However, the two dual Hamiltonians in Eqs 1,2 involve 9 different symbols for couplings so it is easy to get confused. The same couplings are given different names in the dual models, and one coupling seems to be introduced and then immediately set to zero. It might be easier for the reader if the same names were used in the different models?

- Excitation spectrum:

p3 It is stated that the excitation spectrum varies more between theories than do critical exponents. Perhaps the authors could add a sentence of explanation. After all, on the sphere (rather than torus) the spectrum and the exponents are exactly equivalent. Is the added value of the present approach the possibility of accessing high-lying levels, as opposed to only a few low-lying ones?

On this point, can the authors comment on how the finite-size error depends on the index of the level considered? How high can one safely go?

In fact I do not see error bars in Figs 4 and 8. Perhaps this is already commented on somewhere, but can the authors comment on an estimate of finite-size error?

- fig 2 on the x-axis the model is related to the classical 2D model at zero temperature. Is the phase diagram near the axis consistent with that?

- p11 “a strong indicator for a (weakly) first order transition”. This transition is always qualified as “weakly” first order. However, Fig 3b seems to show clear coexistence between the different phases, a conventional sign of a first order transition. Isn’t a weak first order transition one which does not show such strong indicators?

- Emergent XY* symmetry. It is known that 4-fold anisotropy is a very weakly irrelevant perturbation in the XY model

H. Shao, W. Guo, A. W. Sandvik Phys Rev Lett 124, 080602 (2020)

S. Pujari, F. Alet, K. Damle Phys Rev B 91, 104411 (2015)

which might have been expected to lead to large finite size corrections. Despite this, the CTES seems to match well with the XY model without this perturbation. Do the authors have any comment? Do they see any signs of weak irrelevance?

Fig 3 is nice.

Sec 4.1 The reader could be reminded of the correspondence between sigma_x and phi_A phi_B (cf. footnote 3 below)

p20 “it is impossible to satisfy the algebra” can the meaning in this language of the breaking of Rz be explained?

- Repetition - I had the feeling that the paper could be more concise at various points. For example on p15 when it says “this suggests that the nature of the critical point is not Ising^2 for JAT/J>0 anymore” the reader has already been told several times that it is an XY* transition. Also for example some of the basic points in 4.3.1 and 4.3.2 are the same perhaps allowing compression.

- When the idea of a “…*” phase transition is introduced, maybe some references would be useful for readers not familiar with this idea.

- p9 footnote 3 seems worth including in main text.

- typos p3 “cirquit” p22 “explicitely”

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Anonymous Report 1 on 2022-7-4 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202205_00017v1, delivered 2022-07-04, doi: 10.21468/SciPost.Report.5332


This paper presents a thorough study of the toric code with nearest and next-nearest neighbour interactions, as a case in point of a system that exhibits both Z_2 TO and SB phases, as well as a variety of continuous and discontinuous transitions between them. The study uses both the modern CTES technique (based on ED), building on earlier work in PRL 117, 210401 and in PHYSICAL REVIEW B 96, 035142 by some of the authors, as well as QMC simulations.

The most remarkable result is perhaps the discovery of XY universality in the phase transitions of a system that would have naively been assumed in the Ising class of Z_2 symmetry breaking. The presence of fractionalised excitations further promotes this to an XY^* class.

The accomplishments of this work include the extension of the catalogue of chartered CTESs, further developing this innovative technique to characterise continuous phase transitions; a toy model where the fractionalisation-enlarged critical exponent \eta^* can be computed analytically (at the (Ising^2)^* transition); and a calculation of the non-trivial fractionalisation class of the condensing anyons.

While there may not be a single outstanding and innovative result worthy of publication in its own right, I find this overall to be a solid and thorough piece of scientific research that presents a coherent set of advances of interest to various fields of strongly correlated physics, from phase transitions and critical phenomena, to topological phases of matter and the effects of fractionalisation on critical exponents when quasiparticles condense.

I am happy to recommend it for publication in SciPost.

Requested changes

I further offer a couple of minor comments:

1) the authors state that they compute the CTES at the critical point ("We tune the Hamiltonian parameters to the critical values"). However, they use system sizes so small that I would generally expect the location of the "critical point" to exhibit a substantial finite-size drift in parameter space. What values of critical parameters are used here? Do the authors use the "critical" values extrapolated in the infinite size limit? Or the parameters at which remnant features of the thermodynamic criticality are observed at the system size relevant for the CTES calculations? And does it make a difference to the CTES behaviour if one uses the former or the latter?

2) at the end of Sec.3, the authors invoke universality arguments to relate (2+1)D quantum criticality to 3D classical criticality. This analogy is based on the action of a transverse field in imaginary time being akin to a nearest-neighbour interaction in that direction. However, this does not apply to the 4-spin term, that therefore acts only within each of the 2D space slices, and the resulting 3D model is highly anisotropic. One can try to argue about the higher symmetry of a critical point to claim that this issue is immaterial and therefore the model is equivalent to the 3D classical AT model, but it may not be immediately obvious, at least to some readers. It would be good if the authors could comment on the matter, unless there is something I may be missing in my thinking?

3) One of the featured results in the paper is the enhanced XY symmetry on a critical line away from the Ising^2 point. This is reminiscent to me of the critical point of the square lattice quantum dimer model, where O(1) rotational symmetry emerges at the roughening transition, whereas away from it one only has lattice rotational symmetry. Is there something to be learnt by this analogy, or are the two only coincidentally similar? [For illustration purposes, I am thinking of Eqs.(2.27-2/28) in; however, this is not the first paper discussing it, I think.]

4) Finally, I have a personal curiosity about the torus spectroscopy technique -- which admittedly I am not very familiar with. The authors remark that it "allows to characterise the continuous quantum phase transitions with surprising accuracy, given the fact that we only use ED for up to 36 spins". On the one hand, one could say that it must therefore be a more effective probe of critical behaviour, which is a long wavelength properly. On the other hand, cynically someone could worry that we may be looking at a very local probe as proxy for long range behaviour, and thus it could be prone to corrections / errors except for fine tuned models. I guess time will tell once the CTES gets tested on more and more models. However, I was wondering whether the authors can offer any intuition about its effectiveness in systems so small (6x6!) that talking about scale invariance becomes difficult.

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  • clarity: high
  • formatting: perfect
  • grammar: excellent

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