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Quantum robustness of the toric code in a parallel field on the honeycomb and triangular lattice

by Viktor Kott, Matthias Mühlhauser, Jan Alexander Koziol, Kai Phillip Schmidt

This is not the latest submitted version.

Submission summary

Authors (as registered SciPost users): Jan Alexander Koziol · Kai Phillip Schmidt
Submission information
Preprint Link:  (pdf)
Date submitted: 2024-03-08 14:29
Submitted by: Koziol, Jan Alexander
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational


We investigate the quantum robustness of the topological order in the toric code on the honeycomb lattice in the presence of a uniform parallel field. For a field in $z$-direction, the low-energy physics is in the flux-free sector and can be mapped to the transverse-field Ising model on the honeycomb lattice. One finds a second-order quantum phase transition in the 3D Ising$^\star$ universality class for both signs of the field. The same is true for a postive field in $x$-direction where an analogue mapping in the charge-free sector yields a ferromagnetic transverse-field Ising model on the triangular lattice and the phase transition is still 3D Ising$^\star$. In contrast, for negative $x$-field, the charge-free sector is mapped to the highly frustrated antiferromagnetic transverse-field Ising model on the triangular lattice which is known to host a quantum phase transition in the 3D XY$^\star$ universality class. Further, the charge-free sector does not always contain the low-energy physics for negative $x$-fields and a first-order phase transition to the polarized phase in the charge-full sector takes place at larger negative field values. We quantify the location of this transition by comparing quantum Monte Carlo simulations and high-field series expansions. The full extension of the topological phase in the presence of $x$- and $z$-fields is determined by perturbative linked-cluster expansions using a full graph decomposition. Extrapolating the high-order series of the charge and the flux gap allows to estimate critical exponents of the gap closing. This analysis indicates that the topological order breaks down by critical lines of 3D Ising$^\star$ and 3D XY$^\star$ type with interesting potential multi-critical crossing points. We further demonstrate that all findings for the toric code on the honeycomb lattice can be transferred exactly to the toric code on a triangular lattice.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2024-5-8 (Invited Report)


(1) The found behaviors are very interesting.

(2) This is an extremely solid and well written work. The calculations are extensive and very well explained.


There are no real marked weaknesses.


I enjoyed learning of these results. The behaviors found by the authors for the honeycomb and (its dual) triangular lattice toric code models in a field are notable. The rich phenomenology with very different behaviors (3D Ising* and 3D XY*) appearing for positive and negative single parallel fields h_x, including the two distinct transitions for negative fields h_x were quite unexpected and illuminating at this end. The asymmetry between x and z fields and that between the charge and flux excitations that are associated with the different (dual) lattices is important.

The paper is extremely solid work. The detailed duality maps and numerical calculations are well explained and written in a self-contained way.

I strongly recommend the publication of this work with no additional requested changes at this end. Inasmuch as the general listed expectations, this work ~ "opens a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work" and satisfies all of the required acceptance criteria.

Apart from the earlier comments, I had one other remark.

What is meant by "supersymmetry"? At this end, the symmetry that appears to be referred to under this name (that associated with the self-duality of the square lattice) does not immediately translate to conventional supersymmetry. If the intention of the authors was to implicitly discuss holes doped into the toric code model (which may indeed be supersymmetric, then this might be made explicit.


Publish (easily meets expectations and criteria for this Journal; among top 50%)

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2024-4-8 (Invited Report)


This is a nicely structured and well-written paper that presents a study of the honeycomb (equivalently, triangular) lattice toric code model in presence of longitudinal (x,z) fields. For completeness, the paper reviews several known results. The methodology is however relatively novel in this setting (SSE QMC, high-field series expansion and perturbative linked-cluster expansion with full graph decomposition), as are some of the findings -- including a thorough analysis of the scaling exponents, and the uncovering of an intriguing first order transition from the clock-ordered phase to the polarised phase for negative x-field values. The potential for multicritical behaviour is also intriguing.

Overall, I would like to recommend this manuscript for publication in SciPost Physics. It certainly meets all general acceptance criteria. I am asked to justify at least one of the 4 available expectations. I feel that "groundbreaking discovery" and "breathrough on a previous research stumbling bloch" do not apply, nor does really the "synergetic link between different areas". I am happy to make the case for a "new pathway in an existing research direction, with potential for multipronged follow-up", on the grounds of the comprehensive study and the application of techniques not commonly used in this research setting, and the discovery of a first order transition and possible multicritical behaviour. It could lead to interesting follow up work, but the case is admittedly not very strong in my opinion. No matter, I remain happy to recommend publication.

I have a couple of major points that I would like the authors to take into consideration (and a few possible typos spotted on the way):

1) in the abstract and in the manuscript, the authors claim that they "... demonstrate that all findings for the toric code on the honeycomb lattice can be transferred exactly to the toric code on the triangular lattice". I find this statement a bit strong and I would invite the authors to state it as an observation / fact rather than a demonstration. While it is perhaps true that no one has written it down on published paper before (I am not sure), it follows from a well-known and trivial property: the medial (i.e., bond-centred dual) lattice of the honeycomb and triangular lattice is the same (namely, the kagome lattice). If I am not mistaken, this is the key point in Sec.2.2, except then wishing to swap x and z spin components, which requires an additional rotation in spin space, resulting in the relabelling of the x and z components of the applied field (and putting a minus sign to the y component). Such mappings are extensively used in stat mech (thinking of Baxter's work on soluble models, off the top of my head, several decades ago).

2) everywhere the authors are careful with direct and dual lattice / strings / loops, except in between Eq.(4) and (5), where they talk about "any contractible loop of sigma^x or sigma^z matrices". It may be worth clarifying here also that in the latter case one ought to consider loops of the direct lattice (i.e., closed paths along the bonds) whereas in the former case one ought to consider loops of the dual lattice (i.e., closed paths that cross each bond of the direct lattice in its middle point, normal to it).

3) in Sec.3.1 the notation T_0, T_{\pm 2} is used without being properly introduced. It would be good to add a few lines of explanation / clarification for the unfamiliar reader.

4) in Fig.4, why is the extrapolated value visibly *not* on the linear fit line? Shouldn't it be the L-->infinity limit of said line?

5) In Sec.4, the authors find a direct first order transition between the clock-ordered phase and the polarised phase by investigating the behaviour of the charge-free and charge-full sectors only. The authors claim that, intuitively, other charge sectors do not alter the ground-state phase diagram. This is not immediately intuitive to me, and I was wondering if they could give a few further words of explanation.
Given the curvature of the Pade result in Fig.6. One could imagine energies of finite charge density phases to also have positive curvature but sufficiently low overall value to intersect the blue curve to the left of the current transition and the red curve to the right of the current transition, thus inducing an intermediate phase with finite charge density. As a matter of fact, one could intriguingly speculate that there could be a staircase of phases of different finite-charge-density phases in between.
Would it be much work to try, for example, the half-charge-filling case and see where the corresponding energy curve lies in Fig.6?

Possible typos:

Eq.(23) q_m in the denominator -- the m should be capitalised?

"To ensure, that" -- remove the comma?

"approximmants" -- typo

"quantum cluster update cluster in space" -- possible typo?

"descibed" -- typo

"physics of both, the low-field" -- I think that the comma is a typo

"phase transition is shift to large negative" -- typo, maybe "shifts"?

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
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