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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

Submission summary

Authors (as registered SciPost users): Jan Alexander Koziol · Kai Phillip Schmidt
Submission information
Preprint Link: https://arxiv.org/abs/2402.15389v3  (pdf)
Date accepted: 2024-07-04
Date submitted: 2024-06-11 16:11
Submitted by: Koziol, Jan Alexander
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational

Abstract

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. All findings for the toric code on the honeycomb lattice can be transferred exactly to the toric code on the triangular lattice.

Author indications on fulfilling journal expectations

  • Provide a novel and synergetic link between different research areas.
  • Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
  • Detail a groundbreaking theoretical/experimental/computational discovery
  • Present a breakthrough on a previously-identified and long-standing research stumbling block

Author comments upon resubmission

Response to the "Anonymous Report 1 on 2024-4-8 (Invited Report)":

We thank the referee for the careful examination of our manuscript and the positive and constructive feedback.

1.) We thank the referee for this comment. In the revised version we have rephrased the corresponding sentence in the abstract as well as in Sect. 2.2.

2.) We thank the referee for this comment. It indeed is helpful to clarify, that the contractible loops of $sigma^x$ matrices lie on the direct lattice, while contractible loops of $sigma^z$ matrices lie on the dual lattices. In the revised version we specified, on which lattice the loops lie.

3.) We thank the referee for this comment. We have now introduced the $T$-operators in a way that it is easier to grasp for unfamiliar readers.

4.) We thank the referee for this comment. Indeed, the data depicted in the plot was wrong, due to erroneous plotting. We corrected our error and replaced the figure with the correct one.

5.) We thank the referee for this comment. In the revised version we have removed the sentence regarding the intuition that only the charge full and empty sector are relevant. We have further stressed that we only focused on these two sectors in the article. Let us mention that we started working on a quantum Monte Carlo technique that is directly sampling the toric code in a $XZ$-field to explore the role of fractional charge fillings.

We thank the referee for these issues and the list of typos.

We have corrected all suggested typos in the revised version.



Response to the "Anonymous Report 2 on 2024-5-8 (Invited Report) ":

We thank the referee for the careful examination of our manuscript and the positive feedback.

We thank the referee for his comment. We used the term supersymmetry to refer to the symmetry of the toric code on a square lattice when interchanging x and z fields (as done frequently in the literature). In the revised version we have dropped the notion "supersymmetry" and replaced it with "symmetry" to avoid misunderstandings.

List of changes

Line 21-22: Changed "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." to "All findings for the toric code on the honeycomb lattice can be transferred exactly to the toric code on the triangular lattice."

Line 75: Changed "supersymmetry" to "symmetry"

Line 78: Changed "supersymmetry" to "symmetry"

Line 94: Changed "demonstrate" to "observe"

Line 126: Changed "Furthermore, any contractible loop of $\sigma^x$ or $\sigma^z$ matrices corresponds to the product of operators X or Z contained in the loop, respectively [15]." to "Furthermore, any contractible loop of $\sigma^z$ matrices on the honeycomb lattice or $\sigma^x$ matrices on its dual triangular lattice corresponds to the product of operators X or Z contained in the loop, respectively [15]."

Line 167: Changed "First, observe that the triangular lattice is dual to the honeycomb lattice, and that the very same spin sites reside on the links of both lattices." to "Indeed, the triangular lattice is dual to the honeycomb lattice and the very same spin sites reside on the links of both lattices."

Line 194: Changed "In the low-field expansion, we can express the parallel field as $T_0 + T_{+2} + T_{-2}$ , where the commutation relation $[H_{\rm tc} , T_n ] = nT_n$ holds true, with $n$ representing the net change in total charge and flux particle numbers resulting from the action of $T_n$" to "In the low-field expansion, we can express the field term $\sum_i h_x \sigma_i^x$ as $T_0^{\rm f} + T_{+2}^{\rm f} + T_{-2}^{\rm f}$, since a field in $x$-direction either creates a flux-pair $T_{+2}$, annihilates a flux-pair $T_{-2}$ or moves a single flux by $T_0$ on the respective bonds. Similarly, the field in $z$-direction $\sum_i h_z \sigma_i^z$ can be expressed as $T_0^{\rm c} + T_{+2}^{\rm c} + T_{-2}^{\rm c}$ , as it creates a charge-pair, annihilates a charge-pair or moves a charge on the respective bonds. In both cases, $T_n = T_n^{\rm f} +T_n^{\rm c}$ fullfills the commutation relation $[H_{\rm tc} , Tn ] = nT_n$, with $n$ representing the net change in total charge and flux particle numbers resulting from the action of $T_n$."

Line 502: Added "At this point it should be stressed, that we only considered the charge-free and charge-full sectors, while all other sectors of fractional filling were neglected. A more thorough analysis is needed, since the tools presented in this work do not allow for an examination of those sectors."

We updated Figure 4

Current status:
Accepted in target Journal

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

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