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Gaining insights on anyon condensation and 1-form symmetry breaking across a topological phase transition in a deformed toric code model

by Joe Huxford, Dung Xuan Nguyen, Yong Baek Kim

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Submission summary

Authors (as registered SciPost users): Joe Huxford · Dung Nguyen
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-12-12
Date submitted: 2023-11-29 22:34
Submitted by: Huxford, Joe
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


We examine the condensation and confinement mechanisms exhibited by a deformed toric code model proposed in [Castelnovo and Chamon, Phys. Rev. B, 2008]. The model describes both sides of a phase transition from a topological phase to a trivial phase. Our findings reveal an unconventional confinement mechanism that governs the behavior of the toric code excitations within the trivial phase. Specifically, the confined magnetic charge can still be displaced without any energy cost, albeit only via the application of non-unitary operators that reduce the norm of the state. This peculiar phenomenon can be attributed to a previously known feature of the model: it maintains the non-trivial ground state degeneracy of the toric code throughout the transition. We describe how this degeneracy arises in both phases in terms of spontaneous symmetry breaking of a generalized (1-form) symmetry and explain why such symmetry breaking is compatible with the trivial phase. The present study implies the existence of subtle considerations that must be addressed in the context of recently posited connections between topological phases and broken higher-form symmetries.

Author comments upon resubmission

Dear Editor,

We would like to resubmit a revised version of the manuscript to Scipost Physics. We thank the referees for their useful comments and are encouraged by the overall positive feedback our manuscript received. We have addressed the referees’ concerns in detail in responses to their reports and made corresponding adjustments to the manuscript. We believe that after these revisions, we have answered all of the referees’ queries, and that our work is now appropriate for publication in SciPost Physics.

Joe Huxford, on behalf of the authors

List of changes

We made the following changes to the paper:

We added a comment in Section 4 to clarify that the deformed 't Hooft loop, which plays the role of order parameter, does not need to be unitary.

We added a new Section 5 to generalize our formalism to many different commuting projector models, as well as different types of deformation. In particular, we construct a deformed version of the quantum double model and show how the more general formalism can reproduce the deformed toric code.

We added a comment above equation (2.4) to point out the derivation in Appendix D.

We added some additional comments in the conclusion section to clarify the character of the phase transition and to summarize the new section (Section 5).

We also made assorted minor changes to make mathematical notation clearer.

Published as SciPost Phys. 15, 253 (2023)

Reports on this Submission

Anonymous Report 1 on 2023-12-4 (Invited Report)


The revised version addressed my concerns and is good for publication.

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

Login to report


Joe Huxford  on 2023-11-30  [id 4163]


To highlight the changes we made between the original submission and the resubmission, we have attached a file here (differences.pdf) which shows these changes in the manuscript.