Gibbs-state phases beyond microscopic symmetry breaking: A phase invariant for the 3D toric code
Abstract
Do phases of Gibbs states admit a classification beyond symmetry breaking? We study the three-dimensional toric code with transverse and longitudinal fields that explicitly break both microscopic higher-form symmetries. We introduce the connected correlation fW of two locally decoded winding Wilson loops. Nonzero asymptotic fW rules out shallow local preparation from product states. An asymptotic optimization over the observables and physical recovery channels defines an invariant ℱ, which we prove constant within phases defined by two-way local channels with product ancillas and sublinear causal radius. The invariant equals one for the canonical zero-temperature state of the field-free toric code. Quantum Monte Carlo tests the conjectured thermodynamic values fW=1 at weak fields and low temperature and fW=0 elsewhere. The fixed-decoder correlation provides a lower bound on ℱ, so the results suggest an extended region with ℱ=1. Product states, including the infinite-temperature state and fully polarized infinite-field limits, have ℱ=0 exactly and therefore lie in a different phase from states with ℱ=1.
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BibTeX
@misc{watanabe2026gibbsstate,
title = {{Gibbs-state phases beyond microscopic symmetry breaking: A phase invariant for the 3D toric code}},
author = {Haruki Watanabe},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.00134},
eprint = {2607.00134},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.00134},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gibbs-state phases beyond microscopic symmetry breaking: A phase invariant for the 3D toric code
AU - Haruki Watanabe
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2607.00134
UR - https://arxiv.org/abs/2607.00134
AB - Do phases of Gibbs states admit a classification beyond symmetry breaking? We study the three-dimensional toric code with transverse and longitudinal fields that explicitly break both microscopic higher-form symmetries. We introduce the connected correlation f_W of two locally decoded winding Wilson loops. Nonzero asymptotic f_W rules out shallow local preparation from product states. An asymptotic optimization over the observables and physical recovery channels defines an invariant ℱ, which we prove constant within phases defined by two-way local channels with product ancillas and sublinear causal radius. The invariant equals one for the canonical zero-temperature state of the field-free toric code. Quantum Monte Carlo tests the conjectured thermodynamic values f_W=1 at weak fields and low temperature and f_W=0 elsewhere. The fixed-decoder correlation provides a lower bound on ℱ, so the results suggest an extended region with ℱ=1. Product states, including the infinite-temperature state and fully polarized infinite-field limits, have ℱ=0 exactly and therefore lie in a different phase from states with ℱ=1.
ER -