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Slow mixing and emergent one-form symmetries in three-dimensional Z2 gauge theory

Charles Stahl1, Benedikt Placke2, Vedika Khemani1, and Yaodong Li1,3

  • 1Department of Physics, Stanford University, Stanford, California 94305, USA
  • 2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
  • 3Department of Mechanical Engineering and Department of Physics, University of Washington, Seattle, Washington 98195, USA

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Phys. Rev. E 114, 034126 – Published 14 September, 2026

DOI: https://doi.org/10.1103/5ypm-x96s

Abstract

Symmetry-breaking order at low temperatures is often accompanied by slow relaxation dynamics, due to diverging free-energy barriers arising from interfaces between different ordered states. Here we extend this correspondence to classical topological order, where the ordered states are locally indistinguishable, so there is no notion of interfaces between them. We study the relaxation dynamics of the three-dimensional (3D) classical Z2 lattice gauge theory (LGT) as a canonical example. We prove a lower bound on the mixing time in the deconfined phase, tmix=exp[Ω(L)], where L is the linear system size. This bound applies even in the presence of perturbations that explicitly break the one-form symmetry between different long-lived states. This perturbation destroys the energy barriers between ordered states, but we show that entropic effects nevertheless lead to diverging free-energy barriers at nonzero temperature. Our proof establishes the LGT as a robust finite-temperature classical memory. We further prove that entropic effects lead to an emergent one-form symmetry, via a notion that we make precise. We argue that the exponential mixing time follows from universal properties of the deconfined phase, and numerically corroborate this expectation by exploring mixing timescales at the Higgs and confinement transitions out of the deconfined phase. These transitions are found to exhibit markedly different dynamic scaling, even though both have the static critical exponents of the 3D Ising model. We expect this entropic mechanism for memory and emergent symmetry to also bring insight into self-correcting quantum memories.

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Phys. Rev. E 114, 034126– Published 14 September, 2026
  • Received 18 January 2026
  • Accepted 26 August 2026
Article part of CHORUS
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