Slow mixing and emergent one-form symmetries in three-dimensional $\mathbb{Z}_2$ gauge theory

TL;DR

This paper investigates the slow relaxation dynamics and emergent one-form symmetries in the 3D $ ext{Z}_2$ lattice gauge theory, revealing universal entropic effects that lead to robust classical memory at finite temperature.

Contribution

It proves a lower bound on mixing times in the deconfined phase and demonstrates the emergence of one-form symmetry driven by entropic effects, with implications for classical and quantum memories.

Findings

Exponential lower bound on mixing time in the deconfined phase.

Emergent one-form symmetry due to entropic effects.

Different dynamic scaling at Higgs and confinement transitions.

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 $\mathbb{Z}_2$ lattice gauge theory (LGT) as a canonical example. We prove a lower bound on the mixing time in the deconfined phase, $t_{\text{mix}} = \exp [\Omega(L)]$, 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 time scales 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 novel entropic mechanism for memory and emergent symmetry to also bring insight into self-correcting quantum memories.