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

TL;DR

The paper addresses how slow relaxation and robust memory extend from conventional symmetry breaking to classical topological order, focusing on the 3D $Z_2$ lattice gauge theory. It combines a membrane representation of the partition function with rigorous bottleneck/Cheeger-type arguments to prove that Glauber dynamics in the deconfined phase exhibits $t_{\rm mix}=\exp(\Omega(L))$, and shows this slow mixing persists under symmetry-breaking perturbations due to entropic effects that drive emergent one-form symmetry. A central technical advance is the notion of conditional ensembles $Z(x,\Gamma)$ that reveal symmetry between topologically distinct sectors; this emergent symmetry is shown via duality to control bottlenecks and extend beyond the deconfined phase. The authors complement analytic results with large-scale numerics that compare Higgs- and confinement-driven transitions, finding markedly different dynamic scaling despite identical static Ising exponents, and discuss implications for self-correcting memories in both classical and quantum settings. Overall, the work provides a concrete entropic mechanism for memory and emergent symmetry in finite-temperature topological orders and outlines a path toward broader applicability in higher-form-symmetric systems.

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, $t_{\text{mix}} = \exp [Ω(L)]$, 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 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.

Figures (11)

• Figure 1: The "re-entrant" phase diagram of the 3D $\mathbb{Z}_2$ LGT in the presence of a symmetry breaking field, see \ref{['sec:background']} for details. Along the dashed line with a small but nonzero $h$, the model is in a trivial paramagnetic phase (called the Higgs phase) at $T = 0$, but at some $T > 0$ the deconfined phase reappears. We say the one-form symmetry is only emergent in the presence of thermal fluctuations, and this consequently leads to a robust, "entropic" classical memory.
• Figure 2: (a) Geometry of the 3D $\mathbb{Z}_2$ lattice gauge theory. (b) Mapping from spins to membrane representation. (c) Phase diagram of the model [\ref{['eq:Z_def_gauge_fix']}]. We also show a typical configuration for each region of the phase diagram, and indicate ranges of validity of our rigorous results. The illustrations show typical configurations within each phase, as determined by $x$ (the fugacity of membranes) and $y$ (the fugacity of membrane boundaries). That is, the membrane area is suppressed by a small $x$, and the length of membrane boundaries is suppressed by a small $y$.
• Figure 3: Example of the two homologically inequivalent classes of ground states in our model. All membranes illustrated here are boundary-less ($\partial M = \emptyset$). The two classes are distinguished by a homologically nontrivial Wilson loop along the $z$ direction, which takes values $+1$ and $-1$ respectively.
• Figure 4: Illustration of the bottleneck theorem [\ref{['thm:tmix_bottleneck']}].
• Figure 5: Geometry and notation of the two-dimensional Ising model.

Theorems & Definitions (19)

• Theorem 0
• Theorem 1: Slow mixing of $\mathbb{Z}_2$ LGT
• Lemma 2: Emergent one-form symmetry
• Proposition 4
• Proposition 5
• Proposition 6
• proof
• Proposition 7
• proof
• Theorem 8