Perturbatively Stable Self-Correcting Classical Memory from Gauge Averaging

Classical systems that store information in thermal equilibrium without active error correction are known as self-correcting memories. While Wegner's 3d Z2 gauge theory is a known candidate for such memory, it has been unclear whether this property is robust against arbitrary small perturbations to the Hamiltonian. This paper investigates whether the memory phase is perturbatively stable or if it breaks down under general symmetry-breaking interactions.

The author introduces a technique called "gauge averaging," which allows for the systematic replacement of a symmetry-breaking perturbation with an effective, symmetry-preserving interaction. By leveraging the extensive local symmetries of the unperturbed Wegner gauge theory, the author shows that thermal fluctuations can effectively restore these symmetries. The study uses a cluster expansion to prove that as long as the perturbation is weak relative to the temperature, the system remains in a self-correcting memory phase. The author then applies this to the specific case of a magnetic field perturbation, providing a quantitative estimate for the phase boundary that aligns closely with numerical results.

The study proves that the self-correcting memory of 3d Wegner gauge theory is a fluctuation-stabilized phase. Contrary to the intuition that perturbations might destroy the memory, the author demonstrates that the robustness of the memory actually increases with temperature up to a critical point. This confirms that the topological order in the model is not merely a feature of the unperturbed Hamiltonian but defines a stable phase of matter that persists under a wide range of local perturbations.

This work provides a rigorous foundation for understanding the stability of classical topological memories. By showing that local symmetries can be restored by fluctuations, the paper offers a powerful tool for analyzing the robustness of other physical systems. It also bridges the gap between theoretical proofs and numerical observations, offering a new perspective on how topological phases survive in the presence of realistic, non-ideal interactions.