Absorbing state transitions with discrete symmetries

TL;DR

This paper explores the conditions under which stable absorbing phases can exist in one-dimensional classical stochastic systems with discrete symmetries, revealing that nonlocal feedback can stabilize such phases and identifying a new universality class.

Contribution

It demonstrates that nonlocal feedback mechanisms can stabilize absorbing phases in classical systems with multiple discrete symmetries, introducing a new universality class for the phase transition.

Findings

Two-state models exhibit a known absorbing-active transition.

Three-state models with local rules lack a robust absorbing phase due to branching.

Nonlocal feedback stabilizes the absorbing phase, leading to a new universality class.

Abstract

Robust phases of matter, which remain stable under small perturbations, are of fundamental importance in statistical physics and quantum information. Recent advances in interactive quantum dynamics have led to renewed interest in out-of-equilibrium dynamical phases and associated phase transitions in both classical and quantum many-body systems. Motivated by these developments, we investigate whether a stable absorbing phase can exist in one-dimensional classical stochastic systems, with local update rules, in the presence of fluctuations. We study models with multiple absorbing states related by discrete symmetries, such as Z2 for two-state systems, and Z3 or S3 for three-state systems. In these models, domain walls perform random walks and coarsen under local rules, which, if perfect, eventually bring the system to an absorbing state in polynomial time. However, imperfect feedback can cause domain walls to branch, potentially leading to an opposing active phase. While two-state models exhibit a well-known transition between absorbing and active phases as the branching rate increases, in three-state models with only local dynamics, branching is a relevant perturbation, ruling out a robust absorbing phase under purely local rules. However, we discover that by incorporating nonlocal information into the feedback, the absorbing phase can be stabilized, with the transition between the active and absorbing phases belonging to a new universality class. Finally, we outline how these classical rules can be implemented using deterministic quantum circuits and discuss their connections to passive error correction.