Topological phases in discrete stochastic systems

TL;DR

This paper reviews recent advances in understanding topological phases in discrete stochastic systems, highlighting their potential in biological and synthetic contexts, and discussing new theoretical insights and analytical tools.

Contribution

It introduces the study of topological states in non-equilibrium stochastic models, expanding the framework beyond traditional equilibrium quantum systems.

Findings

Topological invariants can characterize non-equilibrium stochastic systems.

Edge states require non-Hermiticity in these models.

Potential applications in biology and reconfigurable materials.

Abstract

Topological invariants have proved useful for analyzing emergent function as they characterize a property of the entire system, and are insensitive to local details, disorder, and noise. They support boundary states, which reduce the system response to a lower dimensional space and, in 2D systems, offer a mechanism for the emergence of global cycles within a large phase space. Topological invariants have been heavily studied in quantum electronic systems and have been observed in other classical platforms such as mechanical lattices. However, this framework largely describes equilibrium systems within an ordered crystalline lattice, whereas biological systems are often strongly non-equilibrium with stochastic components. We review recent developments in topological states in discrete stochastic models in 1D and 2D systems, and initial progress in identifying testable signature of topological states in molecular systems and ecology. These models further provide simple principles for targeted dynamics in synthetic systems and in the engineering of reconfigurable materials. Lastly, we describe novel theoretical properties of these systems such as the necessity for non-Hermiticity in permitting edge states, as well as new analytical tools to reveal these properties. The emerging developments shed light on fundamental principles for non-equilibrium systems and topological protection enabling robust biological function.