Classification of topological phases of matter in stochastic systems

Abstract

Topological phases of matter support edge states protected from noise and perturbations, and the classification of which symmetry and dimension support such phases was developed for quantum systems and analogous platforms. While topological phases have also been discovered in stochastic systems and posited as mechanisms for biochemical processes, a classification consistent with their Markovian constraints remains lacking, impeding generalization to new scenarios. These constraints alter the matrix space preventing the application of previous classification methods, while hosting new properties such as the necessity of non-Hermiticity for non-trivial topological phases. We introduce new methods including a new homotopy approach and redefinition of the point gap and find that only two symmetry classes remain robust: no symmetry and pseudo-Hermiticity. In these symmetry classes, we identify the dimensions with topologically non-trivial phases and their group structure, creating a rigorous framework for predicting robust behavior in active and living matter.