Bulk-Boundary Correspondence in Ergodic and Nonergodic One-Dimensional Stochastic Processe

Stochastic processes are used as models of various nonequilibrium phenomena, including chemical reactions, molecular motors, and biochemical processes. In particular, we focus on Markov processes described by master equations, which can be regarded as non-Hermitian Schrödinger equations.

In this talk, we will present the bulk-boundary correspondence in one-dimensional stochastic processes, which states that the value of the topological index under the periodic boundary condition is equal to the number of the steady states under the open boundary condition. To this end, we define a winding number of master equations and provide a sketch of the proof. The key theoretical observation is the correspondence between bulk and boundary currents of exponentially localized waves, which are also used in the non-Bloch band theory. We will also show the numerical demonstration of the bulk-boundary correspondence, including an example of nonergodic systems that may have hidden symmetries and corresponding conserved quantities. Finally, we extend our results to the asymmetric simple exclusion process (ASEP), a simple example of one-dimensional many-body stochastic processes.