Topological and nonlinearity-induced thermalization in a PT-symmetric split-Langevin bath

Open classical systems with balanced, separated gain and loss, called PT-symmetric systems, have been extensively studied over the past decade. Here, we investigate the properties of a uniform, harmonic chain with spatially separated viscous loss and stochastic gain that are only statistically balanced. We show that such a "split Langevin" bath leads to either the absence of thermalization or non-equilibrium steady states with inhomogeneous temperature profile, both of which are understood in terms of normal modes of the chain. With a Su-Schrieffer-Heeger (SSH) chain, a canonical model with topological edge modes, we show that the steady-state properties reflect the topological phase of the underlying chain. We also show that nonlinearity stabilizes the amplifying modes in a harmonic chain, thereby leading to thermalization irrespective of the gain and loss locations. Our results expand the pool of possible realizations of non-Hermitian models to the stochastic domain.