Non-reciprocal stochastic topological networks show new properties as compared to quantum counterparts

Stochastic topological systems draw from topological invariants first developed for quantum systems. While stochastic and quantum systems can share the same invariant in the bulk, their spectra differ under open boundary conditions, leading to new properties. We systematically investigate how spectra in both systems differ. Solving the spectrum analytically using Chebyshev polynomials, we find that in a 1D uniform chain and SSH model with even sites, the stochastic spectrum is given by a one-site smaller quantum system. The spectral differences are most prominent in small system sizes and large non-reciprocity, where quantum states converge while the gap increases between the steady-state and the slowest decaying state in stochastic systems. In the 2D SSH model, we find that exceptional points emerge in different areas of the spectrum in the topological phase. More broadly, this work characterizes unique physical properties that emerge from identical networks described by Laplacian and adjacency matrices respectively.