Correspondence between Non-Hermitian Topology and Directional Amplification in the Presence of Disorder

In order for non-Hermitian (NH) topological effects to be relevant for practical applications, it is necessary to study disordered systems. Without disorder, a class of driven-dissipative cavity arrays displays directional amplification when associated with a nontrivial winding number of the NH dynamic matrix. In this work, we show analytically that the correspondence between NH topology and directional amplification holds also in the presence of disorder. We first show that a NH topological phase is preserved as long as the size of the point gap, i.e., the minimum distance of the disorderless complex spectrum from the origin, is larger than the maximum amount of disorder; the disorder is assumed to be bounded but otherwise general, i.e., complex and both local (on-site disorder) and nonlocal (disordered couplings). We then derive analytic bounds for the probability distribution of the scattering matrix elements, which show that the key features of nontrivial NH topology, namely, that the end-to-end forward (reverse) gain grows (is suppressed) exponentially with system size, are preserved in disordered systems. Our results prove that NH topology in cavity arrays is robust against disorder.