Unifying framework for non-Hermitian and Hermitian topology in driven-dissipative systems

Recently, a one-to-one correspondence between non-trivial non-Hermitian topology and directional amplification has been demonstrated, theoretically and experimentally, for the case of one complex band. Here, we extend our framework to multiple bands and higher spatial dimension. This proves to be far from trivial. Building on the singular value decomposition, we introduce a new quantity that we dub generalised singular spectrum (GSS). The GSS allows us to define physically meaningful bands related to the system's scattering behaviour and to define invariants for novel notions of point gaps (non-Hermitian topology) and line gaps (Hermitian-like topology), respectively. For both invariants, we prove a bulk-boundary correspondence and show that they give rise to two different kinds of topological edge modes. We illustrate our results with a 1D non-Hermitian Su-Schrieffer-Heeger (SSH) model and a 2D non-Hermitian model that features corner-to-corner amplification. Our work is relevant for many state-of-the-art experimental platforms and it sets the stage for applications such as novel directional amplifiers and non-reciprocal sensors.