Skin effect and winding number in disordered non-Hermitian systems

Unlike their Hermitian counterparts, non-Hermitian (NH) systems may display an exponential sensitivity to boundary conditions and an extensive number of edge-localized states in systems with open boundaries, a phenomena dubbed the ``non-Hermitian skin effect.'' The NH skin effect is one of the primary challenges to defining a topological theory of NH Hamiltonians, as the sensitivity to boundary conditions invalidates the traditional bulk-boundary correspondence. The NH skin effect has recently been connected to the winding number, a topological invariant unique to NH systems. In this paper we extend the definition of the winding number to disordered NH systems by generalizing established results on disordered Hermitian topological insulators. Our real-space winding number is self-averaging, continuous as a function of the parameters in the problem, and remains quantized even in the presence of strong disorder. We verify that our real-space formula still predicts the NH skin effect, allowing for the possibility of predicting and observing the NH skin effect in strongly disordered NH systems. We use our theory to demonstrate the NH Anderson skin effect, in which a skin effect is developed as disorder is added to a clean system, as well as explain recent results in optical funnels.