Non-Bloch edge dynamics of non-Hermitian lattices

The non-Hermitian skin effect, i.e., the localization of nominally bulk modes, not only drastically reshapes the spectral properties of non-Hermitian systems, but also dramatically modifies the real-time dynamics therein. Here we investigate the time evolution of waves (or quantum-mechanical particles) initialized around the edge of non-Hermitian lattices. The non-Hermitian skin effect tends to localize the wave to the edge, meaning that the real-time dynamics differs from the Bloch-theory picture. We focus on the long-time decay or growth rate of wave function, which is quantified by the Lyapunov exponents. These exponents can be obtained from the saddle points in the complex momentum space. We propose an efficient yet unambiguous criterion for identifying the dominant saddle point that determines the Lyapunov exponents. Our criterion can be precisely formulated in terms of a mathematical concept known as the Lefschetz thimble. Counterintuitively, the seemingly natural criterion based on the imaginary part of the energy fails. Our work provides a coherent theory for characterizing the real-time edge dynamics of non-Hermitian lattices. Our predictions are testable in various non-Hermitian physical platforms.