Non-Bloch Edge Dynamics of Non-Hermitian Systems

The non-Hermitian skin effect dramatically alters both the spectrum and real-time dynamics of non-Hermitian systems. We investigate the edge dynamics of waves initialized in non-Hermitian lattices, where the non-Hermitian skin effect tends to localize states, leading to behavior beyond the Bloch picture. By focusing on the Lyapunov exponents that quantify wave growth or decay, we show that they are governed by saddle points in the complex momentum space. We introduce an unambiguous criterion based on "Lefschetz thimbles" to identify the dominant saddle point, overcoming the failure of the conventional energy-based criterion. This framework provides a coherent theory for the non-Hermitian edge dynamics, enabling the prediction of observable phenomena such as the non-Bloch boomerang effect-a wave packet round trip across the lattice-and a refined energy threshold for the self-healing of boundary modes.