Real-Time Edge Dynamics of Non-Hermitian Lattices

We derive the asymptotic forms of the Green's function at the open edges of general non-Hermitian band systems in all dimensions in the longtime limit, using a modified saddle point approximation and the analytic continuation of the momentum. The edge dynamics is determined by the ``dominant saddle point,'' a complex momentum which, contrary to previous conjectures, may lie outside the generalized Brillouin zone. From this result, we obtain the effective edge Hamiltonians that evidently, as demonstrated by extensive numerical simulations, characterize the dynamics on the edges and can be probed in real-time experiments or spectroscopies.