Correspondence between winding numbers and skin modes in non-hermitian systems

We establish exact relations between the winding of “energy” (eigenvalue of Hamiltonian) on the complex plane as momentum traverses the Brillouin zone with periodic boundary condition, and the presence of “skin modes” with open boundary condition in non-hermitian systems. This correspondence is exactly related to by our new understanding of generalized Brillouin zone, which is the curve defined on the complex plane and always surrounding the same number of zeros and poles of characteristic equation of the system. We show that the nonzero winding with respect to any complex reference energy leads to the presence of skin modes, and vice versa. We also show that both the nonzero winding and the presence of skin modes share the common physical origin that is the non-vanishing current through the system.