Analytic calculation of generalized Brillouin zone in non-Hermitian bands

In this talk, we introduce a systematic method to calculate the generalized Brillouin Zone (GBZ) analytically in one dimensional non-Hermitian systems. In general, a m-band non-Hermitian Hamiltonian is constituted by m distinct sub-GBZs, each of which is a piecewise analytic closed loop. All the analytic properties of the GBZ can be characterized by an algebraic equation, which is dubbed as auxiliary GBZ. As an application of our theory, we show a new kind of topological phase under open boundary condition, which is characterized by the non-zero energy winding number for each band.