Non-Bloch Band Theory of Non-Hermitian Systems

Kazuki Yokomizo (Tokyo Institute of Technology)

Non-Hermitian systems, which are described by non-Hermitian Hamiltonians have been attracting much attention. In particular, the bulk-edge correspondence has been intensively studied in topological systems. In contrast to Hermitian systems, it seems to be violated in some cases. The reasons for this violation is that the Bloch wave vector is treated as real in non-Hermitian systems similarly to Hermitian ones.
 In this presentation, we establish a generalized band theory in a one-dimensional tight-binding model. In particular, we explain how to determine the generalized Brillouin zone C_β for the complex Bloch wave number β=e^ik,k∈C. In contrast to Hermitian cases, where C_β is always a unit circle, in non-Hermitian systems, C_β is a closed curve, not necessarily a unit circle. Furthermore, we find that C_β can have cusps, and its shape depends on system parameters. A byproduct of our theory is that one can prove the bulk-edge correspondence between the winding number defined from C_β and existence of topological edge states in the one-dimensional non-Hermitian systems.