Generalized Brillouin zone and non-Hermitian band theory

The energy band theory is one of the cornerstones of condensed matter physics. It also has wide applications in other branches of physics. Recently, a number of questions from non-Hermitian physics call for a generalization of energy band theory to non-Hermitian systems. In the study of non-Hermitian topological states, it has been found that such a generalization necessitates redefinitions of certain fundamental concepts of band theory. In particular, the non-Hermitian skin effect (NHSE) causes the breakdown of Bloch-band picture and conventional bulk-boundary correspondence. To calculate the energy spectra and define topological invariants, the standard Brillouin zone gives way to the generalized Brillouin zone (GBZ). Many intriguing non-Hermitian phenomena, including the non-Hermitian skin effect, can be precisely characterized in terms of the generalized Brillouin zone. The non-Hermitian band theory based on the concept of generalized Brillouin zone, now generally known as the non-Bloch band theory, has successfully described and predicted a number of novel non-Hermitian phenomena. The present article provides a brief introduction to the main concepts of non-Bloch band theory, and its applications in the non-Hermitian bulk-boundary correspondence, Green's functions, wave dynamics, chiral damping, and non-Bloch parity-time symmetry.