Effective Non-Hermitian Hamiltonian is a widely useful language for describing open systems. Intriguingly, non-Hermitian systems exhibit a unique bulk-boundary correspondence beyond the conventional framework of Bloch band theory. To predict the topological edge modes, topological invariants have to be defined in a generalized Brillouin zone rather than in the familiar one. We also show that the consequences of this revision of band theory are not limited to non-Hermitian topology. Among other results beyond topology, we show that the non-Bloch band theory has a natural application to the working of nonreciprocal amplification, a phenomenon that signals are amplified in a preferred propagation direction while suppressed in the reversed direction. Compact formulas for the gain and directionality of one-dimensional nonreciprocal amplifiers are obtained in the framework of non-Bloch band theory. 

References:

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[2] Lei Xiao, et al. Non-Hermitian bulk-boundary correspondence in quantum dynamics. Nat. Phys. 16, 761 (2020)

[3] Wen-Tan Xue, Ming-Rui Li, Yu-Min Hu, Fei Song, Zhong Wang, Non-Hermitian band theory of directional amplification, arXiv:2004.09529