Edge States and Topological Invariants of Non-Hermitian Systems

Abstract

The bulk-boundary correspondence is among the central issues of non-Hermitian topological states. We show that a previously overlooked "non-Hermitian skin effect" necessitates redefinition of topological invariants in a generalized Brillouin zone. The resultant phase diagrams dramatically differ from the usual Bloch theory. Specifically, we obtain the phase diagram of the non-Hermitian Su-Schrieffer-Heeger model, whose topological zero modes are determined by the non-Bloch winding number instead of the Bloch-Hamiltonian-based topological number. Our work settles the issue of the breakdown of conventional bulk-boundary correspondence and introduces the non-Bloch bulk-boundary correspondence.

Perspective on topological states of non-Hermitian lattices

“…This anomalous localization was attributed to the proximity (in parameter space) to exceptional points with an order scaling with the system size in [48], an effect which was interpreted as an environment-mediated interaction effect. The same effect was confirmed analytically in [49] for a non-Hermitian SSH model with asymmetric hoppings and dubbed the non-Hermitian skin effect. Recently, Lee and Thomale [66] presented a characterization of these peculiar boundary modes and provided conditions for their existence.…”

“…Together with Floquet systems, non-Hermitian lattices represent our first strong steps on the land of nonequilibrium, dynamical, topological phases. The departure from the Hermitian paradigm brings many surprises, like the fact that because of the extreme sensitivity to boundary conditions, a pristine non-Hermitian lattice may become devoid of extended states [47,48], an effect termed the non-Hermitian skin effect [49,50].…”

Dynamics and topology of non-Hermitian elastic lattices with non-local feedback control interactions

“…Further observations of a seemingly breakdown of the bulk-boundary correspondence principle [62,63] has led to proposals for a general classification of the topological phases of non-Hermitian systems [55,56,64]. A particular point of interest is the observation of the non-Hermitian skin effect [65][66][67][68][69][70][71], whereby all Eigen states of one-dimensional (1D) systems are localized at a boundary, in sharp contrast with the extend Bloch modes of Hermitian counterparts. This intriguing feature of non-Hermitian lattices has recently been experimentally demonstrated using topo electrical circuits [72] and quantum walks of single photons [73].…”

Wannier-type photonic higher-order topological corner states induced solely by gain and loss

“…Topological phases and topological phase transitions in fermionic and bosonic systems, described by Hermitian Hamiltonians, have attracted great interests in past three decades [1][2][3]. Recent studies have revealed that topological phases can be extended to non-Hermitian systems beyond the scope of closed systems [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Especially, the interplay between non-Hermiticity and topological states leads to unique properties that have no counterparts in Hermitian systems [20,21].…”