Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems

Abstract

Non-Hermitian systems exhibit striking exceptions from the paradigmatic bulk-boundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis)appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in non-Hermitian systems, their combined biorthogonal density penetrates the bulk precisely when phase transitions occur. This leads to generalized bulk-boundary correspondence and a quantized biorthogonal polarization that is formulated directly in systems with open boundaries. We illustrate our general insights by deriving the phase diagram for several microscopic open boundary models, including exactly solvable non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators.

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

“…Hence, the feedback control interactions define distinct phases characterized by winding numbers which exhibit opposite behaviors for lattices with γ c = 0.1 and γ c = −0.1. These behaviors manifest as localized bulk Eigen modes in finite lattices, a phenomenon known as non-Hermitian skin-effect (NHSE) [65][66][67][68][69][70][71]. As an illustration, the Eigen frequencies of a finite lattice with N = 100 masses under free-free boundary conditions are displayed as black dots in figures 4(b) and (h), while representative Eigen modes marked by the blue circles are displayed in figures 4(c) and (i).…”

“…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].…”

Perspective on topological states of non-Hermitian lattices

“…invariants based on a finite system rather than bulk states, or invariants defined in real space [127]. A similar approach, without reference to Bloch states but rather a system with open boundaries, was presented by Kunst and collaborators [59].…”

“…On the other hand, there is an intense activity in the search for a consistent classification [51][52][53][54][55], the definition of the topological invariants [56][57][58], and the search for a bulk-boundary correspondence [47,48,59,60].…”

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].…”