Non-Hermitian systems and topology: A transfer-matrix perspective

Abstract

Topological phases of Hermitian systems are known to exhibit intriguing properties such as the presence of robust boundary states and the famed bulk-boundary correspondence. These features can change drastically for their non-Hermitian generalizations, as exemplified by a general breakdown of bulk-boundary correspondence and a localization of all states at the boundary, termed the non-Hermitian skin effect. In this paper, we present a completely analytical unifying framework for studying these systems using generalized transfer matrices -a real-space approach suitable for systems with periodic as well as open boundary conditions. We show that various qualitative properties of these systems can be easily deduced from the transfer matrix. For instance, the connection between the breakdown of the conventional bulk-boundary correspondence and the existence of a non-Hermitian skin effect, previously observed numerically, is traced back to the transfer matrix having a determinant not equal to unity. The vanishing of this determinant signals real-space exceptional points, whose order scales with the system size. We also derive previously proposed topological invariants such as the biorthogonal polarization and the Chern number computed on a complexified Brillouin zone. Finally, we define an invariant for and thereby clarify the meaning of topologically protected boundary modes for non-Hermitian systems. arXiv:1812.02186v3 [cond-mat.mes-hall]

Perspective on topological states of non-Hermitian lattices

“…Borgnia and collaborators [60] combined local Green's functions with the doubled Hamiltonian construction to bring together skin effects and bridge the gap for a bulk-boundary correspondence. On a different path, but with the same unifying aim, Kunst and collaborators [132] put forward a transfer matrix approach. [115] focused on lattices in one dimension with losses added as on-site terms identifying a winding number.…”

PT -symmetric non-Hermitian Dirac semimetals

“…This kind of discrepancy becomes especially crucial for topological phases, because it leads to the failure of bulk-boundary correspondence. Such discrepancy can be qualitatively captured by the transfer matrix T as it is capable of describing systems with open and periodic boundary conditions [52]. The criterion for the PBC and OBC systems to be consistent with each other is that the transfer matrix is unimodular, i.e., | det T | = 1.…”

Probing non-Hermitian skin effect and non-Bloch phase transitions

“…Among the most relevant features observed in non-Hermitian systems, one should mention the strong sensitivity of the energy spectra on boundary conditions [7,[15][16][17][18][19][20][21], the non-Hermitian skin effect (NHSE) [7,9,17,18,[20][21][22][23][24], i.e. the exponential localization of continuum-spectrum eigenstates to the edges, and the failure of the bulkboundary correspondence based on Bloch band topological invariants [4,17,18,[24][25][26][27][28][29][30][31][32][33][34][35][36][37]. Recently, several attempts have been suggested to restore the bulk-boundary correspondence, such as those based on the biorthogonal bulk-boundary correspondence [17], the non-Bloch bulk topological invariants [18,[25][26][27][28][29]35], the singular value decomposition [30], and the Green functions [31,32].…”