Exceptional topology of non-Hermitian systems

Abstract

We review the current understanding of the role of topology in non-Hermitian (NH) systems, and its far-reaching physical consequences observable in a range of dissipative settings. In particular, we elucidate how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm. An immediate consequence is the ubiquitous occurrence of nodal NH topological phases with concomitant open Fermi-Seifert surfaces, where conventional band-touching points are replaced by the aforementioned exceptional degeneracies. We furthermore discuss new notions of gapped phases including topological phases in single-band systems, and clarify how a given physical context may affect the symmetry-based topological classification. A unique property of NH systems with relevance beyond the field of topological phases consists in the anomalous relation between bulk-and boundary-physics, stemming from the striking sensitivity of NH matrices to boundary conditions. Unifying several complementary insights recently reported in this context, we put together a clear picture of intriguing phenomena such as the NH bulk-boundary correspondence, and the NH skin effect. Finally, we review applications of NH topology in both classical systems including optical setups with gain and loss, electric circuits, mechanical systems, and genuine quantum systems such as electronic transport settings at material junctions, and dissipative cold-atom setups.

Full Spectrum of the Liouvillian of Open Dissipative Quantum Systems in the Zeno Limit

“…3 we show, as a function of Γ, the real part of all Liouvillian eigenvalues of a Heisenberg chain with 2 spins, the first spin being targeted by a z-polarizing dissipation. Depending on the anisotropy, we find up to eight exceptional points, where two or more eigenvalues coalesce [26][27][28][29]. Fully analytical Zeno regime sets in beyond the rightmost branching point, see Fig.…”

Emergence of PT-symmetry breaking in open quantum systems

“…Moreover, there has been considerable interest in the classifications of the symmetries of non-Hermitian Hamiltonians [19][20][21][22] and Liouvillian [49,50] lattice models for dissipative fermions. Connected to the classifications of non-Hermitian Hamiltonians, we emphasize that according to Eq.…”

“…Over the past years, this effect has attracted considerable attention and has been demonstrated in various optical [6][7][8], electrical [9] and mechanical [10] settings. Apart from purely fundamental interest, this mechanism also has many important practical consequences, for example, for the operation of multi-mode lasers [11,12], enhanced measurements [13][14][15][16][17][18], the bandstructure of dissipative lattice systems [19][20][21][22] or energy transport at macroscopic [23] and microscopic [24,25] scales.…”

Bulk-Boundary Correspondence for Non-Hermitian Hamiltonians via Green Functions

“…Moreover, genuinely non-Hermitian Hamiltonians, i.e., Hamiltonians that cannot be deformed to a Hermitian Hamiltonian without energy bands touching, have novel topological properties not found in Hermitian systems. They can be characterized by topological invariants different from those of Hermitian systems [27][28][29][30][31][32][33][34][35][36][37][38], but the extent of a bulk-boundary correspondence is, surprisingly, much less clear [39][40][41][42][43][44][45][46][47][48][49][50].…”