Periodic table for topological bands with non-Hermitian symmetries

Abstract

Classifications of symmetry-protected topological (SPT) phases provide a framework to systematically understand the physical properties and potential applications of topological systems. While such classifications have been widely explored in the context of Hermitian systems, a complete understanding of the roles of more general non-Hermitian symmetries and their associated classification is still lacking. Here, we derive a periodic table for non-interacting SPTs with general non-Hermitian symmetries. Our analysis reveals novel non-Hermitian topological classes, while also naturally incorporating the entire classification of Hermitian systems as a special case of our scheme. Building on top of these results, we derive two independent generalizations of Kramers theorem to the non-Hermitian setting, which constrain the spectra of the system and lead to new topological invariants. To elucidate the physics behind the periodic table, we provide explicit examples of novel non-Hermitian topological invariants, focusing on the symmetry classes in zero, one and two dimensions with new topological classifications (e.g. Z in 0D, Z2 in 1D, 2D). These results thus provide a framework for the design and engineering of non-Hermitian symmetry-protected topological systems. arXiv:1812.10490v2 [cond-mat.mes-hall]

Perspective on topological states of non-Hermitian lattices

“…Since the pioneering TKNN paper [2], this has been a crucial issue where progress is still underway even in Hermitian Hamiltonians [111,112]; the interested reader can find more extensive accounts on this topic in [8,10,113,114]. Different paths are currently being intensively explored in the quest for a bulkboundary correspondence that may allow for classification of the topological phases of non-Hermitian lattices [22,[115][116][117][118]. These forking paths arise because of the many possible options; the very first one is the use of the Hamiltonian or Green's functions as a starting point.…”

“…In most of these works the symmetries of the effective Hamiltonian play a crucial role. As compared with usual Hermitian systems, the symmetries are also much enriched in the non-Hermitian case [116]. Indeed, symmetries such as particle hole symmetry fork into two since complex conjugation and transposition become distinct [54], while others get unified, as antiunitary symmetries which are distinct in the Hermitian case and now can be mapped onto each other [121].…”

Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models

“…Similarly, the extension of the topological concepts developed for Hermitian quantum mechanics to these new systems has been a fruitful field of research 36 . Symmetry-based applications have been proposed [37][38][39][40] , but several notions are still actively discussed-the bulk-boundary correspondence being one 36,[41][42][43][44][45][46][47][48][49] . Indeed, the phase diagram of the same model can vary significantly depending on the choice of boundary conditions (open or periodic), a phenomenom dubbed the non-Hermitian skin effect.…”

“…The correspondence can actually be redefined in two different ways: One can redefine an effective Brillouin zone for the periodic Hamiltonian where the momentum can take complex values 45,50 ; the topological invariants computed on this new Brillouin zone are then in agreement with the phase diagram of the open system. Conversely, the correspondence can be based on the singular value decomposition (SVD) of the Hamiltonian instead of the eigenvalue decomposition 37,39,40,51 . The SVD-based phase diagrams of the open and periodic systems coincide, and topological phases are characterized by the presence of edge-localized singular zero modes.…”

Correlations in non-Hermitian systems and diagram techniques for the steady state

“…Recently, the experimental realization of topological phases in optical waveguides with dissipation [9][10][11] has motivated an intense effort to generalize central ideas of topological band-theory to the case of non-Hermitian Hamiltonians [12][13][14]. While a complete understanding of this topic is still lacking, a number of striking differences from the Hermitian case have already been established.…”