Abstract
Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. While several properties unique to non-Hermitian topological systems were uncovered, the fundamental role of symmetry in non-Hermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects non-Hermitian topological phases. Here we show that two fundamental anti-unitary symmetries, time-reversal and particle-hole symmetries, are topologically equivalent in the complex energy plane and hence unified in non-Hermitian physics. A striking consequence of this symmetry unification is the emergence of unique non-equilibrium topological phases that have no counterparts in Hermitian systems. We illustrate this by presenting a non-Hermitian counterpart of the Majorana chain in an insulator with time-reversal symmetry and that of the quantum spin Hall insulator in a superconductor with particle-hole symmetry. Our work establishes a fundamental symmetry principle in non-Hermitian physics and paves the way towards a unified framework for non-equilibrium topological phases.
Highlights
Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians
Properties intrinsic to topological insulators can appear in the corresponding topological superconductors, and vice versa: a counterpart of the Majorana chain in a non-Hermitian insulator with time-reversal symmetry and that of the quantum spin Hall insulator in a non-Hermitian superconductor with particle-hole symmetry. We emphasize that such topological phases are absent in Hermitian systems; nonHermiticity alters the topological classification in a fundamental manner, and non-equilibrium topological phases unique to nonHermitian systems emerge as a result of the topological unification of time-reversal and particle-hole symmetries
Since a band gap should refer to an energy range in which no states exist, it is reasonable to define a band n to be gapped such that Em (k) ≠ En (k) for all the band indices m ≠ n and wavevectors k (Fig. 1)[31], which is a natural generalization of the gapped band structure in the Hermitian band theory and explains the experimentally observed topological edge states in non-Hermitian systems[36,37,38,39,41,42]
Summary
Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. For non-Hermitian Hamiltonians, of which spectra are not restricted to be real, time-reversal symmetry renders the spectra symmetric about the real axis[46], while particle-hole symmetry makes the spectra symmetric about the imaginary axis[26,33,52]; they are topologically equivalent in the complex energy plane (see Supplementary Note 2 for details). In non-Hermitian systems, by contrast, time-reversal symmetry gives real energies E 2 R or complex-conjugate pairs (E, E*), while particle-hole symmetry gives pure imaginary energies E 2 iR or pairs (E, −E*).
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