![]() ![]() ![]() We emphasize that topological phases are absent in 1D class AI in the presence of Hermiticity 3, 4, 5 non-Hermiticity induces the unique non-equilibrium topological phase as a result of the topological unification of time-reversal and particle-hole symmetries. These edge states are immune to disorder that respects time-reversal symmetry (see Supplementary Note 7 for details), which is a signature of the topological phase. In fact, once we lift the Hermiticity constraint on the Hamiltonian H, the Wigner theorem dictates that an anti-unitary operator \(\mathcal\). Here we point out that two fundamental anti-unitary symmetries, time-reversal symmetry and particle-hole symmetry, are the two sides of the same symmetry in non-Hermitian physics. However, it is yet to be understood how symmetry constrains non-Hermitian systems in general and how symmetry protects non-Hermitian topological phases. Moreover, topological lasers were proposed and realized on the basis of the interplay between non-Hermiticity and topology 39, 41, 42. Recently, a topological band theory for non-Hermitian Hamiltonians was developed, and the topological phase in the quantum Hall insulator was shown to persist even in the presence of non-Hermiticity 31. Here symmetry again plays a key role for example, spectra of non-Hermitian Hamiltonians can be entirely real in the presence of parity-time symmetry 46. In general, non-Hermiticity arises from the presence of energy or particle exchanges with an environment 43, 44, and a number of phenomena and functionalities unique to non-conservative systems have been theoretically predicted 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56 and experimentally observed 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67. Meanwhile, there has been growing interest in non-equilibrium open topological systems, especially non-Hermitian topological systems 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42. Whereas the topological phase in the quantum Hall insulator is free from any symmetry constraint and breaks down in the presence of time-reversal symmetry 6, 7, certain topological phases are protected by symmetry for example, the quantum spin Hall insulator is protected by time-reversal symmetry 8, 9, 10, 11 and the Majorana chain is protected by particle-hole symmetry 14.Äespite its enormous success, the existing framework for topological phases mainly concerns equilibrium closed systems. The AZ classification plays a key role in characterizing the topological phases 3, 4, 5 of condensed matter such as insulators 6, 7, 8, 9, 10, 11, 12 and superconductors 13, 14, 15, 16, as well as photonic systems 17 and ultracold atoms 18, all of which are classified into the periodic table 19, 20, 21, 22. Time-reversal symmetry is complemented by particle-hole and chiral symmetries, culminating in the Altland–Zirnbauer (AZ) ten-fold classification 2. ![]() It was Wigner who showed that all symmetries are either unitary or anti-unitary and identified the fundamental role of time-reversal symmetry in anti-unitary operations 1. Our work establishes a fundamental symmetry principle in non-Hermitian physics and paves the way towards a unified framework for non-equilibrium topological phases. 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. A striking consequence of this symmetry unification is the emergence of unique non-equilibrium topological phases that have no counterparts in Hermitian systems. 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. 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. Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. ![]()
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