Exceptional points are universal level degeneracies induced by non-Hermiticity. Whereas past decades witnessed their new physics, the unified understanding has yet to be obtained. Here we present the complete classification of generic topologically stable exceptional points according to two types of complex-energy gaps and fundamental symmetries of charge conjugation, parity, and time reversal. This classification reveals unique non-Hermitian gapless structures with no Hermitian analogs and systematically predicts unknown non-Hermitian semimetals and nodal superconductors; a topological dumbbell of exceptional points in three dimensions is constructed as an illustration. Our work paves the way toward richer phenomena and functionalities of exceptional points and non-Hermitian topological semimetals.Topology plays a pivotal role in the understanding of phases of matter [1]. In gapless systems such as semimetals and nodal superconductors, topology guarantees stable degeneracies accompanying distinctive excitations [2][3][4][5][6][7]. A prime example is the Weyl semimetal in three dimensions, where each gapless point is topologically protected by the Chern number defined on the enclosing surface. Symmetry further brings about diverse types of topological semimetals. Their unified understanding is developed as the classification theory according to fundamental symmetries such as PT and CP symmetries [8][9][10][11][12][13][14][15].Recently, the interplay between topology and non-Hermiticity has attracted widespread interest in a non-Hermitian extension of topological insulators and semimetals [54-72]. Non-Hermiticity ubiquitously appears, for instance, in nonequilibrium open systems [73] and correlated electron systems [68], leading to unusual properties with no Hermitian counterparts. One of their salient characteristics is the emergence of exceptional points [74-76], i.e., universal non-Hermitian level degeneracies at which eigenstates coalesce with and linearly depend on each other [77]. The past decade has witnessed a plethora of rich phenomena and functionalities induced by exceptional points [73], including unidirectional invisibility [78-81], chiral transport [82-86], enhanced sensitivity [87-92], and unusual quantum criticality [93-99]. Such exceptional points also alter the nodal structures of topological semimetals in a fundamental manner [54-72]. Notably, non-Hermiticity deforms a Weyl point and spawns a ring of exceptional points [75]. This Weyl exceptional ring is characterized by two topological charges [56], a Chern number and a quantized Berry phase, and such a multiple topological structure has no analogs to Weyl and Dirac points in Hermitian systems [2-15]. Its experimental observation has also been reported in an optical waveguide array [69]. Moreover, pseudo-Hermiticity [63], PT symmetry [62, 65], and chiral symmetry [67] enable an exceptional ring (surface) in two (three) dimensions. A symmetry-protected exceptional ring has been experimentally observed in a two-dimensional photonic crystal slab ...

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Discrete quantum walks are periodically driven systems with discrete time evolution. In contrast to ordinary Floquet systems, no microscopic Hamiltonian exists, and the one-period time evolution is given directly by a series of unitary operators. Regarding each constituent unitary operator as a discrete time step, we formulate discrete space-time symmetry in quantum walks and evaluate the corresponding symmetry protected topological phases. In particular, we study chiral and/or time-glide symmetric topological quantum walks in this formalism. Due to the discrete nature of time evolution, the topological classification is found to be different from that in conventional Floquet systems. As a concrete example, we study a two-dimensional quantum walk having both chiral and time-glide symmetries and identify the anomalous edge states protected by these symmetries.

We provide classification of gapless phases in non-Hermitian systems according to two types of complex-energy gaps: point gap and line gap. We show that exceptional points, at which not only eigenenergies but also eigenstates coalesce, are characterized by gap closing of point gaps with nontrivial topological charges. Moreover, we find that bulk Fermi arcs accompanying exceptional points are robust because of topological charges for real line gaps. On the basis of the classification, some examples are also discussed.

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