Non-Hermiticity enriches topological phases beyond the existing Hermitian framework. Whereas their unusual features with no Hermitian counterparts were extensively explored, a full understanding about the role of symmetry in non-Hermitian physics has still been elusive and there has remained an urgent need to establish their topological classification in view of rapid theoretical and experimental progress. Here, we develop a complete theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies the celebrated Altland-Zirnbauer symmetry classification for insulators and superconductors. In particular, charge conjugation is defined in terms of transposition rather than complex conjugation due to the lack of Hermiticity, and hence chiral symmetry becomes distinct from sublattice symmetry. It is also shown that non-Hermiticity enables a Hermitian-conjugate counterpart of the Altland-Zirnbauer symmetry. Taking into account sublattice symmetry or pseudo-Hermiticity as an additional symmetry, the total number of symmetry classes is 38 instead of 10, which describe intrinsic non-Hermitian topological phases as well as non-Hermitian random matrices. Furthermore, due to the complex nature of energy spectra, non-Hermitian systems feature two different types of complex-energy gaps, point-like and line-like vacant regions. On the basis of these concepts and K -theory, we complete classification of non-Hermitian topological phases in arbitrary dimensions and symmetry classes. Remarkably, non-Hermitian topology depends on the type of complex-energy gaps and multiple topological structures appear for each symmetry class and each spatial dimension, which are also illustrated in detail with concrete examples. Moreover, the bulk-boundary correspondence in non-Hermitian systems is elucidated within our framework, and symmetries preventing the non-Hermitian skin effect are identified. Our classification not only categorizes recently observed lasing and transport topological phenomena, but also predicts a new type of symmetry-protected topological lasers with lasing helical edge states and dissipative topological superconductors with nonorthogonal Majorana edge states. Furthermore, our theory provides topological classification of Hermitian and non-Hermitian free bosons. Our work establishes a theoretical framework for the fundamental and comprehensive understanding of non-Hermitian topological phases and paves the way toward uncovering unique phenomena and functionalities that emerge from the interplay of non-Hermiticity and topology.

This review elaborates pedagogically on the fundamental concept, basic theory, expected properties, and materials realizations of topological superconductors. The relation between topological superconductivity and Majorana fermions are explained, and the difference between dispersive Majorana fermions and a localized Majorana zero mode is emphasized. A variety of routes to topological superconductivity are explained with an emphasis on the roles of spin-orbit coupling. Present experimental situations and possible signatures of topological superconductivity are summarized with an emphasis on intrinsic topological superconductors. CONTENTS

It is proposed that in s-wave superfluids of cold fermionic atoms with laser-field-generated effective spin-orbit interactions, a topological phase with gapless edge states and Majorana fermion quasiparticles obeying non-Abelian statistics is realized in the case with a large Zeeman magnetic field. Our scenario provides a promising approach to the realization of quantum computation based on the manipulation of non-Abelian anyons via an s-wave Feshbach resonance. PACS numbers:Introduction -Recently, there has been considerable interest in topological phases of quantum many-body systems, which are characterized by the following features [1,2,3,4,5,6,7,8]: (i) there are topologicallyprotected gapless edge states on surface boundaries of the systems, which are stable against local perturbations, (ii) for two-dimensional (2D) systems, there are quasiparticles with fractional quantum numbers (e.g., fractional charges) termed "anyons". To this time, the possibility of realizing topological phases has been studied for various states realized in condensed matter systems, such as quantum (spin) Hall states [2,3,4,5,6], vortex states of p + ip superconductors [9,10,11,12], and spin liquid states [7,13], and for cosmological systems such as axion strings [14]. The feature (ii) is particularly of interest in connection with the realization of fault-tolerant quantum computation based on the manipulation of non-Abelian anyons [7,13,15,16,17,18]. Since topological phases provide not only a novel paradigm of quantum ground states but also a potential breakthrough for technological advance, it is desirable to pursue various possible schemes for their realization.

A topological superconductor (TSC) is characterized by the topologically protected gapless surface state that is essentially an Andreev bound state consisting of Majorana fermions. While a TSC has not yet been discovered, the doped topological insulator Cu(x)Bi(2)Se(3), which superconducts below ∼3 K, has been predicted to possess a topological superconducting state. We report that the point-contact spectra on the cleaved surface of superconducting Cu(x)Bi(2)Se(3) present a zero-bias conductance peak (ZBCP) which signifies unconventional superconductivity. Theoretical considerations of all possible superconducting states help us conclude that this ZBCP is due to Majorana fermions and gives evidence for a topological superconductivity in Cu(x)Bi(2)Se(3). In addition, we found an unusual pseudogap that develops below ∼20 K and coexists with the topological superconducting state.

The existence of edge states and zero energy modes in vortex cores is a hallmark of topologically nontrivial phases realized in various condensed matter systems such as the fractional quantum Hall states, p + ip superconductors, and Z2 insulators (quantum spin Hall state). We examine this scenario for two dimensional noncentrosymmetric superconductors which allow the parity-mixing of Cooper pairs. It is found that even when the s-wave pairing gap is nonzero, provided that the superconducting gap of spin-triplet pairs is larger than that of spin-singlet pairs, gapless edge states and zero energy Majorana modes in vortex cores emerge, characterizing topological order. Furthermore, it is shown that for Rashba superconductors, the quantum spin Hall effect produced by gapless edge states exists even under an applied magnetic field which breaks time-reversal symmetry provided that the field direction is perpendicular to the propagating direction of the edge modes. This result making a sharp contrast to the Z2 insulator is due to an accidental symmetry inherent in the Rashba model. It is also demonstrated that in the case with magnetic fields, the non-Abelian statistics of vortices is possible under a particular but realistic condition.PACS numbers:

We complete a classification of topological phases and their topological defects in crystalline insulators and superconductors. We consider topological phases and defects described by non-interacting Bloch and Bogoliubov de Gennes Hamiltonians that support additional order-two spatial symmetry, besides any of ten classes of symmetries defined by time-reversal symmetry and particle-hole symmetry. The additional order-two spatial symmetry we consider is general and it includes $Z_2$ global symmetry, mirror reflection, two-fold rotation, inversion, and their magnetic point group symmetries. We find that the topological periodic table shows a novel periodicity in the number of flipped coordinates under the order-two spatial symmetry, in addition to the Bott-periodicity in the space dimensions. Various symmetry protected topological phases and gapless modes will be identified and discussed in a unified framework. We also present topological classification of symmetry protected Fermi points. The bulk classification and the surface Fermi point classification provide a novel realization of the bulk-boundary correspondence in terms of the K-theory.Comment: 58 pages, final versio

A unique feature of non-Hermitian systems is the skin effect, which is the extreme sensitivity to the boundary conditions. Here, we reveal that the skin effect originates from intrinsic non-Hermitian topology. Such a topological origin not merely explains the universal feature of the known skin effect, but also leads to new types of the skin effects -symmetry-protected skin effects. In particular, we discover the Z2 skin effect protected by time-reversal symmetry. On the basis of topological classification, we also discuss possible other skin effects in arbitrary dimensions. Our work provides a unified understanding about the bulk-boundary correspondence and the skin effects in non-Hermitian systems.Recently, non-Hermitian Hamiltonians [1-7] have been extensively studied in open classical [8][9][10][11][12][13][14] and quantum [15][16][17][18][19][20] systems as well as disordered or correlated solids with finite-lifetime quasiparticles [21][22][23][24][25][26][27]. In particular, much research has focused on distinctive characteristics of non-Hermitian topological phases . The rich non-Hermitian topology is attributed to the complex-valued nature of the spectrum, which enables two types of complex-energy gaps [56]: line gap and point gap. Since a non-Hermitian Hamiltonian with a line gap is continuously deformed to a Hermitian one without closing the line gap [56], topology for a line gap describes the persistence of conventional topological phases against non-Hermitian perturbations, which is relevant to topological lasers [40][41][42][43][44], for example. On the other hand, a non-Hermitian Hamiltonian with a point gap is allowed to be deformed to a unitary one [46,56]. As a result, point-gapped topological phases cannot always be continuously deformed into any Hermitian counterparts; topology for a point gap is intrinsic to non-Hermitian systems in sharp contrast to a line gap. A point gap describes unique non-Hermitian topological phenomena such as localization transitions [1,2,46,52,58] and emergence of exceptional points [21-26, 34, 37, 49, 50, 53, 55].A hallmark of topological phases is the presence of the localized states at the boundaries as a result of nontrivial topology of the bulk [61-63]. Remarkably, non-Hermiticity alters the nature of the bulk-boundary correspondence (BBC) . The critical distinction is the extreme sensitivity of the bulk to the boundary conditions, which is called the non-Hermitian skin effect [68]. It accompanies the localization of bulk eigenstates as well as the dramatic difference of bulk spectra according to the boundary conditions, which forces us to redefine the bulk topology so as to be suitable for the open boundary condition [67,68,78,80]. The BBC persists in the presence of a line gap since non-Hermitian Hamiltonians with a line gap can be continuously deformed to Hermitian ones. However, the BBC for a point gap has still remained unclear. Since a point gap describes intrinsic non-Hermitian topology, the nature of the BBC may be disparate from the Hermitian counterpart...

Journal of the Physical Society of Japan FULL PAPERSSuperconductivity is a phenomenon where the macroscopic quantum coherence appears due to the pairing of electrons. This offers a fascinating arena to study the physics of symmetry breaking, i.e., broken gauge symmetry. However, the important symmetries in superconductors are not only the gauge invariance. Especially, the symmetry properties of the pairing, i.e., the parity and spin-singlet / spin-triplet, determine the physical properties of the superconducting state. Recently it has been recognized that there is the important third symmetry of the pair amplitude, i.e., even or odd parity with respect to the frequency. The conventional uniform superconducting states correspond to the even-frequency pairing, but the recent finding is that the odd-frequency pair amplitude arises in the spatially non-uniform situation quite ubiquitously. Especially, this is the case in the Andreev bound state (ABS) appearing at the surface/interface of the sample.The other important recent development is on the nontrivial topological aspects of superconductors. As the band insulators are classified by topological indices into (i) conventional insulator, (ii) quantum Hall insulator, and (iii) topological insulator, also are the gapped superconductors. The influence of the nontrivial topology of the bulk states appears as the edge or surface of the sample, i.e., bulk-edge correspondence.In the superconductors, this leads to the formation of zero energy ABS (ZEABS). Therefore, the ABSs at the surface/interface of the superconductors are the place where the symmetry and topology meet each other which offer the stage of rich physics. In this review, we discuss the physics of ABS from the viewpoint of the odd-frequency pairing, the topological bulk-edge correspondence, and the interplay of these two issues.It is described how the symmetry of the pairing and topological indices determine the absence/presence of the ZEABS, its energy dispersion, and properties as the Majorana fermions. Various related issues such as the Helium 3, transport of Majorana fermions, and tunneling spectroscopies are also briefly discussed.