Symmetry and Topology in Non-Hermitian Physics

Non-Hermiticity enriches topological phases beyond the existing framework for Hermitian topological phases. 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 non-Hermitian topological phases. We demonstrate that non-Hermiticity ramifies the celebrated Altland-Zirnbauer symmetry classification for insulators and superconductors. In particular, charge conjugation is unitary rather than antiunitary 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 class. Taking into account sublattice symmetry or pseudo-Hermiticity as an additional symmetry, the total number of symmetry classes is 38 rather than 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, multiple topological structures appear for each symmetry class and each spatial dimension, which are also illustrated in detail with concrete examples. Recently observed lasing and transport topological phenomena are categorized into our classification. Our theory also provides topological classification of Hermitian and non-Hermitian free bosons.

Here the essential distinction between Hermitian and non-Hermitian systems is the degrees of freedom that we have access to; nonunitary operations forbidden in Hermitian systems can be performed in non-Hermitian systems. In other words, a change in the spectrum from real to complex increases the number of the parameters that describe the system. Since topology crucially depends on the underlying manifold, non-Hermiticity is expected to alter the topological classification of insulators and superconductors [190][191][192][193][194][195][196][197][198][199][200][201][202]. In fact, the emergent non-Hermitian topological phases [118] do imply such a change in the topological classification. Remarkably, a recent work [114] proposed classification of non-Hermitian topological systems on the basis of two antiunitary symmetries. Under this classification, however, topological phases are absent in two dimensions due to its strict definition of the complex-energy gap, which seems to conflict with the recent theoretical [8,9,94,109,111,113,116,118] and experimental [139] works in two dimensions. Moreover, Ref. [114] does not take into account the so-called pseudo-Hermiticity [203][204][205], which is a generalization of Hermiticity and parity-time symmetry [1]. Notably, pseudo-Hermiticity is a possible constraint unique to non-Hermitian systems and may provide a novel topological feature [94]. Therefore, it has still been elusive how non-Hermiticity alters the topological classification of insulators and superconductors. In view of the rapid theoretical and experimental advances in non-Hermitian physics, there has been a great interest and an urgent need for comprehensive topological classification that provides a reference point for experiments and predicts novel non-Hermitian topological phases.
This work provides complete classification of non-Hermitian topological systems based on all the internal symmetries. In non-Hermitian physics, fundamental concepts such as symmetry and energy gaps dramatically change compared with the conventional ones in Hermitian physics. We first organize the internal symmetries in Sec. II; it is shown that symmetry ramifies due to the distinction between transposition and complex conjugation for non-Hermitian Hamiltonians, which culminates in the 38-fold symmetry classification in contrast to the 10-fold AZ symmetry classification in Hermitian systems. In particular, we demonstrate that particle-hole symmetry should be defined with transposition and hence unitary as Eq. (7), rather than complex conjugation in previous literature. Similarly, chiral symmetry and sublattice symmetry become distinct from each other in non-Hermitian physics, although they are equivalent in the presence of Hermiticity. Moreover, the 38-fold symmetry classification naturally includes pseudo-Hermiticity [203][204][205], which provides a novel topological structure unique to non-Hermitian systems. We note that the Bernard-LeClair symmetry classification [206], which was previously considered to describe non-Hermitian random matrices [206,207] and non-Hermitian topological phases [26,94,125], only partially reproduces our 38-fold symmetry classification. In fact, the previous symmetry classification overcounted some and overlooked others of our non-Hermitian symmetry classes, as discussed in detail in Sec. II E. Our 38-fold symmetry classification thus serves as a non-Hermitian generalization of the renowned AZ symmetry classification for Hermitian Hamiltonians. We next show in Sec. III that an extension of the energy gap for non-Hermitian Hamiltonians is not unique due to the complex nature of the energy spectrum: it can be either point-like (zero-dimensional) or line-like (one-dimensional) in the complex-energy plane (Fig. 1). Importantly, the definition that should be adopted depends on individual physical situations, and the two definitions are independent of and complementary to each other. On the basis of the clarified definitions of the symmetry and complex-energy gaps in addition to K -theory [208], we provide in Sec. IV complete topological classification of non-Hermitian insulators and superconductors for all the 38 symmetry classes and two types of the complex-energy gap. The results are summarized as periodic tables III-IX. The crucial idea behind this topological classification is that the complex-spectral-flattening procedures differ according to the type of the complex-energy gap: a non-Hermitian Hamiltonian can be flattened to a unitary matrix in the presence of a point gap, whereas it can be flattened to a Hermitian or an anti-Hermitian matrix in the presence of a line gap (Fig. 2). Remarkably, there appear multiple topological structures in each symmetry class and each spatial dimension as a unique non-Hermitian feature, which is also illustrated with an example in Sec. IV D. As discussed in Sec. V, our classification describes the non-Hermitian topological phases observed in recent experiments [130,131,[133][134][135][136][137][138][139][140][141], which are not fitted into the previous classification [114] for the lack of complete understandings about symmetry and complex-energy gaps in non-Hermitian physics. As a crucial byproduct, our non-Hermitian theory also provides the topological classification of Hermitian and non-Hermitian free bosons as shown in Sec. VI. We conclude this work in Sec. VII.

II. SYMMETRY
For Hermitian Hamiltonians, internal (nonspatial) symmetries fall into the AZ symmetry class [189]: time-reversal symmetry (TRS), particle-hole symmetry (PHS), and chiral symmetry (CS), where TRS and PHS are antiunitary, whereas CS is unitary. These symmetries lead to the 10-fold classification of Hermitian topological insulators and superconductors [190][191][192]. On the other hand, it is nontrivial whether the AZ symmetry fully describes all the internal symmetries even in the presence of non-Hermiticity. In fact, PHS is defined by unitary operation as Eq. (7) and cannot be antiunitary any longer for non-Hermitian Hamiltonians due to the distinction between transposition and complex conjugation. Because PHS is no longer antiunitary, CS does not coincide with sublattice symmetry, although they are equivalent in the presence of Hermiticity. As a consequence, the total number of symmetry classes is 38 as shown below, each of which describes intrinsic non-Hermitian topological phases as well as non-Hermitian random matrices.

A. AZ symmetry
We consider a generic noninteracting fermionic system described by the following second-quantized non-Hermitian Hamiltonian where the matrix H is a first-quantized (single-particle) non-Hermitian Hamiltonian. In addition, (ψ i ) i=1,2,··· is a set of fermion annihilation operators for a normal system and a Nambu spinor for a superconductor, which satisfies the canonical anticommutation relations Time-reversal operation is described by an antiunitary operatorT that acts on the fermion operators aŝ where T + is a unitary matrix (T + T † + = T † + T + = 1). Then time-reversal invariance of the second-quantized Hamil-tonianTĤT −1 =Ĥ leads to in real space, and in momentum space, where H (k) is a Bloch-Bogoliubovde Gennes (BdG) Hamiltonian. This action on a singleparticle non-Hermitian Hamiltonian by TRS is the same as that on a Hermitian one [189]. We here note that our discussion can also be applied to the generalized non-Bloch wave functions [113], as long as the corresponding symmetry is respected with complex wavenumbers. PHS is associated with charge conjugation that mixes fermion creation and annihilation operators and generally appears in superconductors and superfluids. It is described by a unitary operatorĈ that acts on the fermion operators asĈψ where Then the presence of PHS for the second-quantized Hamilto-nianĈĤĈ −1 =Ĥ leads to in real space, and in momentum space. Remarkably, in the presence of Hermiticity (Ĥ † =Ĥ), this PHS condition is equivalent to and hence PHS becomes antiunitary [189]. For non-Hermitian Hamiltonians, however, transposition and complex conjugation do not coincide with each other and thus PHS is not antiunitary but unitary.
As a combination of TRS and PHS, CS is defined by an antiunitary operatorΓ :=TĈ. The invariance of the HamiltonianĤ underΓ imposes the following condition on a single-particle Hamiltonian: in real space, and in momentum space. This CS condition is equivalent to ΓH (k) Γ −1 = −H (k) in the presence of Hermiticity (Ĥ † =Ĥ) [189], but it is not for non-Hermitian Hamiltonians. For instance, the Su-Schrieffer-Heeger model [142] with balanced gain and loss [97,106,133,138] respects CS.
The three symmetries T + , C − , and Γ constitute a natural and physical extension of the AZ symmetry class for non-Hermitian Hamiltonians (Table I), which respectively act on a Bloch-BdG non-Hermitian Hamiltonian as Eqs. (4), (7), and (9). The 10-fold AZ symmetry class is divided into two complex classes that only involve CS and eight real classes where TRS and PHS are relevant. We again emphasize that the physical PHS is not antiunitary but unitary for non-Hermitian Hamiltonians, and the definition of CS also changes correspondingly. I. AZ and AZ † symmetry classes for non-Hermitian Hamiltonians. Time-reversal symmetry (TRS) and particle-hole symmetry (PHS) are defined by T+H * (k) T −1 , respectively. Chiral symmetry (CS) is a combined symmetry of TRS and PHS defined by ΓH † (k) Γ −1 = −H (k) with Γ 2 = 1. The 10-fold AZ symmetry class is divided into two complex classes that only involve CS and eight real classes where TRS and PHS are relevant. Moreover, TRS † and PHS † are respectively defined by C+H T (k) C −1 , which constitute the AZ † symmetry classes. Class AI (AII) in the real AZ symmetry class and D † (C † ) in the real AZ † symmetry class are equivalent to each other.
In contrast to the Hermitian case, there arise internal symmetries other than the AZ symmetry class. In fact, as a result of the distinction between transposition and complex conjugation for a non-Hermitian Hamiltonian (H T = H * ), a variant of TRS can be defined with transposition by where C + is a unitary matrix (C + C † + = C † + C + = 1). Similarly, a variant of PHS can be defined with complex conjugation by where T − is a unitary matrix (T − T † − = T † − T − = 1). In the following, we denote the symmetry described by Eq. (10) as TRS † and the symmetry described by Eq. (11) as PHS † , since TRS † (PHS † ) is defined by Hermitian conjugation of TRS (PHS). For Hermitian Hamiltonians (H = H † ), TRS and PHS respectively coincide with TRS † and PHS † ; however they do not in the presence of non-Hermiticity. We note that PHS † is equivalent to "non-Hermitian particle-hole symmetry" in Refs. [56,60,95,98,112,114,118], which plays an important role in a single-mode laser [56] and a flatband [60] in photonics.
TRS † and PHS † in addition to CS also constitute the 10-fold symmetry class, which we call the AZ † symmetry class (Table I). This AZ † symmetry class is again divided into two complex classes that only involve CS and eight real classes where TRS † and PHS † are relevant. Here each complex AZ † class coincides with the corresponding complex AZ class. Moreover, class AI in the real AZ class and class D † in the real AZ † class are equivalent, since when a non-Hermitian Hamiltonian H respects TRS, another non-Hermitian Hamiltonian iH respects PHS † [118]. Similarly, class AII in the real AZ class and class C † in the real AZ † class are equivalent.

C. Sublattice symmetry
Another important internal symmetry is sublattice symmetry (SLS), which is defined for a Bloch-BdG Hamiltonian by where S is a unitary matrix (SS † = S † S = 1). For instance, SLS appears in a bipartite lattice where particle hopping only connects sites on different sublattices, such as the Su-Schrieffer-Heeger model [142] with asymmetric hopping [100,106,108,110,111,113]. Remarkably, SLS coincides with CS defined by Eq. (9) in the presence of Hermiticity (H = H † ) [189], but it does not for non-Hermitian Hamiltonians. SLS can be considered as an additional symmetry to the AZ symmetry [197] (see Tables XI and XII in Appendix A for details). There are 3 symmetry classes for the complex AZ class with SLS (Table XI) and 19 symmetry classes for the real AZ class with SLS (Table XII). Here classes AI, BDI, and CII with SLS that anticommutes with TRS are respectively equivalent to classes AII, DIII, and CI with SLS that obeys the same algebra. Moreover, each real AZ class with SLS is equivalent to the corresponding real AZ † class with SLS (see Table XIII in Appendix A for details).

D. Pseudo-Hermiticity
In non-Hermitian physics, pseudo-Hermiticity serves as another key internal symmetry [203][204][205], which is defined by with a unitary and Hermitian matrix η (ηη † = η † η = 1 and η † = η). Here pseudo-Hermiticity is a generalization of Hermiticity, in that it is trivially satisfied with η = 1 in the presence of Hermiticity. In addition, it is also a generalization of parity-time symmetry [1] since positivity of η is equivalent to the real spectrum of a non-Hermitian Hamiltonian [203]. Pseudo-Hermiticity can also be considered as an additional symmetry to the AZ or AZ † symmetry class. Moreover, the AZ or AZ † class with pseudo-Hermiticity is equivalent to the AZ or AZ † class with SLS (see Table XIV in Appendix B for details).

E. 38-fold classification
The symmetries discussed above constitute all the internal symmetries in non-Hermitian physics, which generalizes and extends the AZ symmetry classification [189] for Hermitian Hamiltonians to that for non-Hermitian ones. This symmetry classification is 38-fold: the 10 AZ symmetry classes with the additional 6 AZ † symmetry classes, as well as the 22 AZ symmetry classes with SLS. Notably, the 4 symmetry classes in the AZ † symmetry class also appear in the AZ symmetry class, and each AZ symmetry class with SLS is equivalent to the corresponding AZ † symmetry class with SLS (see Appendix A for details) or the AZ symmetry class with pseudo-Hermiticity (see Appendix B for details). Our 38-fold symmetry classification is applicable to a number of non-Hermitian systems that are theoretically investigated and experimentally realized, as discussed in detail below. II. Relationship between the Bernard-LeClair (BL) symmetries and the non-Hermitian AZ symmetries discussed in the present work. Here TRS, PHS, CS, and SLS respectively stand for time-reversal symmetry, particle-hole symmetry, chiral symmetry, and sublattice symmetry; TRS † (PHS † ) denotes the symmetry defined by Hermitian conjugation of TRS (PHS).
TRS, PHS † Our 38-fold classification is basically equivalent to the Bernard-LeClair symmetry classification that describes non-Hermitian random matrices [26,94,125,206,207]: P sym. : with c = ±1 and unitary operators c, p, q, and k. Table II summarizes the relationship between the Bernard-LeClair symmetries and ours. Whereas our classification is 38-fold, Bernard-LeClair's one is 43-fold. This disagreement originates from overcounting and overlooking non-Hermitian symmetry classes in their classification.
In particular, they distinguished the pseudo-Hermiticity [Q symmetry defined by Eq. (16)] with positivity from generic pseudo-Hermiticity without positivity. However, it is known that the pseudo-Hermiticity with positivity is equivalent to Hermiticity [203][204][205]. Thus, the former pseudo-Hermiticity just gives the Hermitian symmetry classes. Here the following 5 symmetry classes distinguished in the Bernard-LeClair classification are considered to be the same in our classification: We recall that the Hermitian symmetry class is the 10fold AZ symmetry class. Subtracting these Hermitian 10 classes from their 43 classes, we only have 33 classes as intrinsic non-Hermitian symmetry classes. However, they overlooked the following 5 symmetry classes, which should be added when the aforementioned distinction is made: Adding these 5 classes to 33 classes reproduces our 38fold symmetry class. We complete the non-Hermitian 38-fold symmetry class, in which the 5 classes in Eq. (19) were overlooked by Bernard and LeClair. Remarkably, our 38 classes present different classifying spaces and give different topological phases, as shown in Sec. IV. Importantly, although their symmetries are mathematically the same as ours after correctly including the 5 classes in Eq. (19), the physical insight of these symmetry classes has remained elusive until the present work. Therefore, our symmetries give a more fundamental framework in the study of non-Hermitian physics.

III. COMPLEX-ENERGY GAP
In the topological classification of Hermitian insulators and superconductors, two Hermitian Hamiltonians are defined to be topologically equivalent if and only if they are smoothly deformed to each other with symmetry and an energy gap. In the non-Hermitian case, on the other hand, it is nontrivial how the energy gap is defined since the spectrum is complex for a generic non-Hermitian Hamiltonian.
Here we recall that an energy gap means the energy region where no states are present. In the Hermitian case, such a vacant region in the spectrum should be a zero-dimensional point E = E F called the Fermi energy since the spectrum is entirely real and one-dimensional. Thus it is naturally and uniquely defined to have an energy gap if and only if its energy bands do not cross the Fermi energy E = E F [ Fig. 1 (a)]. In the non-Hermitian case, by contrast, the forbidden energy range where no states exist is not necessarily contractible to a zero-dimensional point since the complex spectrum of a generic non-Hermitian Hamiltonian is two-dimensional. As a result, such a forbidden energy region can be either a zero-dimensional point or a one-dimensional line, and accordingly the definition of the complex-energy gap in a non-Hermitian Hamiltonian is not unique. It can be defined to have a zero-dimensional point gap if and only if its complex-energy bands do not cross a point E = E P in the complex-energy plane [ Fig. 1 (b)], and independently, it can also be defined to have a one-dimensional line gap if and only if its complex-energy bands do not cross a line in the complex-energy plane [ Fig. 1 (c)]. The precise definitions of these complex-energy gaps are provided later in this section.
Importantly, two definitions are independent of each other and the definition that should be adopted depends on the individual physical situations that we are interested in. For instance, the Anderson localization transition in a one-dimensional non-Hermitian system [10,15,114] can be captured by the point gap. On the other hand, the topologically protected edge states experimentally observed in non-Hermitian optical and photonic systems [130,131,[133][134][135][137][138][139] can be understood by the line gap. The two definitions of the complex-energy gaps are thus complementary to each other. Moreover, the topological classification drastically changes according to the definition of the complex-energy gap, as discussed in detail in the next section. In the absence of symmetry, for example, a topological phase characterized with a point gap is present only in odd spatial dimensions, whereas a topological phase characterized with a line gap is present only in even spatial dimensions (see Table III in Sec. IV for details). We note that Refs. [94,109,118] explicitly adopt the line gap, whereas Ref. [114] adopts the point gap.

A. Point gap
Although a complex-energy point E = E P that serves as an obstacle in the complex-energy plane is arbitrary in the absence of symmetry, it is subject to restrictions in the presence of symmetry. For instance, it should be taken as Im E P = 0 in the presence of TRS since eigenenergies come in (E, E * ) pairs; it should be taken as E P = 0 in the presence of SLS since eigenenergies come in (E, −E) pairs. Thus it is convenient to choose E P as zero energy, which leads to the precise definition of the point gap as follows: Under this definition, a gapless system possesses a zero-energy state for some k. The point gap helps understand the localization transition in a non-Hermitian system in one dimension [10,15,114] that occurs due to the competition between disorder and non-Hermiticity. Moreover, when we regard H (k) as a non-Hermitian dynamical matrix that determines the structure of mechanical systems, topological boundary modes in isostatic lattices [30] can be captured under the point gap.

B. Line gap
A complex-energy line that serves as an obstacle in the complex-energy plane can also be subject to restrictions in the presence of symmetry, whereas such a line is arbitrary in the absence of symmetry. In particular, it should be either the imaginary axis (Re E = 0) or the real axis (Im E = 0) when symmetry imposes a real structure on the complex spectrum. For instance, the real axis should be considered when pairs of eigenenergies (E, E * ) appear with TRS; the imaginary axis should be considered when pairs of eigenenergies (E, −E * ) appear with CS. In contrast to the point gap, there are no restrictions in the presence of SLS, since SLS does not give the complex spectrum real structures [eigenenergies just come in (E, −E) pairs]. Thus it is convenient to choose the line that determines the complex gap as the imaginary axis (real gap) or the real axis (imaginary gap), which leads to the precise definition of the line gap in the following: Under this definition of the real (imaginary) gap, a gapless system includes an eigenenergy with Re E (k) = 0 (Im E (k) = 0) for some k. The line gap is employed explicitly in Refs. [94,109,118] and implicitly in many other pieces of work, and characterizes topologically protected boundary states, which were also observed in experiments [130,131,[133][134][135][137][138][139]. Remarkably, the presence of an imaginary gap has a significant influence on the nonequilibrium wave dynamics [118], although it has no counterparts in the Hermitian band theory.

IV. TOPOLOGICAL CLASSIFICATION
We provide topological classification of non-Hermitian insulators and superconductors according to all the 38 symmetry classes discussed in Sec. II and two types of the complex-energy gaps discussed in Sec. III. Here non-Hermitian Hamiltonians H 0 (k) and H 1 (k) are defined to be topologically equivalent if and only if there exists a non-Hermitian Hamiltonian with certain symmetries and a complex-energy gap for all λ ∈ [0, 1]. Our strategy is to reduce this non-Hermitian problem to the established topological classification of Hermitian Hamiltonians in the AZ symmetry class without [190][191][192] and with [197] additional symmetries. In particular, we demonstrate that a non-Hermitian Hamiltonian can be smoothly deformed into a unitary matrix and hence a larger Hermitian matrix in the presence of a point gap [ Fig. 2 (b); see also Theorem 1 below and its proof in Appendix C for details] and a Hermitian or an anti-Hermitian matrix in the presence of a line gap [ Fig. 2 (c); see also Theorem 2 below and its proof in Appendix D for details]. The K -theory classification for the point gap is also discussed in Appendix E.
Our results are listed in the periodic tables for the complex AZ symmetry class (Table III), the real AZ symmetry class (Table IV), the real AZ † symmetry class (Table V), the complex AZ symmetry class with SLS (Table VI), and the real AZ symmetry class with SLS (Table VII). In addition to this 38-fold topological classification, we provide the periodic tables for the AZ symmetry class with pseudo-Hermiticity (Table VIII and Table IX). The 7-fold periodic table based on two antiunitary symmetries (T + and T − ) and unitary symmetry (S) are also shown in Table XV in Appendix F.

A. Point gap: unitary flattening
In the presence of a point gap, a non-Hermitian Hamiltonian can be flattened into a unitary matrix without gap closing. This property is guaranteed by the following theorem (see Appendix C for a proof): Here the presence of symmetry for the original non-Hermitian Hamiltonian H (k) discussed in Sec. II imposes the following constraints on the extended Hermitian HamiltonianH (k): Moreover,H (k) respects additional CS (SLS): Importantly, there exists a one-to-one correspondence between a unitary matrix U (k) and an extended Hermitian matrixH (k) that satisfies Eq. (27) [114,209], and hence the topology of H (k) can also be captured by the extended Hermitian HamiltonianH (k). Therefore, the topological classification of a non-Hermitian Hamiltonian H (k) with a point gap and symmetry reduces to that of a Hermitian Hamiltonian that respects symmetry given by Eqs. (22)- (27), which was already obtained in Refs. [190][191][192]197]. In this manner, the periodic tables under the point gap are obtained as Tables III-IX. Notably, a similar theorem was proved in Ref. [114]. However, it cannot be applicable in the presence of C ± , and Theorem 1 in the present work is a nontrivial generalization of the theorem in Ref. [114]. Let us consider class DIII as an example (Table IV). The original non-Hermitian Hamiltonian H (k) respects both TRS and PHS: As a result, the extended and flattened Hermitian Hamil-tonianH (k) respects TRS described by Eq. (22) with T +T * + = −1 and PHS described by Eq. (23) withC −C * − = +1, as well as additional CS (SLS) described by Eq. (27). Therefore, the topological classification of the original non-Hermitian Hamiltonian reduces to that of the Hermitian Hamiltonian in class DIII with additional CS that commutes with TRS and anticommutes with PHS; the topology of such Hermitian Hamiltonians is characterized by the classifying space R 4 [197]. Theorem 2 also reduces the topological classification of a non-Hermitian Hamiltonian to that of a Hermitian matrix [190][191][192]197]. Here we note that the topology of an anti-Hermitian Hamiltonian H (k) [i.e, H † (k) = −H (k)] under an imaginary gap is equivalent to that of a Hermitian Hamiltonian iH (k) under a real gap [118]. The periodic tables under the line gap are also obtained  as Tables III-IX. Let us again consider class DIII as an example (Table IV). The original non-Hermitian Hamiltonian H (k) respects both TRS and PHS as Eqs. (28) and (29). In the presence of a real gap, H (k) can be flattened to a Hermitian HamiltonianH (k) that belongs to class DIII, which is characterized by the classifying space R 3 [190][191][192].
In the presence of an imaginary gap, on the other hand, H (k) can be flattened to an anti-Hermitian Hamiltonian H (k) that respects Eqs. (28) and (29). Importantly, the topology ofH (k) is equivalent to that ofH (k) := iH (k), which respects Hermiticity and Here transposition and complex conjugation coincide with each other due to the presence of Hermiticity, and Eq. (31) reduces to the antiunitary constraint given by Thus, the non-Hermitian Hamiltonian H (k) under an imaginary gap reduces to the Hermitian Hamiltonian H (k) that respects two antiunitary symmetries as Eqs. (30) and (32); the topology of such Hermitian Hamiltonians is characterized by the classifying space C 0 [197].      . Non-Hermitian topological phases are classified according to the AZ symmetry class with additional SLS, the spatial dimension d, and the definition of a complex-energy point (P) or line (L) gap. The subscript of L specifies the line gap for the real or imaginary part of the complex spectrum. The subscript of S specifies the commutation (+) or anticommutation (−) relation to time-reversal symmetry (TRS) and/or particle-hole symmetry (PHS); for the symmetry classes that involve both TRS and PHS (BDI, DIII, CII, and CI), the first subscript specifies the relation to TRS and the second one to PHS.

C. Dirac Hamiltonian
Hermitian topological insulators and superconductors can be understood with continuum models that have the massive Dirac Hamiltonian representation [202]: where k = (k 1 , · · · , k d ) is the momentum deviation from a relevant momentum reference point, and Γ 1 , · · · , Γ d are Dirac matrices that satisfy the Clifford algebra (i.e., {Γ i , Γ j } = 2δ ij ). The mass term mΓ 0 anticommutes with all the Dirac matrices Γ 1 , · · · , Γ d in the kinetic term and determines the topology of the classifying space.
Our classification suggests that non-Hermitian topological systems can also be described by a non-Hermitian generalization of the Dirac Hamiltonian. However, the complex-spectral-flattening procedures distinct from the Hermitian case imply that non-Hermiticity can modify the proper representation of Dirac matrices. In fact, in the presence of a point gap, non-Hermitian Dirac matrices Γ P i (i = 1, · · · , d) are defined so that their Hermitianized matricesΓ obey the Clifford algebra (i.e., {Γ P i ,Γ P j } = 2δ ij ). This in turn leads to the relations for Γ P i , This set of relations determines the proper non-Hermitian Dirac matrices in the presence of a point gap. Table X shows an example of the representations of Γ P i 's (i = 1, · · · , n) for small n, which is clearly distinct from the conventional Hermitian Dirac matrices. In the presence of a line gap, on the other hand, Dirac matrices take the same representation as the Hermitian case, since a non-Hermitian Hamiltonian can be flattened into a Hermitian or an anti-Hermitian Hamiltonian. The non-Hermitian Dirac Hamiltonian provides a systematic way to have a model for the periodic tables. In 1D class A, for instance, a non-Hermitian Dirac Hamiltonian can be expressed as in the presence of a point gap. With this continuum model, the Z topological invariant (winding number) [114] in Table III can be readily obtained as Here the fractional topological invariant W is common to continuum models for both Hermitian and non-Hermitian cases and should be complemented by the structure of wave functions away from the relevant momentum point; it becomes an integer W = sgn[m] when we regularize the mass m as m − k 2 .

D. Real and imaginary gaps
For each symmetry class and each spatial dimension, there appear multiple topological structures in the classification tables. For instance, since CS acts as an anti-Hermitian conjugation for non-Hermitian Hamiltonians as Eq. (9), it distinguishes between real and imaginary gaps, both of which give the different topological structures (Table III). To understand this unique non-Hermitian feature in detail, we consider a 2 × 2 non-Hermitian Hamiltonian in one-dimension where σ 0 is the 2 × 2 identity matrix and σ = (σ x , σ y , σ z ) is a set of Pauli matrices. Imposing CS with Γ := σ z , we have the following constraints on h i (k) (i = 0, x, y, z): which implies that h 0 (k) and h z (k) [h x (k) and h y (k)] are pure imaginary (real) for all k. Therefore, by redefining h 0 (k) and h z (k) as h 0 (k) → ih 0 (k) and h z (k) → ih z (k), we obtain the Hamiltonian with CS as where h i (k)'s (i = 0, x, y, z) are real functions. The eigenenergies of H (k) are given by and thus the system supports a real (an imaginary) gap for h 2 First we consider the case with a real gap. After the Hermitian flattening with E (k) = ±1, h i (k)'s obey These conditions define a surface in the parameter space (h x , h y , h z ) of the Hamiltonian [ Fig. 3 (a)]. The surface is open in the h z -direction and circular in the other directions. Each Hamiltonian with a real gap gives an image from the one-dimensional Brillouin zone through h (k), which draws a circle on the surface. Obviously, a one-dimensional winding number can be defined just by counting how many times the circle winds the surface. By contrast, we cannot have such a winding number in the case with an imaginary gap. After the anti-Hermitian flattening with E (k) = ±i, we have which gives a surface in Fig. 3 (b). Since topologically stable loops are absent on this surface, no onedimensional winding number is well-defined. The above observation is fully consistent with the periodic table III: for 1D class AIII, the topological invariant is characterized by an integer for a real gap, while it is trivial for an imaginary gap.

E. Pseudo-Hermiticity and topology
Although there are topological phases characterized by the Chern number in a two-dimensional system without symmetry (Table III), these topological phases vanish in the presence of TRS with T + T * + = +1 under a real gap (Table IV). However, in the presence of pseudo-Hermiticity η − that anticommutes with TRS (η − T + = −T + η * − ), a different type of topological phases emerges that is described by the time-reversal-invariant Chern number [94], as shown in detail below. Therefore, pseudo-Hermiticity provides a novel topological structure as a unique non-Hermitian feature. To see this unique property of pseudo-Hermiticity, we start with the standard procedure for diagonalization of a non-Hermitian Hamiltonian H. Let |u n (|u n ) be a right (left) eigenstate of H H |u n = E n |u n , H † |u n = E * n |u n .
The eigenstates form the biorthonormal basis [33], which obey with the completeness condition n |u n u n | = n |u n u n | = 1.
We compactly express these biorthonormal relations as which diagonalize H as Now let us see how pseudo-Hermiticity imposes an additional constraint. From pseudo-Hermiticity defined by η −1 H † η = H, we have which yields that η |u n is a right eigenstate of H with eigenenergy E * n : Therefore, E n in general has a complex-conjugate partner E * n in the spectrum of H. This simple structure leads to significant consequences. In particular, an isolated real eigenenergy of H remains real for any smooth deformation of H unless it coalesces with other eigenenergies. In fact, the above constraint implies that eigenenergies should come in complexconjugate pairs, and thus an isolated real eigenenergy cannot become complex by itself. Such reality of the eigenenergy is important to obtain stable states in non-Hermitian systems. Moreover, it is shown that the entire real spectrum is equivalent to the presence of pseudo-Hermiticity with positivity [203].
Pseudo-Hermiticity also gives a nontrivial topological structure when the system has a real gap [94]. In the presence of a real gap at Re E = E F , we can define an "empty" ("occupied") state as a state with Re E n > E F (Re E n < E F ). If |u n is an occupied (empty) state, so is η |u n , since |u n and η |u n have eigenenergies with the same real part. From the completeness of the basis, we can relate them as with c mn := u m |η|u n . Here we have c mn = 0 for E m = E * n , and c mn is Hermitian with respect to the indices m and n (i.e., c mn = c * nm ) since η is Hermitian. Thus, we can diagonalize c mn by a unitary matrix G without mixing between occupied and empty states, with real eigenvalues λ m . Taking the following new biorthogonal basis with φ m |φ n = φ m |φ n = δ mn , we have η |φ n = sgn (λ n ) |φ n .
Therefore, both occupied states and empty states in the new basis are divided into two subsectors, i.e., states with η |φ n = +|φ n , and states with η |φ n = −|φ n .
We denote these states as |φ ± n . This is a non-Hermitian generalization of the η-eigensector. In fact, for Hermitian H, the right and left eigenstates coincide with each other, and thus the above equations reduce to the eigenequations of η: η |φ ± n = ± |φ ± n . Importantly, the new basis {|φ ± n } is no longer eigenstates of H unless E n is real. Indeed, G mn mixes the eigenstate with E n and that with E * n . However, since G mn does not mix occupied and empty states, the new basis {|φ ± n } keeps the same topology as the original one {|u n }, while the subsector structure manifests itself only in the new basis.
The presence of this subsector structure enables us to introduce a topological invariant for each subsector. For instance, in the case of class A with pseudo-Hermiticity in two dimensions, the Chern number can be defined for each subsector: where F ± n (k) := ∂ kx A ±y n (k) − ∂ ky A ±x n (k) with A ±i n := i φ ± n |∂ ki φ ± n . As shown in Table VIII, these two independent Chern numbers agree with two integer topological invariants (Z ⊕ Z) in 2D class A with pseudo-Hermiticity in the presence of a real gap.
This subsector structure also makes it possible to define the nonzero Chern numbers even in time-reversal symmetric systems. From time-reversal symmetry, the total Chern number vanishes (i.e., C + n + C − n = 0), but their difference can be nonzero [i.e., (C + n − C − n )/2 = 0 ∈ Z], which is referred to as the time-reversal-invariant Chern number in Ref. [94]. As shown in Table IX, our classification correctly captures this integer invariant (Z) in the presence of a real gap. It is also found that pseudo-Hermiticity is naturally imposed on free bosonic systems, which is discussed separately in Sec. VI.

V. EXPERIMENTAL RELEVANCE
Our topological classification of non-Hermitian systems based on internal symmetry has a direct relevance to various experiments in nonequilibrium open systems with gain and/or loss [130,131,[133][134][135][137][138][139]. In fact, the observed topologically protected edge states are justified by the periodic tables III-IX. For instance, topologically protected bound states were observed in a passive dimerized photonic crystal in one dimension [133]. Moreover, lasing topological edge states were observed in an active (pumped) array of microring resonators also in one dimension [138]. Both systems are essentially described by the Su-Schrieffer-Heeger model [142] with balanced gain and loss [97,106]: whereâ i (â † i ) andb i (b † i ) annihilate (create) a photon on site i in sublattices, respectively, v, w ∈ R denote the hopping amplitudes, and γ ∈ R denotes the balanced gain and loss. In momentum space, the Bloch Hamiltonian is obtained as Although this system no longer respects SLS due to the presence of gain and loss, it remains to respect CS defined by Eq. (9) with Γ := σ z : This system thus belongs to AZ symmetry class AIII, and our classification table (Table III) predicts the topological phase characterized by integers under the definition of the line gap in the real part of its complex spectrum. This Z topological phase is characterized by the winding number discussed in Sec. IV D, which is precisely defined by where q (k) := v + w e −ik / |v + w e −ik | 2 − γ 2 is the off-diagonal part of the Q matrix (see Appendix G for details) [94]. The nonzero winding number implies the emergence of topologically protected bound states with Re E = 0, which are indeed observed in experiments [133,138]. In stark contrast to Hermitian systems, these bound states can have eigenenergies with positive imaginary parts, which leads to their amplification (lasing) with time [138]. We emphasize that the topological phase in this system cannot be captured by the classification in Ref. [114], which considers neither line gap nor CS that is essential in this non-Hermitian topological phase. Another prime example is a topological-insulator laser [139], which is a non-Hermitian extension of the Haldane model [145] with energy gain. This topological laser possesses topologically protected edge states even in the presence of non-Hermiticity, which qualitatively enhances the lasing efficiency due to the topological immunity against defects and disorder. The topologicalinsulator laser does not rely on any symmetry and hence belongs to AZ symmetry class A in two dimensions; the topologically protected edge states are attributed to the topological invariant characterized by an integer (Table III). This Z topological invariant is given by the Chern number [96,109,113,116] where F n (k) := ∂ kx A y n (k) − ∂ ky A x n (k) is the Berry curvature and A i n (k) := i u n (k) | ∂ ki u n (k) (i = x, y) is the Berry connection for a non-Hermitian Hamiltonian whose right (left) eigenstates [33] are denoted as |u n (k) (|u n (k) ). Here again the topological phase of this non-Hermitian system cannot be explained by the classification in Ref. [114], which only considers a point gap instead of a line gap.

A. Topological classification
Whereas the topological classification of Hermitian free fermions was well established [190][191][192], its bosonic counterpart has been absent even in Hermitian systems. Our theory of non-Hermitian systems provides such topological classification of Hermitian and non-Hermitian free bosons. We consider a generic noninteracting (quadratic) bosonic system with a set of bosonic annihilation (creation) operatorŝ a := (â 1 , · · · ,â N ) [â † := (â † 1 , · · · ,â † N )], which satisfies Here the non-Hermitian BdG Hamiltonian H BdG is described by where M and ∆ ± are N × N non-Hermitian matrices, and ∆ ± are required to be symmetric (i.e., ∆ T ± = ∆ ± ) because of Bose statistics. In the presence of Hermiticity, M becomes Hermitian and ∆ ± satisfies ∆ † + = ∆ − . In contrast to fermionic systems whose BdG Hamiltonians are diagonalized with unitary matrices, bosonic BdG Hamiltonians should be diagonalized with paraunitary matrices so that their quasiparticles fulfill the bosonic commutation relations [29]. In other words, we should diagonalize not the original BdG Hamiltonian H BdG but the effective matrix Here the effective matrix H σBdG is generally non-Hermitian even if the original BdG Hamiltonian H BdG is Hermitian. Importantly, the non-Hermiticity results from the bosonic commutation relations, which can induce dynamical instability [29].
To consider the topological phases of free bosons, symmetry imposed on the effective non-Hermitian matrix H σBdG is relevant. In general, owing to ∆ T ± = ∆ ± , H σBdG respects PHS with C − := σ y , which reduces to Eq. (7) in momentum space. Moreover, in the presence of Hermiticity for H BdG , H σBdG respects pseudo-Hermiticity with η := σ z , which reduces to Eq. (13) in momentum space. Therefore, the topological classification of Hermitian and non-Hermitian free bosons reduces to that of the non-Hermitian matrix H σBdG that respects Eqs. (67) and/or (68) in addition to some other symmetries, which is already obtained as Table IX. PHS and pseudo-Hermiticity in Eqs. (67) and (68) satisfy C − C * − = −1 and {η, C − } = 0. Therefore, in the absence of TRS and other additional symmetries, a noninteracting bosonic BdG system naturally belongs to class C (class C with η − ) for non-Hermitian (Hermitian) H BdG . On the other hand, in the presence of TRS, which usually obeys T + T * + = 1 for a bosonic system, the natural symmetry class is class CI (class CI with η +− ). To apply our classification to bosonic systems, however, a more careful consideration for an energy gap is necessary. For Hermitian fermionic systems with PHS, we usually assume a gap at zero energy. In the case of Hermitian superconductors, for instance, we take a superconducting gap at zero energy since all states below the gap are fully occupied in the ground state at zero temperature, and the lowest excited state appears in the gap. For free bosons, on the other hand, this assumption is not obvious since any states are not fully occupied in the ground state because of Bose statistics. Thus, we can consider an energy gap away from zero energy. In this case, the choice does not respect PHS, and hence the topological classification effectively neglects PHS. Therefore, the relevant symmetry class is class A (class A+η) or class AI (class AI+η + ) for non-Hermitian (Hermitian) H BdG , instead.
Our topological classification describes topological phenomena of free bosons [21][22][23][24][25][26][27][28][29]. In class A with η, Table VIII predicts the topological phase characterized by an integer in two dimensions in the presence of a real gap, which corroborates the magnon Hall effect [21] as a bosonic counterpart of the quantum Hall effect. It should be noted here that H BdG in Ref. [22] is positive definite as well as Hermitian, so that the Z ⊕ Z invariant reduces to the single Chern number (Z), as explained later. Recently, a bosonic analogue of the Z 2 topological insulator was also proposed in Ref. [28]. In addition to Eqs. (67) and (68), this system respects pseudo-time-reversal symmetry given by Eq. (4) with T + T * + = −1, which leads to symmetry class AII with pseudo-Hermiticity η + . Table IX predicts the Z 2 topological phase in two dimensions in the presence of a real gap, which is consistent with the Z 2 topological invariant constructed in Ref. [28]. Again, we note that the Z 2 ⊕ Z 2 invariant reduces to the single Z 2 number since H BdG is positive definite. Remarkably, our topological classification not only justify the known bosonic topological phenomena but also may lead to novel topological phases of free bosons.

B. Pseudo-Hermiticity and paraunitary condition
Due to pseudo-Hermiticity, H σBdG may host real eigenvalues despite non-Hermiticity and additional topological structures appear, as was explained in Sec. IV E. Meanwhile, as mentioned above, H σBdG is diagonalized by a paraunitary rather than unitary matrix. In fact, this unique feature of bosonic systems is a consequence of the real spectrum and pseudo-Hermiticity, as described in detail below.
Let |u n (|u n ) be a right (left) eigenstate of H σBdG . Using the same procedure as in Sec. IV E, we take the biorthonormal basis (|φ ± n , |φ ± n ) in which c mn := u m |η|u n is diagonal. When the eigenvalues of H σBdG are real, this basis also diagonalizes H σBdG , and thus we have Therefore, introducing the following matrices, we have with E ± := diag E ± 1 , . . . , E ± N . Recalling η = σ z and the biorthonormal relation LR † = 1, the latter equation in Eq. (71) yields nothing but the paraunitary condition given by The original bosonic BdG Hamiltonian H BdG is often supposed to be positive definite as well as Hermitian. In this case, we can construct R as follows [22]. From the Cholesky decomposition [210], H BdG is recast into the product of an invertible upper triangle matrix K and its Hermitian conjugate K † as Then, we introduce the Hermitian matrix Kσ z K † and diagonalize it by a unitary matrix U , where ε := diag (ε 1 , . . . , ε N ) is a diagonal matrix consisting of positive eigenvalues of Kσ z K † . From Sylvester's law of inertia [210], the numbers of positive and negative eigenvalues of Kσ z K † are the same, and thus Kσ z K † can be diagonalized in the form of Eq. (74). Rewriting the right-hand side of Eq. (74) as we obtain where satisfies the paraunitary condition in Eq. (72). From the above construction, we can see that the positive definite Hermitian condition for H BdG provides a strong constraint. By comparing Eq. (71) with Eq. (76), we have E ± = ±ε. Therefore, a positive (negative) energy state |u in Eq. (76) always satisfies η |u = |u (η |u = −|u ). We can also show that positive-energy and negative-energy eigenstates are related to each other by PHS in Eq. (67). Thus, the sector with η |u = |u and that with η |u = −|u are not independent of each other, and thus they are characterized by the same topological invariant. This constraint reduces possible independent topological invariants.

VII. CONCLUSION
We have clarified symmetry and complex-energy gaps in non-Hermitian physics and sorted out all the non-Hermitian topological phases.
Whereas antiunitary symmetries are unified in non-Hermitian physics [118], symmetry can also ramify due to the distinction between transposition and complex conjugation for non-Hermitian Hamiltonians. As a result, the non-Hermitian symmetry class is 38-fold beyond the celebrated 10fold AZ symmetry class [189], each of which describes intrinsic non-Hermitian topological phases as well as non-Hermitian random matrices. Moreover, a complexenergy gap can be either point-like (zero-dimensional) or line-like (one-dimensional) due to the complex nature of the energy spectrum, which enriches the topology of non-Hermitian systems. On the basis of these clarification, we have classified all the non-Hermitian topological phases as summarized in the periodic tables III-IX. This classification corroborates the unique lasing and transport phenomena recently observed in experiments [130,131,[133][134][135][137][138][139]. Although these experiments cannot be described by the classification provided in Ref. [114], our work provides a more general and comprehensive framework, so that the book on non-Hermitian topological systems has now been closed.
The theoretical framework developed in the present work opens up new applications in non-Hermitian topological physics. One of the most crucial ones is to find and design a novel symmetry-protected topological laser even in three dimensions beyond the two-dimensional one without symmetry protection [139]. Our framework can also be applied to find topological phases in non-Hermitian superconductors, which has potential to be of use in topological quantum computation [211]. We hope that the classification of non-Hermitian topological phases completed by this work will lead to such novel phenomena and functionalities that originate from the interplay of non-Hermiticity and topology. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas "Topological Materials Science" (KAKENHI Grant No. JP15H05855) from the Japan Society for the Promotion of Science (JSPS). K.K. was supported by the JSPS through Program for Leading Graduate Schools (ALPS). K.
Note added. -After this work had been submitted, there appeared a related work by Zhou and Lee [212]. Although Ref. [212] initially counted the number of symmetry classes as 42, Ref. [212] has corrected it as 38 after knowing our 38-fold symmetry classification [213].
Appendix A: Sublattice symmetry as an additional symmetry SLS can be considered to be an additional symmetry to the AZ symmetry [197] as shown in Tables XI and XII. Moreover, Table XIII shows the equivalence between the  TABLE XIII. Equivalence between the real AZ symmetry class with sublattice symmetry (SLS) and the real AZ † symmetry class with SLS. The subscript of S specifies the commutation (+) or anticommutation (−) relation to TRS/TRS † and/or PHS/PHS † ; for the symmetry classes that involve both TRS/TRS † and PHS/PHS † (BDI, DIII, CII, and CI; BDI † , DIII † , CII † , and CI † ), the first subscript specifies the relation to TRS/TRS † and the second one specifies the relation to PHS/PHS † .
real AZ symmetry class with SLS and the real AZ † symmetry class with SLS. Let us take class DIII † as an example. In this symmetry class, the Hamiltonian respects both TRS † and PHS † : We consider adding SLS that satisfies with c , t ∈ {±1}. Then TRS can be constructed by combining PHS † and SLS as T + := ST − , which satisfies Similarly, PHS can be constructed by combining TRS † and SLS as C − := SC + for c t = +1 and C − := iSC + for c t = −1, which satisfy Here C − is chosen so that it commutes with T + : Thus class DIII † with SLS is equivalent to one of the AZ symmetry classes with SLS.
Appendix B: Pseudo-Hermiticity as an additional symmetry Table XIV shows the equivalence between pseudo-Hermiticity and SLS as an additional symmetry to the AZ symmetry. Let us take class DIII as an example. In this symmetry class, the Hamiltonian respects both TRS and PHS: As a combination of TRS and PHS, the Hamiltonian also respects CS: with Γ := i C − T * + . Then we consider adding pseudo-Hermiticity that satisfies with t , c ∈ {±1}. Here SLS can be constructed by combining CS and pseudo-Hermiticity. In the case of t c = +1, SLS is defined as S := ηΓ with S 2 = 1, which satisfies ST + = − t T + S * and SC − = − c C − S * ; in the case of t c = −1, SLS is defined as S := iηΓ with S 2 = 1, which satisfies ST + = t T + S * and SC − = c C − S * . To prove Theorem 1, we introduce the following Hermitian HamiltonianH 0 (k) constructed from the non-Hermitian Hamiltonian H (k):  Equivalence between pseudo-Hermiticity and sublattice symmetry as an additional symmetry in the AZ symmetry class. For the complex classes, the subscript of η and S specifies the commutation (+) or anticommutation (−) relation to chiral symmetry. For the real classes, the subscript of η and S specifies the commutation (+) or anticommutation (−) relation to time-reversal symmetry (TRS) and/or particle-hole symmetry (PHS); for the symmetry classes that involve both TRS and PHS (BDI, DIII, CII, and CI), the first subscript specifies the relation to TRS and the second one to PHS.

AZ class
We note thatH 0 (k) is identical toH (k) in Eq. (21), except that the off-diagonal component H (k) is not unitary (H 2 0 (k) = 1). For H (k) with certain symmetries,H 0 (k) has the corresponding symmetries defined by Eqs. (22)- (26), whereH (k) is replaced byH 0 (k). Moreover, as long as H (k) retains a point gap,H 0 (k) also has an energy gap, and vice versa [114,209]. Therefore, we can perform a smooth deformation of H (k) while keeping its symmetries and point gap just by the corresponding deformation ofH 0 (k). Now we show that we can smoothly deformH 0 (k) so that it satisfiesH 2 0 (k) = 1, which immediately leads to Theorem 1. SinceH 0 (k) is Hermitian, it can be diagonalized as where Φ (k) is unitary and ε i (k)'s are real. We also have ε i (k) = 0 asH 0 (k) is gapped. Then, using the following Ω we introduce a one-parameter family of Hamiltonians, with λ ∈ [0, 1]. HereH λ (k) is Hermitian and keeps a gap because Ω provides an adiabatic path with H 2 0 (k) →H 2 1 (k) = 1. Therefore, ifH λ (k) has the same symmetry asH 0 (k), we have Theorem 1.
In fact,H λ (k) has the same symmetry. For instance, we considerH 0 (k) with PHS, In this case, we havẽ As a result, we also have PHS forH λ (k), In a similar manner, we can show thatH λ (k) has the same symmetry asH 0 (k) for the other symmetries.

Spectral flattening for the line gap
Let us consider a non-Hermitian Hamiltonian H(k) with a line gap and denote the right and left eigenstates as |u n (k) and |u n (k) , respectively: For our purpose, it is sufficient to consider the case without exceptional points since they can be pair-annihilated without closing a line gap. Then, |u n (k) and |u n (k) satisfy the biorthonormal condition [33] u m (k)|u n (k) = u m (k)|u n (k) = δ mn , (D2) and the completeness condition For later convenience, we collect these eigenstates as row vectors R(k) and L(k): The above biorthonormal and completeness conditions are compactly written as and the right eigenequations become which is recast into Now we flatten the complex spectrum of H(k) without closing a line gap. As long as the system does not close the gap, it keeps the same topological structures, and hence this flattening procedure does not affect the classification of topological phases in the presence of a line gap. Importantly, the flattening process depends on the symmetry class. If any symmetry operation does not include complex or Hermitian conjugation, the line gap merely implies the presence of two disconnected parts of the band energies in the complex-energy plane. While keeping the line gap, we can smoothly change one part of the spectrum into +1 and the other into −1: where p (q) is the number of bands contained in one (the other) part of the spectrum. After this flattening procedure, we obtain a non-Hermitian Hamiltonian H(k) with H 2 (k) = 1. On the other hand, if a symmetry operation with complex or Hermitian conjugation is relevant, we have a real structure in the complex-energy spectrum. The real part of the spectrum can be distinguished from the imaginary one, and thus we have two distinct types of line gaps, i.e., a real gap and an imaginary gap, where a real (an imaginary) gap implies a gap in the real (imaginary) part of the complex spectrum. Correspondingly, there are two different flattening processes as follow. (i) For a system with a real gap, one can smoothly change the band energies with a larger (smaller) real part into +1 (−1) without closing the real gap. The resultant Hamiltonian has the same form as Eq. (D8). (ii) For a system with an imaginary gap, one can smoothly change the band energies with a larger (smaller) imaginary part into +i (−i) without closing the imaginary gap, Then, by multiplying H(k) by −i, the Hamiltonian takes the form of Eq. (D8) again [118]. However, this procedure gives an additional minus sign to the symmetry operations with complex or Hermitian conjugation. Therefore, after the flattening procedure, TRS (CS) becomes PHS † (pseudo-Hermiticity), and vice versa. Thus, the classification problem reduces to the non-Hermitian Hamiltonian with the form subject to proper symmetry constraints. Below we show that the above non-Hermitian Hamiltonian can be deformed into a Hermitian one while keeping the symmetry constraints.

Symmetry constraints
To fulfill the above purpose, we solve the symmetry constraints for H(k) in terms of R(k) and L(k).

a. PHS and TRS †
First we consider PHS in Eq. (7). Taking complex conjugation of the Bloch-BdG equation, we have so that the Hermitian conjugate of Eq. (7) leads to Therefore, C − |u * n (−k) gives a left eigenstate of H(k). Since |u n (k) forms a complete basis of H(k), we have with [C − ] mn := u m (k)| C − |u * n (−k) . Here we choose a gauge of the biorthonormal basis in which C − is a unitary matrix independent of k (such a gauge can be taken at least locally). In terms of R(k) and L(k), the above relation is compactly summarized as Multiplying Eq. (D15) by C † − and C † − from the left and right, respectively, we also have Thus, Eq. (D15) is equivalent to Using Eqs. (D15) and (D17), we can rewrite PHS as a constraint on C − and E. Equations (D15) and (D17) yield from Eq. (D5). The latter equation in the above also implies so that we have Thus, the relation C − C * − = ±1 of PHS reduces to In a similar manner, we can also show that PHS reduces to In the above, we derive the symmetry constraints on C − and E [Eqs. (D21) and (D22)] from PHS for H(k). Conversely, we can also show that H(k) in the form of Eq. (D10) has PHS with C − C * − = ±1 when R(k), L(k), C − , and E satisfy Eqs. (D5), (D15), (D21), and (D22). Therefore, when we keep a set of relations the Hamiltonian given by retains PHS with C − C * − = ±1. In a similar manner, we can obtain the following relations from TRS † in Eq. (10). As long as we keep these relations, we can also retain TRS † with C + C * + = ±1 for H(k) in Eq. (D24).

b. TRS and PHS †
In a manner similar to the above argument, one can show that TRS in Eq. (4) and PHS † in Eq. (11) with T ± T * ± = t ∈ {±1} can be obtained, provided that the following relations hold:

Relations between symmetries
When there are two or more symmetry operations, we have commutation relations between them. By choosing phases of operators, TRS and PHS (TRS † and PHS † ) can always be commutative: For SLS and pseudo-Hermiticity, we have respectively. These relations can be satisfied when we have (D33)

Hermitianization
Now we show that a non-Hermitian Hamiltonian in the form of Eq. (D10) can be smoothly deformed into a Hermitian Hamiltonian while keeping symmetry constraints. For this purpose, we perform the polar decomposition of R(k): Here Λ R (k) is given as where the root of R † (k)R(k) is defined as follows. Since R(k) is invertible, R † (k)R(k) is a positive definite Hermitian matrix, and thus R † (k)R(k) can be diagonalized as where V (k) is a unitary matrix and λ i (k) is a positive number. Then, Λ R (k) := [R † (k)R(k)] 1/2 is defined as with λ(k) := diag [λ 1 (k), λ 2 (k), . . . ]. From this equation, we also have Therefore, U R (k) is uniquely determined as U R (k) = Λ −1 R (k) R(k), which is found to be unitary: As easily seen, we can make a non-Hermitian Hamiltonian in the form of Eq. (D10) Hermitian by just deforming Λ R (k) as Λ R (k) → 1. This process also retains a line gap. However, we need to check whether this process can be done while keeping symmetry.
From the symmetry constraints for R(k) in Eqs. (D23), (D25), (D26), (D27), (D28), and (D29), we have the following constraints on R † (k)R(k): for PHS/TRS † , for TRS/PHS † , for CS, for SLS, and for pseudo-Hermiticity. It can also be shown that these constraints are equivalent to the following ones: For instance, the first equation in the above is derived as follows. We first rewrite Eq. (D40) as which leads to This equation is equivalent to for [V † (k) C ∓ V * (−k)] mn = 0, which yields the first equation in Eq. (D45). From Eq. (D45), we also have Here it should be noted that λ n (k) can be smoothly deformed into 1 while keeping Eq. (D48), and thus we can retain the first equation in Eq. (D45) during the process of Λ R (k) → 1. Combining it with the first equation in Eq. (D49), we obtain the correct symmetry constraint on R(k) in Eq. (D23) for PHS/TRS † . This means that we can deform a non-Hermitian Hamiltonian to a Hermitian one while keeping PHS/TRS † . In a similar manner, we can make a non-Hermitian Hamiltonian Hermitian while keeping any other symmetries.

Patching different momentum regions
To derive Eq. (D15), we have taken a special gauge in which C − is independent of k (we have also chosen similar gauges for the other symmetries). Generally, such a gauge can be taken not globally but locally; if we take such a gauge globally, there arises a singularity in R(k) and L(k). To avoid this singularity, we divide the whole momentum space into several subregions and take a proper gauge in each region. Whereas the matrix C − can take the same form in all regions, R(k) and L(k) are given locally so that they can be different in different regions. From the arguments in Sec. D 4, we can deform in each region a non-Hermitian Hamiltonian into a Hermitian one while keeping a line gap and relevant symmetries. Now we show that this Hermitianization process can be performed globally. For definiteness, we focus on the case with PHS below, but the generalization to the other cases is straightforward.
First, let us consider two regions I and II in momentum space, and denote R(k) in the region I (II) as R I (k) [R II (k)]. Since both R I (k) and R II (k) are well-defined on the boundary between the regions I and II, they are related to each other by a gauge transformation with an invertible matrix G(k). Here G(k) is unitary for Hermitian systems, but not necessarily for non-Hermitian systems. Since both R I (k) and R II (k) obey Eqs. (D10) and (D15) on the boundary, G(k) is found to satisfy We then perform the polar decomposition of G(k) with a unitary matrix U G (k). Here Λ G (k) is defined as follows. Since G † (k)G(k) is a positive definite Hermitian matrix, it is diagonalized as with a unitary matrix W (k) and positive real numbers ω 2 i (k)'s. Then, Ω G (k) is defined as with ω i (k) > 0. Using the polar decomposition of G(k), we recast Eq. (D51) into Here it should be noted that Ω G (k) can be extended to the whole region I without encountering a singularity. In fact, ω i (k) can be rewritten as ω i (k) = e −ρi(k) because of the positivity of ω i (k), and we can extrapolate ρ i (k) as ρ i (k) → 0 from the boundary to the center of the region I while keeping Eq. (D55). As a result, we have a well-defined Ω G (k) in the whole region I. Using this Ω G (k), we can construct another welldefined R(k) in the region I, i.e., R I (k) := R I (k) Ω −1 G (k). Although the new matrix R I (k) satisfies Eqs. (D10) and (D15) again, there is an important modification. Now the gauge transformation between the regions I and II becomes unitary, which yields Therefore, Λ R (k) := (R I (k)R I † (k)) 1/2 for the region I; (R II (k)R † II (k)) 1/2 for the region II (D59) defines a smooth single-valued matrix function in the union of the regions I and II. In a similar manner, all the gauge transformations between the different regions can be made unitary, indicating that Λ R (k) can be defined smoothly in the whole momentum space. This means that the Hermitianization process in Sec. D 4 can be performed globally.
We first formulate possible symmetries of non-Hermitian fermionic systems in the many-body Hilbert space. Let G be a symmetry group andφ : G → Z/2 = {±1} be a homomorphism specifying whether g ∈ G is unitary or antiunitary, i.e., g ∈ G acts on the imaginary unit as In addition, letĉ : G → Z/2 = {±1} be a homomorphism specifying whether g ∈ G is particle-hole type or not, i.e., g ∈ G acts on complex fermion operators as whereψ k (ψ † k ) is a complex fermion annihilation (creation) operator in the BZ, and U g (k) is a unitary matrix. Based onφ andĉ, there are four types of symmetries: 1. Unitary symmetryÛ :φ g = +1 andĉ g = +1.
It is notable that particle-hole symmetryĈ is unitary in the many-body Hilbert space. Furthermore, we fix the factor system of the symmetry G that indicates a U (1) phase among two symmetry actions gh ∈ G and hg ∈ G as U g (hk) U h (k) (φ gĉg = +1) U g (hk) U * h (k) (φ gĉg = −1) = e iτ g,h (ghk) U gh (k) , where the twist τ = τ g,h (k) specifies the projective representation for internal degrees of freedom and nonprimitive lattice translations of space group symmetry [214]. For a free fermion HamiltonianĤ = kψ † k H (k)ψ k , the symmetry gĤg −1 =Ĥ is recast aŝ for the single-particle Hamiltonian H (k). Here we assume tr [H (k)] = 0. We then provide the K -theory classification of H (k) under the unitary flattening [i.e., det H (k) = 0 for all k].
Since H (k) can be assumed as a unitary matrix due to Theorem 1 in Sec. IV A, H (k) is identified with an adiabatic time evolution of a Hermitian system with a period, which implies that the classification of non-Hermitian Hamiltonians H (k) under the unitary flattening is the same as that for unit adiabatic time evolutions. Here the unit adiabatic time evolutions are described by the Kgroup with the shift of integer degree n by +1 [198]. We thus expect that the non-Hermitian Hamiltonians H (k) under the unitary flattening are classified by the K -group φ K (τ,c)−1 G (T d ) with φ :=φĉ and c :=ĉ. In fact, for the extended Hermitian HamiltonianH (k) with CS (SLS) Σ [Eqs. (21) and (27) in Sec. IV A], symmetry g ∈ G is represented as so that it satisfies These conditions determine nothing but the symmetry class for Hermitian Hamiltonians with the integer grading n = 1 [198]. Therefore we conclude that non-Hermitian Hamiltonians H (k) under the unitary flattening are classified by the K -group φ K (τ,c)−1 G (T d ). As a consequence, the periodic table for the AZ symmetry class is obtained  as Tables III and IV. Using the non-Hermitian Dirac matrices developed in Sec. IV C, we can also define the symmetry class (G, φ, c, τ, n) with the integer grading n > 0 [198] for non-Hermitian Hamiltonians as follows. As in the Hermitian case, the shift of integer grading is defined by adding CS for γ i 's (i = 1, · · · , n) satisfying u g (k)γ i u g (k) † (φ gĉg = 1) u g (k)γ * i u g (k) † (φ gĉg = −1) From the Hermitinization given by Eq.
Appendix F: Topological classification based on two antiunitary symmetries Topological classification based on two antiunitary symmetries T + , T − and one unitary symmetry S was considered in Ref. [114]. This classification assumes a point gap, and the corresponding classification table for both complex-energy gaps is shown in Table XV. Notably, two antiunitary symmetries T + and T − are topologically equivalent to each other for non-Hermitian Hamiltonians [118], whereas they are clearly distinct for Hermitian Hamiltonians. As a result, some symmetry classes are equivalent to others in non-Hermitian physics. For instance, the symmetry class only having T + with T 2 + = +1 (T 2 + = −1) is equivalent to that only having T − with T 2 − = +1 (T 2 − = −1).

Appendix G: Winding number in 1D class AIII
We define the winding number that characterizes a generic one-dimensional system with CS [94], which is discussed in Secs. IV D and V. Here an eigenenergy is denoted as E n (k), and the corresponding right (left) eigenstate is denoted as |u n (k) (|u n (k) ): H (k) |u n (k) = E n (k) |u n (k) , H † (k) |u n (k) = E * n (k) |u n (k) .

(G1)
We assume the presence of a real gap, i.e., Re E n (k) = 0 for all n and k. Due to CS described by Eq. (9), when |u n (k) is a right eigenstate with E n (k), Γ |u n (k) is a left eigenstate with −E * n (k). Therefore, eigenstates and eigenenergies can be expressed as |u −n (k) = Γ |u n (k) , E −n (k) = −E * n (k) , where an eigenstate with a positive (negative) n is chosen to satisfy Re E n (k) > 0 (Re E n (k) < 0). With these right and left eigenstates, we introduce the following projection operators: which satisfy P 2 R/L (k) = P R/L (k) and P † R (k) = P L (k). In addition, we have Γ P R (k) Γ = where Eq. (G2) and the completeness condition n |u n (k) u n (k)| = 1 are used. With the above P R/L (k), a non-Hermitian extension of the Q matrix is defined as This Q matrix is Hermitian [Q † (k) = Q (k)] and respects CS [Γ Q (k) Γ = −Q (k)] as can be seen from Eq. (G4). As a result, when the chiral-symmetry operator Γ is diagonal, the Q matrix can be expressed as where the off-diagonal part q (k) is an invertible matrix.
Here q (k) is not necessarily unitary in contrast to the Hermitian case due to Q 2 (k) = 1. Nevertheless, the determinant of q (k) does not vanish for all k since it is invertible; the winding number W can thus be defined with this invertible matrix q (k) as This invariant is relevant for the emergence of the topologically protected edge modes [94], as experimentally observed in photonic systems [133,138].