Exceptional band touching for strongly correlated systems in equilibrium

Quasi-particles described by Green's functions of equilibrium systems exhibit non-Hermitian topological phenomena because of their finite lifetime. This non-Hermitian perspective on equilibrium systems provides new insights into correlated systems and attracts much interest because of its potential to solve open questions in correlated compounds. In this paper, we provide a concise review of the non-Hermitian topological band structures for quantum many-body systems in equilibrium as well as their classification.

The aim of this article is to provide a concise review of these advances in the non-Hermitian perspective in correlated systems in equilibrium. As a 2 × 2 Hamiltonian describes the essential properties, we start with this simplest case and review numerical results demonstrating the emergence of exceptional band touching.
The rest of this paper is organized as follows. In Sec. II, we demonstrate the emergence of EPs for a heavy-fermion system by applying the DMFT to a heavy-fermion system. In Sec. III, we show that SPERs and SPESs can emerge for correlated systems with chiral symmetry. In Sec. IV, we address the ten-fold way classification of the exceptional band touching for single-particle spectrum by taking into account P T -(CP -) and chiral symmetry, where P T -(CP -) symmetry denotes the symmetry under the product of time-reversal and inversion (charge conjugation and inversion), respectively. A short summary and remaining open questions appear at the end of this paper.

II. EXCEPTIONAL POINTS FOR STRONGLY CORRELATED SYSTEMS
In this section, we elucidate that the EPs emerge due to finite lifetimes of quasi-particles for strongly correlated systems 59 . Specifically, the origin of the above non-Hermitian topological phenomena is the imaginary part of the self-energy [see Eqs. (11) and (12)] which describes the lifetime of quasi-particles. The emergence of EPs results in the significant difference of the single-particle spectrum.
In the following, after a brief explanation of EPs (Sec. II A) and the single-particle Green's function (Sec. II B), we demonstrate the emergence of EPs for heavy-fermion systems and see that EPs significantly change the single-particle spectrum. A.
Topological properties of EPs

Case of a 2 × 2 Hamiltonian
Let us first analyze a non-Hermitian 2 × 2 Hamiltonian, which elucidates the essential properties of EPs. It is well-known that a generic 2 × 2 matrix can be expanded by the Pauli matrices τ 's and the identity matrix τ 0 where b µ and d µ (µ = 0, 1, 2, 3) are continuous functions taking real values. One can numerically and analytically confirm that the above non-Hermitian matrix may show EPs. In Fig. 1, energy eigenvalues taking complex numbers are plotted for a specific choice of b's and d's. At the EPs, the Hamiltonian becomes non-diagonalizable. Correspondingly, as one can see in Fig. 1, the band touching occurs both for the real and imaginary parts of the energy eigenvalues. Band structure of a continuum model for b0, b1, b2, b3 = 0, kx, 0, ky and d0, d1, d2, d3 = 0, 0, 0, 0.3 . In panel (a) [(b)] the real (imaginary) part of the energy is plotted, respectively. In these figures, the band touching points marked with green dots correspond to the EPs.
In order to see the details, we diagonalize the Hamiltonian (1), which yields The above equation indicates that band touching occurs both for the real and imaginary parts when the following conditions are satisfied In other words, the above conditions are necessary conditions for the emergence of EPs. One can see that the above conditions are indeed sufficient conditions; supposing that Eq. (3) is satisfied, we can see that the Hamiltonian can be rewritten as with a proper choice of the basis. In this basis, one can see that the Hamiltonian is generically non-diagonalizable for d = 0.
In the above, we have seen the following facts. At the EP, the 2 × 2 Hamiltonian (1) becomes non-diagonalizable, resulting in the exceptional band touching. For the 2×2 Hamiltonian, the EP emerges if and only if Eqs. (3a) and (3b) are satisfied. We note that the band touching is protected by non-trivial topology whose topological invariant is discussed in the next subsection.

Topological invariant characterizing EPs
As shown in Fig. 1, the band touching occurs at the EPs. Such band touching for two-dimensional systems can be topologically characterized by the vorticity, akin to the winding number; Here, we have considered that the band touching occurs at energy E 0 . H(k) (dimH ≥ 2) denotes a generic non-Hermitian matrix. ∇ k := (∂ kx , ∂ ky ). The path of the integral is chosen so that it encloses the EP. For dimH = 2, the vorticity can be written as 74,129 where E ± is the energy eigenvalue [see Eq. (2)].
In the following, we see how the vorticity defined in Eq. (5) characterizes the EPs. Consider a generic non-Hermitian matrix H(k) with dimH ≥ 2 which shows the band touching at energy E 0 . The band touching point can be formulated as where k 0 denotes the EP in the momentum space. Mapping the non-Hermitian Hamiltonian to the Hermitian matrix H, we can rewrite the above condition as Here, we have extended the Hilbert space on which the Pauli matrices ρ's act. This can be easily confirmed by noticing that Eq. (8a) can be written as |det[H(k 0 ) − E 0 ]| 2 = 0. The above fact means that the exceptional band touching can be described by the zero modes of the Hermitian matrixH which is chiral symmetric {H,Σ} = 0 withΣ := 1l ⊗ ρ 3 . Therefore, remembering that the zero modes of the chiral symmetric Hermitian Hamiltonian are characterized by the winding number, we can see that the EPs can be characterized by the vorticity (5); substitutingΣ = 1l ⊗ ρ 3 to Eq. (9) yields Eq. (5) 130 . We note that the vorticity is half-quantized due to the extra prefactor 1/2, which is just a convention.
In this section, we have considered two-dimensional systems. We note, however, that the vorticity is well-defined along a one-dimensional path in the three-dimensional BZ. In this case, the vorticity characterizes exceptional loops in the BZ (see also Table I). For a 2 × 2 Hamiltonian, there is complimentary understanding. The EPs appear when both of Eqs. (3a) and (3b) are satisfied, meaning that one degree of freedom is left in the three dimensions. This remaining degree of freedom forms a loop which is nothing but the exceptional loop in three dimensions.

B.
EPs appearing in the single-particle spectrum In the above, we have seen that a non-Hermitian matrix may show EPs which are characterized by the vorticity (5). In this section, we see that a non-Hermitian matrix governs the single-particle excitation spectrum of correlated systems in equilibrium (i.e., the energy is conserved).
In addition to EPs, the non-Hermiticity of the effective Hamiltonian yields low energy excitations. The energy gap can be pure imaginary because of the non-Hermiticity of H eff . In this case, even when the Bloch Hamiltonian is gapped, the system may show Fermi arcs connecting EPs.
We finish this section by making a comment on an additional condition for the EPs in the single-particle spectrum. The effective Hamiltonian appears in the denominator of the spectral weight (12a), meaning that the EPs are seriously smeared when the denominator is large. Therefore, in order for EPs to emerge as a peak in the single-particle spectral function, the frequency ω should satisfy an additional condition [for instance see Eq. (17a)].

C.
EPs for two-dimensional heavy-fermion systems In this section, we demonstrate that EPs emerge in the single-particle spectrum of a heavy-fermion system, by employing the DMFT. In particular, we analyze the Kondo lattice in two dimensions. The Hamiltonian reads, whereĉ † iαs creates an electron with spin s =↑, ↓ in orbital α = a, b of site i.ŝ ib := 1 2ĉ † ibs σ ss ĉ ibs with the Pauli matrices σ's acting on the spin space.Ŝ is the spin 1/2 operator for the localized spins. Here, the Kondo coupling of electrons in orbital a is neglected for simplicity. The hopping t iα,jβ is defined so that the Bloch Hamiltonian is written as where 0 , t, and t take real values, respectively. The Pauli matrices τ 's act on the orbital space. In the non-interacting case, this model shows two Dirac cones for t = 1 and 0 < 0 < 4. In order to analyze the above correlated electron system, we employ the DMFT 132-135 which treats local correlation exactly. In the DMFT framework, the lattice model is mapped to an effective impurity model where the self-energy of spin s [Σ s (ω + iδ) := diag 0, Σ bs (ω + iδ) ] is computed self-consistently 136 . Here, Σ bs (ω + iδ) denotes the selfenergy for orbital b and spin s. In order to compute the self-energy for the effective impurity model, we employ the numerical renormalization group method (NRG) [137][138][139] . This method directly provides the single-particle spectral function, while other methods based on Monte Carlo calculations [140][141][142] require the analytic continuation.
Once the self-energy is obtained, the single-particle spectrum is obtained as Here, we have omitted the subscript s [Σ(ω + iδ) := diag 0, Σ b (ω + iδ) ] by assuming that the system is in the paramagnetic phase. We note that the effective Hamiltonian is a 2 × 2 matrix. Expanding it with the Pauli matrices as Eq. (1), we obtain the following coefficients Therefore, the conditions for EPs appearing as the peak of the single-particle spectral function are written as Here, the second and the third equations are obtained from Eq. (3), specifying the position of the EP k 0 in the BZ. The first equation specifies the energy ω 0 where the EPs emerge as peaks of the spectral function. We note that in the DMFT framework, the momentum dependence of the self-energy is neglected. However, the EPs should emerge even in calculations beyond the DMFT framework because they are topologically protected.
Let us now analyze the Kondo lattice model (13). In the rest of this section, we set the parameters to t, t , 0 = 1, 0.667, 0.667 . The obtained phase diagram is shown in Fig. 2. When the Kondo coupling is small, an antiferromagnetic phase emerges because the Ruderman-Kittel-Kasuya-Yosida interaction 143-145 becomes dominant. Increasing the interaction J, itinerant electrons and localized spins form singlets due to the Kondo effect. As a result, the anti-ferromagnetic phase is suppressed in the region of strong J. We numerically observe the EPs in the paramagnetic phase. The Kondo effect plays an important role for the emergence of EPs. The self-energy is shown in Figs. 2(b) and 2(c) for T = 0.048t which corresponds to the horizontal line in Fig. 2(a). For small J (J = t), the real and imaginary parts of the self-energy take small values because the electrons are almost decoupled from the localized spins. Increasing the coupling J enhances the Kondo effect, which results in a dip structure of ImΣ b (ω + iδ) in the low-energy region (i.e., around ω ∼ 0). This dip structure of the self-energy induces the EPs. The single-particle spectral function for J = 1.8t is plotted in Fig. 3. Firstly, we show the data obtained by assuming that the imaginary part of the self-energy is zero [see Fig. 3(a)] in order to show that the imaginary part of the self-energy is essential for the EPs. In this figure, we can see a single peak due to the existence of a Dirac cone. Fig. 3(b) shows the spectral function obtained by the DMFT. In this figure, we can see that the dip structure of the imaginary part splits the Dirac cone into two EPs as represented with green dots. Furthermore, we can see that the EPs are connected by the Fermi arc where the bulk gap ∆ c = E + − E − becomes pure imaginary. The emergence of the Fermi arc enhances the local density of states around ω ∼ 0 [see Fig. 4(a)]. In the above, we have seen that the imaginary part of the self-energy splits each of two Dirac cones into a pair of EPs connected with the bulk Fermi arc. As we see below, these bulk Fermi arcs are robust because the EPs are topologically protected. Here, we address the characterization of the above EPs. Because the vorticity is written as Eq. (6) for the 2 × 2 Hamiltonian, we can compute its value by plotting the argument of ∆ 2 c [see Fig. 4(b)]. In this figure, the branch cut of ∆ c is represented with white dashed lines which end at EPs. Therefore, taking the integral along the green line illustrated in Fig. 4(b), we can see that the vorticity takes ν = −1/2. We note that the vorticity takes ν = 1/2 for the EP around k x = −π/2.
Changing the Kondo coupling results in pair annihilation of EPs. Here, we note that there are two scenarios: (i) a pair of EPs originating from a Dirac point are annihilated by themselves; (ii) two pairs EPs exchange the pairs and are annihilated. The former scenario can be observed by decreasing the Kondo coupling J. In Fig. 5(a), we can see that two EPs approach and are annihilated. Correspondingly, the Fermi arc vanishes. The latter scenario can be observed by increasing the interaction J. When the Kondo effect is enhanced, the EPs approach the boundary of the BZ specified by k x = π. On this boundary, the pair of EPs arising from two distinct Dirac cones annihilate each other [see

III. SYMMETRY-PROTECTED EXCEPTIONAL RINGS AND SURFACES IN CORRELATED SYSTEMS
In the previous section, we have seen that electron correlations induce EPs in the absence of symmetry. In addition, it is well-known that the symmetry enriches the topological structures for Hermitian systems [146][147][148][149] . Therefore, it should be valuable to analyze the effects of symmetry on EPs, which is the main subject of this section.
Ref. 60 has revealed that many-body chiral symmetry results in novel types of exceptional band touching, SPERs in two dimensions and SPESs in three dimensions. In the following, after elucidating the topological properties of SPERs and SPESs, we demonstrate the emergence of them in correlated systems. Firstly, we analyze a case of the 2 × 2 Hamiltonian [see Eq. (1)] which captures the essential properties. Here, let us suppose that the Hamiltonian for a two-dimensional system satisfies the following relation which indicates that the system is chiral symmetric [see Eq. (29a)]. The above condition imposes the following symmetry condition on the coefficients, b's and d's: Now, let us consider effects of the symmetry constraint on the EPs. As we have seen in Sec. II A 1, EPs emerge when the two conditions Eqs. (3a) and (3b) are satisfied. We note, however, that one of the conditions, Eq. (3b), is always satisfied by the symmetry constraint, meaning that the number of the conditions for the EPs is reduced. This fact indicates that for the two-dimensional BZ, fixing one degree of freedom is sufficient to obtain the EPs. Therefore, the remaining degree of freedom forms a ring of EPs which is denoted as a SPER 60 . On an arbitrary point of the SPERs, the band touching occurs both for the real and imaginary parts.
We can apply the same argument to a three-dimensional system where SPESs emerge 60 . In this case, the two degrees of freedom are left in the BZ.

2.
Topological invariant characterizing SPERs and SPESs with chiral symmetry In the above, we have seen that the symmetry constraint results in SPERs or SPESs where the exceptional band touching occurs. In this section, we show that the band touching is topologically characterized by the zero-th Chern number, a zero-dimensional topological invariant.
Let us suppose that the 2n × 2n Hamiltonian satisfies the following relation where U Γ is a unitary matrix satisfying U 2 Γ = 1l. The above equation is a generic form of the symmetry constraint (18). We now consider the following Hermitian Hamiltonian composed of H(k); where we have assumed that the exceptional band touching occurs at energy E 0 ∈ iR. In a similar way to the case of Sec. II A 2, we can define the topological invariant characterizing the SPERs and SPESs by addressing topological characterization of zero energy excitations described by the Hermitian HamiltonianH. The essential difference from the previous case (Sec. II A 2) is that the Hermitian Hamiltonian preserves the two distinct constraints of chiral symmetry;ΣH The additional chiral symmetry allows us to define the zero-th Chern number. Due to two distinct constraints of chiral symmetry, the Hamiltonian can be block-diagonalized with a unitary operatorŨ = iΣŨ Γ (Ũ 2 = 1l), Here, H + (H − ) denotes the Hamiltonian acting on the subspace where the operatorŨ is reduced to 1l (−1l), respectively. We denote these subspaces by plus and minus sectors. We note that applying eitherΣ orŨ Γ exchanges the plus and minus sectors because of the anti-commutation relation {Ũ Γ ,Σ} = {Σ,Ũ } = 0. Namely, letting |+ be a state of the plus sector (Ũ |+ = |+ ), we obtainŨΣ|+ = −Σ|+ , which means thatΣ|+ belongs to the minus sector. The above facts indicate that the block-diagonalized Hamiltonians H + and H − are related to each other and belong to symmetry class A. Therefore, the characterization of the zero energy excitations ofH(k) can be done with the zero-th Chern number for the plus sector which corresponds to the number of the eigenstates with negative eigenvalues of H + . This fact suggests Z classification of zero-dimensional Hermitian systems belonging to class A. We note that the block-diagonalized Hamiltonian is rewritten as H + = iU Γ H, which can be seen as follows. Noticing that the unitary matrix V block-diagonalizes the unitary operatorŨ = U Γ ⊗ ρ 2 , we can block-diagonalize the Hermitian HamiltonianH: Here, we have used the relation U 2 Γ = 1l. Therefore, the SPERs and the SPESs are characterized by the zero-th Chern number which is the number of negative eigenvalues of the Hermitian Hamiltonian H + (k) = iH(k)U Γ at each point in the BZ.
The above result indicates that the dimension of the objects composed of the exceptional band touching becomes one-dimensional higher by chiral symmetry [Eq. (20)] compared to system without the symmetry (see Table I). Table I   dimension  1  2  3  no symmetry  -point loop  with chiral symmetry point ring surface   TABLE I. Objects formed by EPs in the BZ for each case of spatial dimensions. In the presence of chiral symmetry, exceptional band touching forms objects which are one-dimensional higher than the ones in the absence of symmetry. also indicates that the EPs emerging in one-dimensional systems are either unstable or symmetry-protected.
We finish this section with the complementary understanding for the 2 × 2 Hamiltonian. Namely, exceptional band touching appears at points where both of Eqs. (3a) and (3b) are satisfied. Thus, in the absence of symmetry, exceptional band touching forms (d − 2)-dimensional objects in the d-dimensional BZ. On the other hand, in the presence of chiral symmetry, exceptional band touching forms (d − 1)-dimensional objects because Eq. (3b) is always satisfied by symmetry.

B.
SPERs for a correlated honeycomb lattice The SPERs can emerge for strongly correlated systems in equilibrium. In order to demonstrate the emergence of the SPERs, we apply the DMFT+NRG to a Hubbard model of a honeycomb lattice. The Hamiltonian readŝ whereĉ † iαs creates a fermion with spin s =↑, ↓ at site i and sublattice α = A, B.n iα↑ :=ĉ † iαsĉ iαs . The first term describes the nearest-neighbor hopping with t iα,jβ ∈ R. The second term describes the on-site repulsion (U α ≥ 0). Applying the Fourier transformation for U α = 0, we obtain the Bloch Hamiltonian which is written as Here, we have assumed the hopping t (t := rt) between sites connected with gray (brown) lines, respectively. Vectors a's are illustrated in Fig. 7. We consider that this model can be fabricated for cold atoms because the inhomogeneous Hubbard interaction is implemented with the optical Feshbach resonance 150,151 . The above model preserves the chiral symmetry for an arbitrary value of the interaction U α :

A B
with sgn(α) taking 1 (−1) for α = A (α = B), respectively. This symmetry imposes the following constraint on the Green's function In particular, for ω = 0, the above condition can be rewritten as in terms of the effective Hamiltonian H eff (ω, k). This constraint is nothing but the symmetry discussed in the previous section [see Eq. (18)]. Therefore, the chiral symmetry of the correlated systems (28) protectes the SPERs emerging in the single-particle spectrum. The DMFT results elucidate the emergence of the SPERs. In Fig. 8, the spectrum at ω = 0 is plotted for several values of the temperature. In the non-interacting case, it is well-known that the Dirac cones appear at the corners of the BZ illustrated with the white hexagon. In the presence of the correlations, the Dirac cones split into rings [see green rings in Fig. 8(a)]. Increasing the temperature suppresses the lifetimes of quasi-particles. Correspondingly, the SPERs become large [ Fig. 8(b)]. In this figure, we can also see the effect of symmetry on the Fermi arcs shown in Fig. 3(b). Because of the chiral symmetry, the Fermi arcs change into the Fermi planes. This is because the energy eigenvalues E n appear in a pair (E n , −E * n ) or become pure imaginary in the presence of the chiral symmetry (29b) 152 . For higher temperatures, the SPERs, arising from distinct Dirac cones, merge into the single loop [see Fig. 3(c)]. We also note that the presence of the Dirac cones is not a necessary condition for the SPERs. Introducing the anisotropy . We note that the SPERs are topologically stable; Fig. 9(a) shows that the numerical characterization of the SPERs with the zero-th Chern number can be done. Finally we show that the emergence of Fermi plane accompanying the SPER enhances the specific heat C = d H /dT because the Fermi plane induces additional low energy excitations. In Fig. 9(b), the specific heat is shown with the red line. For comparison, we also plot data with the blue line by assuming that the imaginary part of self-energy for A-and B-sublattices takes the average value Im[Σ A (ω + iδ) + Σ B (ω + iδ)]/2. We note that the system does not show SPERs when the imaginary part for A-sublattice is identical to that for B-sublattice. In this figure, we can see that the specific heat is enhanced because of the Fermi planes accompanying SPERs.

C. SPESs for a correlated diamond lattice
The emergence of the SPESs can also be demonstrated by applying the DMFT to a Hubbard model of a diamond lattice, which is a three-dimensional extension of the honeycomb Hubbard model (26). The lattice structure and the BZ is shown in Fig. 10(a) and 10(b), respectively. In a similar way as the previous section, we introduce an inhomogeneity of the interaction.
In the following, we see the details. For U A = 8t, U B = 0, and T = 0.8t, the SPESs emerge as shown in Fig. 10(c). Here, we have employed the iterative perturbation method 153,154 as the impurity solver of the DMFT. In the following, we see the results in detail. In Fig. 11(a), the single-particle spectral function at zero energy A(ω = 0, k) is plotted for the k xy -k z plane [i.e., the blue plane in Fig. 10(b)]. The green dots plotted in Fig. 11(a) correspond to the sections of SPESs. We note that in the region enclosed with the SPESs, the energy gap becomes pure imaginary, meaning that the zero energy excitations appear in this region. Thus, the Fermi volume appears instead of the Fermi arc discussed in Sec. II C. Fig. 11(b) shows the single-particle spectral function A(ω, k) along the lines connecting the  high symmetry points in the BZ. In this figure, we can confirm the emergence of the Fermi volume by the presence of the zero energy excitations between X and K points. Outside of the SPESs, the zero energy excitations disappear.
We finish this section with a comment concerning the effect of SPESs on the magnetic response. As shown in Fig. 11(c), the LDOS of the B-sublattice is enhanced by the Fermi volume accompanying the SPESs. We note that the LDOS of the A-sublattice is just renormalized. This imbalance of the LDOS can induce a counterintuitive behavior of the local magnetic susceptibility. In Fig. 11(b), the local magnetic susceptibility computed with the random-phase approximation (RPA) 155 is plotted. As shown in Fig. 11(d), due to the imbalance of the LDOS, the magnetic susceptibility of the B-sublattice becomes larger than that of the A-sublattice, although the interaction of the B-sublattice is weaker than that of the A-sublattice.

IV. TEN-FOLD WAY CLASSIFICATION OF THE EXCEPTIONAL BAND TOUCHING IN EQUILIBRIUM SYSTEMS
In Sec. III, we have seen that the correlated systems with chiral symmetry may show SPERs and SPESs in two and three dimensions, respectively. These SPERs and SPESs are characterized by the zero-th Chern number, a zero-dimensional topological invariant taking an arbitrary integer (see Sec. III A 2). In other words, the topological classification of the exceptional band touching is Z for the system with chiral symmetry.
In this section, by generalizing the argument in Sec. III, we address the topological classification of the exceptional band touching. Specifically, we carry out the ten-fold way classification 146-149 of exceptional band touching in the presence/absence of P T -, CP -, and chiral symmetry for correlated systems. This is because P T -(CP -) symmetry is closed at each point in the BZ as well as the chiral symmetry (i.e., the corresponding symmetry transformation does no flip the momentum). We note that the 38-fold way classification for exceptional band touching is carried out in Ref. 94 for a generic Bloch Hamiltonian. However, our analysis clarifies which symmetry classes are relevant for correlated systems. Our ten-fold way classification is consistent with the corresponding classification results for 38 symmetry classes.
In what follows, we address the classification of exceptional band touching after a brief description of the relevant symmetry.

A.
Symmetry constraints

P T -symmetry
For the correlated systems preserving P T -symmetry (i.e., symmetry under the product of time-reversal and spatial inversion), the second quantized HamiltonianĤ satisfies Here, the anti-unitary operator P T is written as whereĉ † iα creates a fermion with state α at site i.Û P T is a unitary operator. K is an operator taking complex conjugation. U P T is a matrix satisfying U P T U * P T = ±1l. Here we have supposed that under the inversion, site j is mapped to −j.
For P T -symmetric systems, the Green's function satisfies 20 which can be rewritten as Eq. (32a) can be seen by a straightforward calculation 156 .

CP -symmetry
For correlated systems preserving CP -symmetry (i.e., symmetry under the product of charge conjugation and inversion), the second quantized HamiltonianĤ satisfies with CP corresponding to the unitary operator CP =Û CP which transformsĉ iα aŝ Here, U CP is a unitary matrix satisfying U CP U * CP = ±1l. For CP -symmetric systems, the Green's function satisfies which can be rewritten as Eq. (34a) can be obtained by using the following relations: and We note that applying the Fourier transformation, G(−ω − iδ, k) is rewritten as G A (−t, k) which is defined as Eqs. (35) and (36) are obtained by straightforward calculations 157,158 .

Chiral symmetry
For the correlated systems preserving chiral symmetry, the second quantized HamiltonianĤ satisfieŝ Here,Û Γ is a unitary operator transforming the annihilation operator aŝ where U Γ is a matrix satisfying U 2 Γ = 1l. For chiral symmetric systems, the Green's function satisfies which can be rewritten as Eq. (32a) can be obtained by a straightforward calculation 159 . This equation can also be obtained from Eqs. (32a) and (34a) by noticing that applying the operatorΓ is equivalent to applying the product of the operators P T and CP .

B. Ten-fold way classification
Prior to the topological classification of exceptional band touching, we note the following two facts. (i) Exceptional band touching of the non-Hermitian Hamiltonian H eff (ω = 0, k) can be described by a Hermitian Hamiltonian satisfying {H,Σ} = 0 withΣ = 1l ⊗ ρ 3 [see e.g., Eq. (8b)] 87,89,90,94,160,161 . (ii) For Hermitian systems, the classification of d EP -dimensional gapless excitations in d spatial dimensions is accomplished by classifying the δ − 1 dimensional gapped Hermitian Hamiltonian with δ = d − d EP 162,163 . Thus, the problem is reduced to classifying gapless excitations of the Hermitian HamiltonianH in the presence/absence of the following symmetry constraints: andŨ P TŨ * P T = ±1l,Ũ CPŨ * CP = ±1l, andŨ ΓŨΓ = 1l. The above relation can also be written with the two anti-unitary operators ( P T =Ũ P T K, and CP =Ũ CP K) and a unitary operator (Γ :=Ũ Γ ). We note that the above unitary matrices (Ũ P T ,Ũ CP , andŨ Γ ) satisfy the following commutation/anti-commutation relations: Therefore, exceptional band touching can be classified by addressing the classification of gapless excitations in Hermitian systems with additional chiral symmetry whose operatorΣ satisfies Eq. (41). We address the classification based on the method of the Clifford algebra 147,164 . The specific procedure of the classification is summarized in Sec. IV B 2. In the next section, we discuss the classification results.

Classification results
Classification results of d EP -dimensional exceptional band touching for H eff (ω = 0, k) are summarized in Table II. Here, we consider the d-dimensional BZ.
For each case of δ = d − d EP and symmetry class, this table elucidates the presence/absence of the δ − 1 dimensional topological invariant in the BZ;"Z" ("Z 2 ") indicates the presence of a topological invariant taking an arbitrary integer (0 or 1), respectively; "0" appearing as the classification result (i.e., from sixth to 13-th column) indicates the absence of such topological invariants.
These classification results explain the exceptional band touching reported so far. For instance, this table indicates the Z classification for class A with δ = 2, meaning that there exists exceptional band touching characterized by a one-dimensional topological invariant. This classification result explains the presence of EPs observed in Fig. 3(b) (d = 2 and d EP = 0). We note that the emergence of EPs for class A is also reported for systems with disorder 58,61 or electron-phonon coupling 57 . With d = 3 and d EP = 1, we obtain the same δ, resulting in the Z classification for class A. This fact also explains the emergence of exceptional loops in three-dimensional systems 64 . The classification results for symmetry classes AI, AII, D, and C elucidate the stability of these band touching points in the presence/absence of P T -or CP -symmetry.
The emergence of SPERs observed in Fig. 8 is also consistent with Table II (d = 2 and d EP = 1). For class AIII with δ = 1, we obtain the Z classification, implying the presence of the zero-th Chern number. The Z classification for class AIII with δ = 1 is also consistent with the emergence of SPESs observed in Fig. 10(c) (d = 3 and d EP = 2). We note that the classification results for symmetry classes BDI, DIII, CII, and CI elucidate the stability of the exceptional band touching in the presence of P T -or CP -symmetry.
While we have mainly analyzed exceptional band touching for symmetry class A or AIII in the previous sections, the classification results summarized in Table II imply the existence of novel exceptional band touching. The verification of exceptional band touching for other cases of symmetry is still missing as well as the material realization.
It is also worth noting that the above table may explain the exceptional band touching away from ω = 0 for class A by recognizing the frequency as an additional momentum, although we have restricted ourselves to ω = 0 so far. Indeed, the emergence of exceptional rings in the ω-k space has been demonstrated for two-dimensional heavy fermions 63 (d = 3 and d EP = 1), which is consistent with the Z classification for symmetry class A with δ = 2. The above fact allows us to interpret the Z classification for class A with δ = 4; it implies the presence of novel EPs in the ω-k space for three spatial dimensions. Further analysis in this direction should be addressed.

2.
Details of the classification for the Hermitian Hamiltonian As discussed in the beginning of this section, classification of the d EP -dimensional exceptional band touching in d spatial dimensions is accomplished by classifying the δ −1-dimensional gapped Hermitian Hamiltonian with additional chiral symmetry satisfying Eq. (41). Here, δ denotes codimension (δ = d − d EP ). In this section, we address the classification of the gapped Hermitian Hamiltonian based on the method of the Clifford algebra 147,164 .
In what follows are technical details of the derivation of Table II. Thus, readers, who are interested in physical interpretation of the classification results rather than the technical details, can skip this section.
Specifically, the topological classification based on the Clifford algebra can be carried out by the following steps 147,164 .
(i) Deform the Hermitian HamiltonianH to the Hermitian Dirac Hamiltonian H 0 where γ's satisfy {γ i , γ j } = 2δ i,j for i, j = 0. · · · , δ − 1. Because such deformation is possible for an arbitrary gapped Hamiltonian, the problem is reduced to classifying the possible mass term γ 0 .
(iii) By adding the mass term γ 0 , consider the extension problem to obtain the corresponding classifying space which turns out to be C q (R q−p ) when the extension problem is Cl q → Cl q+1 (Cl p,q → Cl p,q+1 ), respectively. Here, we note that the corresponding classifying space of the extension problem Cl p,q → Cl p+1,q is R 2+p−q 164 . (iv) By making use of the relation summarized in Table III, obtain the classification result π 0 (C q ) [π 0 (R q )]. We note that the relations π 0 (C q+2 ) = π 0 (C q ) and π 0 (R q+8 ) = π 0 (R q ) hold, which are known as the Bott periodicity.
With the above procedure, (i)-(iv), we can obtain the classification results shown in Table II. In the last column, the Clifford algebra, which is generated by the mass term, the kinetic terms and symmetry operators, is shown for each symmetry class. Although one can reproduce the classification results from the last column, we explicitly apply the above procedure for class A and AII as examples.
class A-. Remembering that the HamiltonianH in δ − 1 dimensions is chiral symmetric {H,Σ} = 0, we obtain the Clifford algebra C δ generated by Introducing the mass term γ 0 results in the extension problem which is written as Cl δ → Cl δ+1 . Here, the Clifford algebra Cl δ+1 is generated by which is shown in the last column of Table II. Therefore, the corresponding classifying space is C δ , which indicates that the classification result is computed with π 0 (C δ ). By making use of the Bott periodicity and the relation summarized in Table III, we obtain the classification results for δ = 1, · · · , 8.
Introducing the mass term γ 0 results in the extension problem which is written as Cl δ+1,1 → Cl δ+2,1 . Here, the Clifford algebra Cl δ+2,1 is generated by {Jγ 0 , Jγ 1 , · · · , Jγ δ−1 , P T , J P T ;Σ}, (48) which is shown in the last column of Table II. Therefore, the corresponding classifying space is R 2+δ , which indicates that the classification result is computed with π 0 (R 2+δ ). By making use of the Bott periodicity and the relation summarized in Table III, we obtain the classification results for δ = 1, · · · , 8. We note that for δ = d + 1, Table II indicates the classification results for the d-dimensional gapped Hamiltonian with additional chiral symmetry satisfying Eq. (41). In this case, the classification results are given by the homotopy group π 0 (C q−1+d ) or π 0 (C q−1+d ) with an integer q while the original ten-fold way classification for topological insulators/superconductors is given by π 0 (C q−d ) or π 0 (C q−d ). This is due to the fact that applying P T or CP does not flip the momentum k 160,165,166 while applying time-reversal or particle-hole operator does (k → −k).

V. SUMMARY AND OUTLOOK
In this paper, we have briefly reviewed the recently developed non-Hermitian perspective of the band structure in equilibrium systems. We have seen that the finite lifetime of quasi-particles induces EPs. In addition, we have seen that the symmetry of the many-body Hamiltonian results in SPERs (SPESs) in two (three) dimensions, respectively. While the above non-Hermitian perspective has been developed recently, there are several open questions to be addressed.
For instance, effects of EPs on transport properties should be further analyzed. As seen in this paper, the exceptional band touching induces low energy excitations such as Fermi arcs. The emergence of these low energy excitations may change the conductivity or other electromagnetic responses.
The experimental observation of EPs in electronic systems is also a crucial issue along this direction. Topological Kondo insulators such as SmB 6 167-173 and YbB 12 174,175 might serve as a platform of the EPs because they are strongly correlated materials and show Dirac cones at surfaces. Prior to the experimental observation, the quantitative analysis such as LDA+DMFT calculations should be carried out as well as the theoretical proposal of how to experimentally observe the EPs.
While this paper focuses on exceptional band touching, non-Hermiticity induces richer topological physics. Non-Hermitian skin effect is the another representative unique phenomenon 77,[82][83][84][85][86][176][177][178][179] ; the energy spectrum of a non-Hermitian matrix significantly depends on the boundary condition when the skin effect occurs. Elucidating whether the non-Hermiticity by the finite lifetimes induces the skin effect is an intriguing theoretical open question to be addressed.
Finally, we comment on another significant issue of non-Hermiticity and correlations. Recently, a fractional quantum Hall phase, a topologically ordered phase, has been extended to non-Hermitian systems 114 . The extension of topologically ordered phases to non-Hermitian systems is further addressed for a non-Hermitian toric code 180,181 . Developing the effective field theory to describe these non-Hermitian topologically ordered phases should be addressed as well as extending them to systems with symmetry (e.g., time-reversal symmetry).
where En (n = 1, · · · , dimH) denotes the energy eigenvalues of H. Substituting Eq. (1) to the above equation, we obtain where E± is the energy eigenvalues [see Eq. (2)]. This can be rewritten as with ∆ := (E+ − E−)/2 = √ b 2 − d 2 + 2ib · d. Here, we have omitted the term proportional to b0 + id0 by assuming that it is canceled with E0. The last line of the above equation corresponds to the right hand side of Eq. (6) . 130 Eq. (5) is obtained as follows. With Σ = 1l ⊗ ρ3, Eq. (9) is rewritten as The last line corresponds to Eq. (5) up to the prefactor; ν = −νW /2 . where Hint denotes the local interaction term Hint := Js 0b · S. δ(τ − τ ) is the delta function. trS denotes taking trace for the localized spin. Gs(τ − τ ) denotes the Green's function of the effective bath.c bs (c bs ) is a Grassmannian variable which corresponds to the creation operator c † 0bs (annihilation operator c 0bs ) at site i = 0. Solving the above model with an impurity solver, we obtain the self-energy Σ bs , which allows us to compute the Green's function as G αβ (iωm + i n, q + k)G βα (iωn, k).
Here, ωn and n denote the Matsubara frequency [ωn = (2n + 1)πT and n = 2nπT with n ∈ Z]. N denotes the number of unit cells. The local magnetic susceptibility χ s α is obtained as χ s A = (χ RPA AA + χ RPA AB )/2, χ s B = (χ RPA BB + χ RPA BA )/2, with q = 0. We set n → 0 instead of doing analytic continuation . 156 Eq. (32a) can be obtained as follows. Firstly, we note that the following relations hold