Topological classification of defects in non-Hermitian systems
Chun-Hui Liu Shu Chen

      
      
      
      
      
      liuchunhui@iphy.ac.cn
         Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

schen@iphy.ac.cn
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China

          
           We classify topological defects in non-Hermitian systems with point gap, real gap and imaginary gap for all the Bernard-LeClair classes or generalized Bernard-LeClair classes in all dimensions. The defect Hamiltonian H(k,  r) is described by a non-Hermitian Hamiltonian with spatially modulated adiabatical parameter r surrounding the defect. While the non-Hermitian system with point gap belongs to any of 38 symmetry classes (Bernard-LeClair classes), for non-Hermitian systems with line-like gap we get 54 non-equivalent generalized Bernard-LeClair classes as a natural generalization of  point gap classes. Although the classification of defects in Hermitian systems has been explored in the context of standard ten-fold Altland-Zirnbauer symmetry classes, a complete understanding of the role of the general non-Hermitian symmetries on the topological defects and their associated classification are still lacking.
           By continuous transformation and homeomorphic mapping, these non-Hermitian defect systems can be mapped to topologically equivalent Hermitian systems with associated symmetries,  and we get the topological classification by classifying the corresponding Hermitian Hamiltonians. We discuss some non-trivial classes with point gap according to our classification table, and give explicitly the topological invariants for these classes. We also study some lattice or continuous models and discuss the correspondence between the topological number and zero modes at the topological defect.
          
          Topological classification of defects in non-Hermitian systems
    Shu Chen
    December 30, 2023
==============================================================



§ INTRODUCTION

            Topological band theory has  achieved great success in the past decades <cit.> and topological classification according to standard ten-fold Altland-Zirnbauer (AZ) symmetry classes, defined by time-reversal symmetry, particle-hole symmetry and their combination, has been well established <cit.>.
            In the scheme of AZ classification, a topological phase can be characterized by either ℤ or ℤ_2 number. Two well-known examples are Haldane model and Kane-Mele model characterized by Chern number <cit.> and ℤ_2 number<cit.>, respectively. In general, the bulk topological number has a correspondence with it's stable edge gapless modes.
            This correspondence leads to quantized Hall conductance and spin current <cit.>. The stable gapless modes also occur at the non-trivial topological defects, which have been studied in field theory <cit.> and condensed matter systems <cit.>. Systematic classifications of topological defects in insulators and superconductors have been carried out for AZ symmetry classes <cit.>.

             Recent experimental studies of non-Hermitian properties in optic systems, electrical systems and open quantum systems  <cit.>  have stimulated the development of non-Hermitian physics <cit.>. Motivated by these progresses, the scope of topological phase of matter has also been extended to non-Hermitian systems <cit.>.              It has been unveiled that non-Hermitian systems exhibit many different properties from the Hermitian systems, e.g., biorthonormal eigenvectors <cit.>, the existence of exception points <cit.>, unusual bulk-edge correspondence <cit.> and emergence of non-Hermitian skin effect in non-reciprocal systems  <cit.>. Effects of non-Hermiticity on defect states are also studied for some specific non-Hermitian topological models <cit.>.
             Except the fundamental interest of understanding these differences,  novel topological properties of non-Hermitian systems may bring potential application in topological lasers and high-sensitive sensors.
             Besides extensive studies on various topological non-Hermitian models <cit.>, classification of non-Hermitian topological phases has been carried out by different groups <cit.>. In a recent work, Gong et. al went an important step for the classification of non-Hermitian phases with non-spatial symmetries <cit.> by considering only time-reversal, pseudo particle-hole and  sublattice symmetries. After that, the topological classification of non-Hermitian systems in the AZ classes with an additional reflection symmetry was given by us <cit.>. In general, non-Hermitian systems have more types of fundamental non-spatial symmetries than their Hermitian counterparts, i.e., there are totally 38 different classes according to Bernard-LeClair (BL) symmetry classes <cit.>, which can be viewed as a natural non-Hermitian generalization of the ten AZ random matrix classes <cit.>.  Very recently, the full classifications for non-Hermitian topological band systems are obtained by Sato et.al <cit.> and Zhou et.al<cit.>.

             So far, the topological classification of non-Hermitian systems focused on the band systems, classification of topological defects in these systems is still lacking. Aiming to fill this gap, in this work we make a full classification of topological defects for non-Hermitian systems. Due to the complex nature of spectra,  non-Hermitian systems have the point-like and line-like gaps, which corresponds to different complex-spectral-flattening procedure <cit.>.
             For non-Hermitian systems with point-like gaps, the classification of topological defect can be carried out in the scheme of BL symmetry classes, which yield 38 different classes with the help of the equivalent transformation H→ iH <cit.>. However, for non-Hermitian systems with line-like gaps, we notice that the transformation H→ iH
             can not be treated as  an equivalent transformation as a real (imaginary) gap is changed to  an imaginary (real) gap by such a transformation. Consequently, we construct 54 generalized Bernard-LeClair (GBL) classes for these systems. For both types of complex-energy gaps, we can transform the classification problem of non-Hermitian Hamiltonian to the classification problem of Hermitian Hamiltonian by continuous transformation and homeomorphic mapping. Then we classify the Hermitian Hamiltonian by using the K theory and Clifford algebra <cit.>.
               And we also construct the topological number of topological defects for some BL classes with point gap. Then we study some continuous models and lattice models, and find that there is a correspondence between defect's topological number and gapless modes at the defect in some of these models.

The paper is organized as follows. In section II, we first give an introductory description of symmetry of non-Hermitian system in the AZ, BL and GBL classes, and also a brief introduction of different gap types. In section III, we complete topological classification of defects in non-Hermitian systems with point gap, real gap and imaginary gap, respectively, for all BL classes or GBL classes. In Section IV, we construct topological numbers of topological defects for some BL classes. In Section V,we study several example systems. A summary is given in the last section.




§ BRIEF REVIEW OF NON-HERMITIAN SYMMETRIES AND GAP TYPES

Before presenting our approach of topological classification of defects in non-Hermitian systems, we first review briefly symmetries of non-Hermitian systems in the AZ and BL classes and the definitions of point gap, real and imaginary gaps, and then introduce the GBL classes. An introduction to these notations and definitions is necessary for our further classification of defects.



 §.§ Altland-Zirnbauer Class

For Hermitian topological systems, the topological classification according time-reversal symmetry, particle-hole symmetry and chiral symmetry is called the AZ symmetry class, which leads to the standard 10-fold symmetry classification. It is known that AZ classes are not able to fully describe the internal symmetries of non-Hermitian systems, which belong to  more general BL classes <cit.>. For a non-Hermitian Hamiltonian in momentum space, the time-reversal symmetry is defined by

    T=U_T 𝒦,     
        T H(- k)=  H( k) T,

where 𝒦 is a complex conjugation operator and U_K is a unitary matrix. Similarly, the particle-hole symmetry is defined by
    
    A=U_A ℛ,     
            A H(- k)= -H( k)A,

where ℛ is a transpose operator and U_A is a unitary matrix <cit.>. Given that T is the time-reversal operator and A
the particle-hole operator, Γ=TA is the chiral symmetry operator. Similar to the Hermitian case, AZ classes are constructed by T, A and Γ (see Table II).



 §.§ Bernard-LeClair Class

 Bernard and LeClair made a full classification for non-Hermitian random matrix <cit.>, and it is the basic building block of non-Hermitian symmetries. In general, there are four types of symmetries in the non-Hermitian case denoted by K, Q, C and P, which are defined by
 
    H=kH^*k^-1   ,   kk^*=η_k 𝕀,    K  sym.
    
        H=qH^† q^-1   ,   q^2=𝕀,    Q  sym.
    
        H=ϵ_c cH^Tc^-1   ,   cc^*=η_c 𝕀,   C  sym.
    
        H=-pHp^-1   ,   p^2=𝕀,    P  sym.

where k, q, c and p are unitary matrices and η_k,ϵ_c,η_c= ± 1. The four unitary matrices satisfy:
      
    c=ϵ_pcpcp^T,    k=ϵ_pkpkp^T,    c=ϵ_qcqcq^T,    p=ϵ_pqqpq^†,

with ϵ_pc, ϵ_pk, ϵ_qc, ϵ_pq=± 1. We can construct 63 symmetry classes by these symmetries and the sign of η_k,ϵ_c,η_c,ϵ_pc,ϵ_pk,ϵ_qc and ϵ_pq.
For the point gap system, one can redefine H → iH as it does not change the properties of point gap. Consequently, H=-kH^*k^-1 and H=-qH^† q^-1 transform to K and Q symmetries, respectively.
The 63 symmetry classes reduce to 38 topological classes (Non, P, Q, K1-2, C1-4, PQ1-2, PK1-3, PC1-4, QC1-8, PQC1-12) up to a redefinition of equivalence <cit.> (see Table III).
For line gap systems,  the transformation H → iH can not be taken as an equivalent  transformation, and then we need more than 38 classes beyond the BL classes. To understand this, we would like to introduce definition of gap before going to the discussion of the generalization of BL classes.





 §.§ Point gap, real gap and imaginary gap 

In general, an energy gap in the band theory means a forbidden energy region with no occupancy of states. For the non-Hermitian systems, the definition of energy gap is nontrivial as the spectrum becomes complex. According to Kawabata et. al <cit.>, non-Hermitian systems should have two different types of complex-energy gaps, i.e., the point-like and line-like gaps. Here we follow the definitions of Kawabata et. al.
Consider the complex energy plane, if a system has a point gap, it means that the band spectra can't cross the zero point (i.e., ∀ k, det(H( k))  0) as schematically shown in Fig.(<ref>a).
        If a system has a real gap, it means that the band energies can't cross the imaginary axis (i.e., ∀ j,  k, Re( E_j( k))  0) as shown in Fig.(<ref>b). If a system  has an imaginary gap, it means that the band energies can't cross the real axis (i.e., ∀ j,  k, Im( E_j( k) ) 0) as shown in Fig.(<ref>c).
        Based on the three different constraints, we can get three different classifications.
        



 §.§ Generalized Bernard-LeClair Class

Since the transformation H → iH  isn't an equivalent transformation for line gap systems, we now construct all nonequivalent non-Hermitian classes without considering H → iH as an equivalent transformation. In this case:

    H=ϵ_kkH^*k^-1   ,   kk^*=η_k 𝕀,    K  sym.
    
        H=ϵ_qqH^† q^-1   ,   q^2=𝕀,    Q  sym.
    
        H=ϵ_c cH^Tc^-1   ,   cc^*=η_c 𝕀,   C  sym.
    
        H=-pHp^-1   ,   p^2=𝕀,    P  sym.

    were ϵ_k=±1, ϵ_q=±1 and other notations are the same as the BL class. We can construct 54 nonequivalent symmetry classes (Non, P, Qa-b, K1-2a-b, C1-4, PQ1-2, PK1-2, PK3a-b, PC1-4, QC1-8a-b, PQC1-8, PQC9-12a-b) by these symmetries and the sign of ϵ_k,ϵ_q,η_k,ϵ_c,η_c,ϵ_pc,ϵ_pk,ϵ_qc and ϵ_pq and equivalence relations (see Table IV,V). The equivalence relations are similar to the equivalence relationships described by Zhou et al<cit.>, except that ϵ_k̃=-ϵ_k and ϵ_q̃=-ϵ_q when we consider P and K symmetries or P and Q symmetries (k̃=pk and q̃=√(ϵ_pq)qp) <cit.>.
   To distinguish with the previous BL class which supports 38 different classes, we call this class as GBL class.




§ TOPOLOGICAL CLASSIFICATION 

A non-Hermitian periodic system with topological defects is described by H( k,  r), where k is defined in a d-dimensional Brillouin zone T^d, and r is defined on a D-dimensional surface S^D surrounding the defect (Fig.<ref>). The defect Hamiltonian is a band Hamiltonian slowly modulated by a parameter r, which
includes spatial coordinates and/or a temporal parameter and changes slow enough so that the bulk system separated far
from the defect core still can be characterized by momentum k <cit.> .



 §.§  Point gap classification 

If the system has a point gap, we can map the non-Hermitian Hamiltonian to a Hermitian one by the doubling process <cit.>:
          
    H̃( k,  r)=[ [           0   H( k,  r); H( k,  r)^†           0 ] ].

The doubled Hamiltonian H̃( k,  r) fulfills an enforced additional chiral symmetry: 
              
    ΣH̃( k,  r)=  -H̃( k,  r) Σ ,

where Σ = σ_z ⊗ 1 and Σ^2=1. And the constraint on det(H( k,  r))  0 is equivalent to det(H̃( k,  r))  0. Then H( k,  r) is homeomorphic to H̃( k,  r), and it is equivalent to classify Hermitian Hamiltonian H̃( k,  r) with chiral symmetry. If
H( k,  r) has T, A, Γ, K, Q, P and C symmetries,  H̃( k,  r) has the corresponding symmetries:
 
    T̃H̃( k,  r)= H̃(- k,  r)T̃,

 
    ÃH̃( k,  r)= - H̃(- k,  r)Ã,


    Γ̃H̃( k,  r)= - H̃( k,  r)Γ̃,


    K̃H̃( k,  r)=H̃(- k,  r)K̃,


    Q̃H̃( k,  r)=H̃( k,  r)Q̃,


    P̃H̃( k,  r)=-H̃( k,  r)P̃,


    C̃H̃( k,  r)=ϵ_c H̃(- k,  r)C̃,

  with T̃=σ_0⊗ U_T𝒦, Ã=σ_x⊗ U_A𝒦, Γ̃=T̃Ã, K̃=σ_0 ⊗ k𝒦, Q̃=σ_x⊗ q, P̃=σ_0 ⊗ p and C̃=σ_x⊗ c𝒦.

  Generally, we can represent the defect Hamiltonian as:



      
    H̃( k,  r)=γ̃_0+ k_1γ̃_1^k+...+k_d γ̃_d^k +r_1γ̃_1^r+...+r_Dγ̃_D^r ,

where
        γ̃_0,γ̃_i^k(i=1,...,d) and γ̃_j^r(j=1,...,D) anticommute with each other and their squares equal to the identity operator. Using the commutation relations of symmetry operators and Hamiltonian, we can construct the  Clifford algebra's extension for each symmetry class. Then we can get the space of the mass term (γ̃_̃0̃) by the correspondence between the Clifford algebra's extension and space of the mass term (Table I).


        For convenience, here we briefly introduce Clifford algebra. There are two types of Clifford algebra, i.e., the complex Clifford algebra and real Clifford algebra. Complex Clifford algebra is defined in the complex domain, and its generators {γ_1,...,γ_n} satisfy that {γ_i,γ_j}=2δ_ij. Complex Clifford algebra can be represented as Cl_n. On the other hand, real Clifford algebra is defined in the real domain, and its generators {γ_1^-,...,γ_n1^-,γ_1^+,...,γ_n2^+}  anticommute with each others. While γ_i^-(i=1,...,n1) are squared to -1, γ_j^+(j=1,...,n2) are squared to 1. The real Clifford algebra can be represented as Cl_n1,n2.



Once we know the space of the mass term, we can get the topological classification by calculating the zero-order homotopy group of the space of the mass term. For the case of point gap,
we give the classification according to the symmetry classes labeled by both standard AZ classes and  BL classes, despite that the former ones are the subclasses of the latter ones. For convenience, when we discuss the AZ classes, we also label the corresponding BL classes simultaneously.
Firstly, we consider the complex classes:

        Class A (Non): The generators of this class are {γ̃_0,γ̃_1^k,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ} (The generators of class are different from generators of Clifford algebra). The Clifford algebra's extension of this class is {γ̃_1^k,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ}→{γ̃_0,γ̃_1,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ}=Cl_d+D+1→ Cl_d+D+2=Cl_1-(d-D)→ Cl_2-(d-D). The space of the mass term (γ̃_̃0̃) is C_1-δ (δ=d-D), and the topological classification of defects for the non-Hermitian class A is characterized by π_0(C_1-δ)=0  (ℤ) for  even (odd) δ. And the classifying space (CL) is C_1 (classifying space is equivalent to the space of mass term when δ=0).

        Class AIII (Q): The generators of this class are {γ̃_0,γ̃_1^k,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ,Γ̃}. The Clifford algebra's extension of this class is {γ̃_1^k,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ,Γ̃}→{γ̃_0,γ̃_1^k,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ,Γ̃}=Cl_d+D+2→ Cl_d+D+3=Cl_2-(d-D)→ Cl_3-(d-D). The space of the mass term (γ̃_̃0̃) is C_2-δ (δ=d-D), and the classification is characterized by π_0(C_2-δ)=ℤ(0) for  even (odd) δ.


        Then we classify the real classes:

        Class AI (K1) : The generators of this class are {γ̃_0,γ̃_1^k,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ ,T̃, J }, where J is the imaginary unit. The Clifford algebra's extension of this class is {γ̃_1^k,...,γ̃_d^k,Jγ̃_1^r,...,Jγ̃_D^r,JΣ,T̃,JT̃}→{ Jγ̃_0,γ̃_1^k,...,γ̃_d^k,Jγ̃_1^r,...,Jγ̃_D^r,JΣ,T̃, JT̃}=Cl_D+1,d+2→ Cl_D+2,d+2. The space of the mass term is R_1-δ.

        Class C (C4): The generators of this class are {γ̃_0,γ̃_1^k,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ ,Ã, J }. The Clifford aglebra's extension of this class is {Jγ̃_1^k,...,Jγ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ,Ã,JÃ}→{γ̃_0,Jγ̃_1^k,...,Jγ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ,Ã, JÃ}=Cl_d+2,D+1→ Cl_d+2,D+2. The space of the mass term is R_7-δ.

        Class BDI (QC5): The generators of this class are {γ̃_0,γ̃_1^k,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ ,Ã ,Γ̃ , J }. The Clifford algebra's extension of this class is {Jγ̃_1^k,...,Jγ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ, JΓ̃, Ã, JÃ}→{γ̃_0,Jγ̃_1^k,...,Jγ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ, JΓ̃, Ã, JÃ}=Cl_d+1,D+3→ Cl_d+1,D+4. The space of the mass term is R_2-δ.

Similarly, we can determine the space of mass term for the other AZ classes.
It was shown that the classification of topological defects depends only on the topological dimension δ=d-D. Once the classifying space is known, e.g. R_n, we can get the space of mass term in a given topological dimension by a shift of R_n to R_n-δ, and the topological classification is obtained by calculating π_0 (R_n-δ). The topological classification of defects for all AZ classes are summarized in table II.

In the same way, we can get the topological classification of BL classes. For example,
for the  class P, the generators are {γ̃_0,γ̃_1^k,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ ,P̃}, and the Clifford algebra's extension of this class is {γ̃_1^k,...,γ̃_d^k,γ̃_1^r,...,γ̃_D^r,Σ}⊗{ΣP̃}→{γ̃_0, γ̃_1^k,..., γ̃_d^k, γ̃_1^r,...,γ̃_D^r,Σ}⊗{ΣP̃}=Cl_d+D+1× Cl_d+D+1→ Cl_d+D+2× Cl_d+D+2. It follows that the space of the mass term is C_1-δ× C_1-δ.
The space of mass term for other BL classes can be obtained under the same scheme. The results are listed in table III.  For classes Cn and C5-n (n=1,⋯,4), the Hermitianized Hamiltonians fulfill the same symmetry constraints, i.e., Eq.(<ref>) and Eq.(<ref>). Then classes Cn and C5-n have the same topological classification. Similarly, classes QCn and QC9-n (n=1,⋯,8) have the same topological classification. When D=0, we have δ=d and our results can be applied
to describe the classification of point gap systems in the absence of defect, which are consistent with those in the reference <cit.>. For the convenience of comparing with results in references <cit.>, we list the correspondence between different notations of BL class in the table VII of the appendix C.

    

 §.§ Real gap classification

    If a non-Hermitian Hamiltonian has a real gap, it has been demonstrated that the Hamiltonian can continuously transform to a Hermitian Hamiltonian H while keeping its symmetry and real gap <cit.>. Then the non-Hermitian GBL class classification is the same with the corresponding Hermitian  classification. To classify 54 GBL classes, the K, Q, C and P symmetries reduce to:
    
    H=ϵ_k kH^*k^-1   , kk^*=η_k 𝕀   K  sym.
    
          H=ϵ_q qH q^-1   , q^2=𝕀   Q  sym.
    
          H=ϵ_c cH^* c^-1   , cc^*=η_c 𝕀   C  sym.
    
          H=-pHp^-1   , p^2=𝕀   P  sym.

    Here H is a Hermitian Hamiltonian. The classification problem reduces to a Hermitian Hamiltonian classification problem. And we can classify each class by Clifford algebra (See Appendix A). The results are listed in Table IV.  When D=0, our results can be applied to describe the classification of real gap systems in the absence of defect. In order to compare with results in the reference <cit.>, we list the correspondence between our notations and those in the reference <cit.> in the table VIII of the appendix C.

    

 §.§ Imaginary gap classification

    If a non-Hermitian Hamiltonian has an imaginary gap, the Hamiltonian can continuously transform to an anti-Hermitian Hamiltonian iH <cit.>, where H is a Hermitian Hamiltonian. To classify 54 GBL classes, the K, Q, C and P symmetries reduce to:
    
    H=-ϵ_k kH^*k^-1   , kk^*=η_k 𝕀   K  sym.
    
          H=-ϵ_q qH q^-1   , q^2=𝕀   Q  sym.
    
          H=ϵ_c cH^*c^-1   , cc^*=η_c 𝕀   C  sym.
    
          H=-pHp^-1   , p^2=𝕀   P  sym.

    The classification problem reduces to a Hermitian Hamiltonian classification problem. And we can classify each class by Clifford algebra (See Appendix B). The results are listed in Table V. Similarly, our results can be applied to describe the classification of imaginary gap systems in the absence of defect by taking D=0. The correspondence between our notations and those in the reference <cit.> can be found in the table VIII of the appendix C.

    














    

    

  

§ CONSTRUCTION OF EXPLICIT TOPOLOGICAL INVARIANTS FOR POINT DEFECTS


    Although the periodic tables of topological classification are given,  they do not provide explicit forms of topological invariants. In this section, we discuss the construction of explicit topological invariants for point gap systems by considering some specific examples. To begin with, we note that the open boundary condition in one dimension is the simplest example of topological defects. For this case, our topological classification of topological defects (d=1, D=0) is consistent with classification of one-dimensional gapped system (d=1). 
    Next we discuss several one-dimensional and two-dimensional examples.

    Class A (Non), Class AI (K1) and Class AII (K2).  Given the Hamiltonian H(k,r), which describes a point defect in d dimensions and is a function of d momentum variables and D=d-1 position variables. For any invertible Hamiltonian, it has a polar decomposition H=UP, where U is a unitary matrix and P is a positive definite Hermitian matrix. Such a decomposition is unique, and the Hamiltonian H(k,r) can be continuously deformed to U(k,r) with the same symmetry constraint <cit.>. Since H(k,r) is topologically equivalent to U(k,r), the ℤ topological number is defined as:
    
    n=(d-1)!/(2d-1)!(2π i)^d∫ _T^d× S^d-1Tr[(UdU^†)^2d-1] ,

which is the winding number associated with the homotopy group π_2d-1[U(n)]=ℤ <cit.>.

    Class P, Class PK1 and Class PK2.  Suppose that the Hamiltonian is H(k,r) and the system has P symmetry described by p=σ_z.  The polar decomposition is H=UP. Consider the symmetry constraint σ_z U= -Uσ_z <cit.>, and U can be represented as
    
    U(k,r)=[ [         0 U_1 (k,r); U_2 (k,r)         0 ] ].

       Then we can define the ℤ⊕ℤ topological number as
        
    n_j=(d-1)!/(2d-1)!(2π i)^d∫ _T^d× S^d-1Tr[(U_jdU_j^†)^2d-1],

        where j=1 and 2.

       Class PK3.  Suppose that the Hamiltonian of a point defect is H(k,r), and the system has both P and K symmetries described by p=σ_z and k=σ_x, respectively. The polar decomposition is H=UP. Consider the symmetry constraint σ_z U(k,r)= -U( k,r)σ_z and σ_x U^*( -k,r)=U( k,r) σ_x, and U can be represented as
    
    U(k,r)=[ [            0    U_1 (k,r); U_1^* (-k,r)            0 ] ] .

       Then we can define the ℤ topological number as
       
    n=(d-1)!/(2d-1)!(2π i)^d∫ _T^d× S^d-1Tr[(U_1dU_1^†)^2d-1].


       Class PQ1. The Hamiltonian of a point defect is H(k,r) with the P and Q symmetries described by p=q=σ_z. Consider the symmetry constraints σ_z H(k,r)= -H(k,r)σ_z and σ_z H^† (k,r)= H(k,r)σ_z as well as the polar decomposition is H=UP, and thus U can be written as
       
    H(k,r)=[ [            0    U_1 (k,r); -U_1^† (k,r)            0 ] ] .

         Then we can define the ℤ topological number by Eq.(<ref>).
         

         Class PC1. The Hamiltonian of a point defect is H(k,r) with the P symmetry described by p=σ_z and C symmetry described by  c=σ_0  as well as the polar decomposition H=UP. Consider the symmetry constraints σ_z U(k,r)= -U(k,r)σ_z and U^T(-k,r)= U(k,r), and thus U has the following form:
       
    U(k,r)=[ [             0     U_1( k,r); U_1^T ( -k,r)             0 ] ] .

         Then we can define the ℤ topological number  by Eq.(<ref>).
         

         Class PC3. The Hamiltonian of a point defect is H(k,r) with P symmetry described by p=σ_z⊗σ_0 and C symmetry described by c=σ_0 ⊗σ_y as well as the polar decomposition H=UP. Consider the symmetry constraints p U(k,r)= -U(k,r)p and c U^T(-k,r)= U(k,r)c, and thus U has the following form:
       
    U(k,r)=[ [                     0             U_1( k,r); σ_y U_1^T (-k,r) σ _y                     0 ] ] .

         Then we can define the ℤ topological number  by Eq.(<ref>).
         

         Class PQC1 and Class QC5. If we chose q=𝕀, the Hamiltonian reduces to a Hermitian Hamiltonian. And the non-Hermitian classes of PQC1 and QC5 reduce to Hermitian classes of BDI and D. So we can define the ℤ and ℤ_2 topological numbers for classes of PQC1 and QC5 the same as the Hermitian classes of BDI and D <cit.>,  which is consistent with our classification.

           

§ EXAMPLES OF POINT-DEFECT MODELS


           To gain an intuitive understanding of nontrivial topological defects in non-Hermitian systems, we construct some examples with a point gap. Through the study of some concrete models, we discuss the correspondence between topological number and zero modes at the defect.
           

 §.§ Point defects in 1D PC1 class

           Consider the one-dimensional (1D) lattice models described by:
           
    H_1=∑ _n [t_1 (a_n ^† b_n+b_n ^† a_n) + t_2e^iθ _1(b_n ^† a_n+1+a_n+1 ^† b_n)]

            and
            
    H_2=∑ _n [t_3 (a_n ^† b_n+b_n ^† a_n) + t_4e^iθ _2(b_n ^† a_n+1+a_n+1 ^† b_n)].

            These two Hamiltonians have the same forms but with different parameters. If we consider the periodic boundary condition, both of them can be represented in the momentum space after a Fourier transformation. The corresponding Hamiltonians in the k space are
           
    H_1(k)=[ [                 0 t_1+t_2e^iθ _1-ik; t_1+t_2e^iθ _1+ik                 0 ] ]

            and
            
    H_2(k)=[ [                 0 t_3+t_4e^iθ _2-ik; t_3+t_4e^iθ _2+ik                 0 ] ] .

Both the Hamiltonians belong to the PC1 Class as they have P and C symmetries with p=σ_z and c=σ_0. The topological properties of the systems are characterized by the ℤ topological numbers, i.e., n_1= 1/2π i∫ _0 ^2π dk∂_kln(t_1+t_2e^iθ _1-ik and n_2=1/2π i∫ _0 ^2π dk∂_kln(t_3+t_4e^iθ _2-ik) for H_1 and H_2, respectively.
                


           Now we connect H_1 and H_2 together end to end to form a new Hamiltonian H. As schematically displayed in Fig.(<ref>a), H_1 and H_2 are coupled together with a coupling constant t. Each connected point can be viewed as a topological defect. The topological invariant for the defect is characterized by  W=n_2-n_1. In Fig.(<ref>b), we display W versus t_1 by fixing t=t_2=t_3=1, t_4=0.5, θ_1=π/3 and θ_2=π/4. In the region of t_1<1, we have W=1, which indicates H_1 and H_2 in topologically different phases. Correspondingly, we observe the existence of zero mode states in the region of t_1<1 as shown in Fig.(<ref>c). The zero mode states are found to be localized at the places of defects, according to Fig.(<ref>d) for t_1=0.5.

           Next we consider the continuous model:
          
    H(k,x)=k̂σ_z +[ m(x)+iα ] σ_x,

          where k̂ represents the momentum operator (k̂=-i ∂_x in the coordinate representation), α is a real constant and m(x) is a function of x, which changes sign as x crosses the zero point.
          Here we take |α| ≪ |m(x)| with x ∈ S^0, where S^0 is the 0D loop around the defect. When α=0, this model reduces to the Jackiw-Rebbi model <cit.>. The Hamiltonian has P and C symmetries with ϵ_c=-1, η_c=1, c=σ_z and p=σ_y, and belongs to the PC1 class. The topological defect is characterized by a ℤ topological number. When α continuously changes from 0 to nonzero, det(H) 0 on S^0 because |α| ≪ |m(x)| on S^0. Then the topological number is the same as that of Jackiw-Rebbi model, i.e.,
           
    W =1/2[ sgn(m(x_1))-sgn(m(x_2)) ],

           where x_1>0 and x_2<0 with x_1,x_2 ∈ S^0.

          When m=b x (b>0 is a constant), 
          in the momentum representation, x̂= i∂_k and the Hamiltonian can be written as:
          
    H(k,i∂_k)=kσ_z +(bi∂_k+iα)σ_x .

            Define H'=e^α k/bHe^-α k/b, then H' is given by
            
    H'(k̂,x̂)=k̂σ_z +bx̂σ_x,

          which is a special form of  Jackiw-Rebbi Hamiltonian and has a topologically protected zero mode at the defect. Denote the zero mode state of H'  as Φ '_0(x)=⟨ x| Φ '_0 ⟩, then H has a topologically protected zero mode Φ_0(x)=e^iα∂_x/bΦ '_0(x) at the defect. Our results show that there is a correspondence between zero modes of defect and the topological number of defect.

           For the non-Hermitian continuous model described by
          
    H(k,x)= (-i∂_x +i β) σ_z + m(x) σ_x,

         we can make a similarity transformation  H'=e^-β xH e^β x and get
          
    H'(k̂,x̂)=-i∂_x σ_z + m(x) σ_x,

   which is a standard Jackiw-Rebbi Hamiltonian. Similarly,  we conclude that H has a topologically protected zero mode Φ_0(x)=e^β xΦ '_0(x), where Φ '_0(x) is the zero mode wavefunction of H'.
   
    

 §.§ Point defects in 1D K1 class


              Consider a toy model described by
              
    H={
    e^ik̂,       x<0 
    
    1        ,        x>0
    .

 where k̂ represents the momentum operator.  The model belongs to the K1 class. The ℤ topological number is given by W= 1/2π i∫ _0 ^2π dk∂_kln(e^ik)=1, which indicates the system belonging to a topological non-trivial class. It is straightforward to check that Ψ=const is an eigenfunction of the Hamiltonian with the eigenvalue 1. For this case, we do not observe the correspondence between the zero mode and ℤ topological number.

   


            

 §.§ Point defects in 2D PC1 class

           Consider a two-dimensional (2D) Hamiltonian described by
            
    H(k̂,r̂)   =    v k̂_xγ_1+ vk̂_y γ_2 +  (Δ _1+iα_1)γ_3+  (Δ_2+iα_2)γ_4    
       =   [ [      σ· k     Δ +iα; Δ ^*+iα^*     -σ· k ] ],

           where (γ_1,γ_2,γ_3,γ_4)=(τ_zσ_x,τ_zσ_y,τ_x,-τ_y), k=(k̂_x,k̂_y) and σ=(σ_x,σ_y). σ_x,y,z and τ_x,y,z are Pauli matrices in the spin and particle-hole spaces. Here Δ_1 and Δ_2 are functions of r̂=(x̂,ŷ), α_1 and α_2 are real constants,  Δ_1+iΔ_2=Δ=|Δ|e^iϕ and α_1+iα_2=α.  We take |α_1| ≪ |Δ_1|, |α_2| ≪ |Δ_2| on S^1, where S^1 is the loop around the defect. When α_1=α_2=0, this model reduces to the Jackiw-Rossi model <cit.>.
            The Hamiltonian has P and C symmetries with ϵ_c=-1, η_c=1, c=τ_yσ_y and p=τ_zσ_z, and thus belongs to the PC1 class. The topological defect is characterized by a ℤ topological number. When α_1 and α_2 continuous change from 0 to nonzero, the det(U(k,r)) 0 on S^1 in Eq.(<ref>) because |α_1| ≪ |Δ_1| and |α_2| ≪ |Δ_2| on S^1. Then the topological number is same as that of Jackiw-Rossi model with
            
    n=1/2π∫_s^1dϕ.


            When Δ_1=bx̂ and Δ_2=bŷ (b is a real constant), 
            in momentum representation, the Hamiltonian takes the following form:
            
    H(k,i∂_k)=vγ_1 k_x+ vγ_2 k_y+ γ_3 (bi∂_kx+iα_1)+ γ_4 (bi∂_ky+iα_2).

              Define H'=e^α_1 k_x+α_2 k_y/bHe^-α_1 k_x+α_2 k_y/b, then H' is given by
              
    H'(k̂,r̂)=vγ_1 k̂_x+ vγ_2 k̂_y+ γ_3 b x̂+ γ_4 b ŷ,

             which is a special form of Jackiw-Rossi Hamiltonian and has a topologically protected zero mode at the defect. Denote the zero mode state of H' as Φ '_0(r)= ⟨r|Φ '_0 ⟩, then H has a topologically protected zero mode Φ_0(r)=e^iα_1 ∂_x /b +i α_2 ∂_y/b Φ '_0(r) at the defect.

             When we take Δ _1=b(x̂^2-ŷ^2) and Δ_2=2bx̂ŷ, the corresponding topological number is given by n=2. We can prove that there are two topologically protected zero modes at the defect by a similar method.
            In the momentum space, we can write the Hamiltonian explicitly as
                
    H(k,i∂_k)=   vγ_1 k_x+ vγ_2 k_y+ γ_3 [b(-∂ _kx^2+∂_ky^2)  +iα_1]
       + γ_4 (-2b∂_kx∂_ky+iα_2).

                Define H'=e^√(i)(β_1 k_x+β_2 k_y)He^-√(i)(β_1 k_x+β_2 k_y), where β_1 and β_2 fulfill that: -bβ _1^2+bβ_2^2+α _1=0 and -2bβ_1β_2+α_2=0, then H' is given by
               
                
               
                
    H'(k̂,r̂) =vγ_1  k̂_x+ vγ_2  k̂_y+ γ_3 b( x̂^2- ŷ^2) + γ_4 2 b  x̂ŷ .

                The Hermitian Hamiltonian H' has two zero modes, then H also has two zero modes at the defects. The study can be directly extended to situations with n>2 . There is a correspondence between the zero modes of defects and the topological number in the 2D PC1 class.

                For the non-Hermitian continuous model described by
          
    H(k,r)    =   γ_1(-iv∂_x+iβ_1)+ γ_2 (-iv∂_y+iβ_2) 
           + bΔ_1γ_3 + bΔ_2γ_4 ,

          where Δ_1 and Δ_2 are functions of r, Δ=Δ_1+iΔ_2 has non-trivial winding on S^1, and S^1 is the loop around the defect. we can make a similarity transformation  H'=e^-β_1x+β_2y/vH e^β_1x+β_2y/v and get
          
    H'(k̂,r̂)=vk̂_xγ_1+ vk̂_yγ_2 + bΔ_1γ_3 + bΔ_2γ_4 y,

   which is a standard Jackiw-Rossi Hamiltonian. Similarly,  we conclude that H has a topologically protected zero mode Φ_0(r)=e^β_1x+β_2y/vΦ '_0(r), where Φ '_0(r) is the zero mode wavefunction of H'.


                

 §.§ Point defects in 2D C3 class

                Consider the Hamiltonian:
                
    H(k̂,r̂)=    v k̂_x γ_1+ v k̂_y γ_2+  (Δ _1+iα_1)γ_3+  (Δ_2+iα_2)γ_4 
       +hσ_z-μτ_z  
    
                                        =   [ [  σ· k+σ_zh-μ        Δ +iα;    Δ ^*+iα^* -σ· k+σ_zh+μ ] ],

                where the first four terms are the same as Eq.(<ref>), h is a  magnetic field and μ is a chemical potentia. When α_1=α_2=0, this model is Fu-Kane model <cit.>.
                The Hamiltonian has C symmetries with ϵ_c=-1, η_c=1 and c=τ_yσ_y and belongs to the C3 class. The h and μ terms couple the zero modes that we discussed in the above section. The topological properties of the system are characterized by a ℤ_2 topological number. Consider α_1 and α_2 continuously changing from 0 to nonzero. In this process, det(H(k,r)) 0 on S^1 (loop round the defect) because |α_1| ≪ |Δ_1| and |α_2| ≪ |Δ_2| on S^1. Then the topological number is the same as the Hermitian case <cit.> given by
               
    n=1/2π∫_s^1dϕ mod  2.


                When Δ_1=bx̂ and Δ_2=bŷ (b is a real constant), the topological invariant is given by n=1. Next, we demonstrate the existence of a zero mode at the defect.
                In momentum representation, the Hamiltonian is written as
                
    H(k,i∂_k)=   vγ_1 k_x+ vγ_2 k_y+ γ_3 (bi∂_kx+iα_1)+ 
       γ_4 (bi∂_ky+iα_2)+hσ_z-μτ_z .
                         .

                  Define H'=e^α_1 k_x+α_2 k_y/bHe^-α_1 k_x+α_2 k_y/b, then H' is given by
                  
    H'(k̂,r̂)=vγ_1 k̂_x+ vγ_2 k̂_y+ γ_3 b x̂+ γ_4 b ŷ+hσ_z-μτ_z,

                  which is a special form of Fu-Kane Hamiltonian and has a topological protected zero mode at the defect. Denote the zero mode state of the Fu-Kane Hamiltonian H' as Φ '_1(r)=⟨r|Φ '_1 ⟩, then H has a topological protected zero mode Φ_1(r)=
                  e^iα_1 ∂_x /b +i α_2 ∂_y/b Φ '_1(r) at the defect. When we choose Δ _1=b(x̂^2-ŷ^2) and Δ_2=2bx̂ŷ, the corresponding topological number is n=0. And we can demonstrate that there is no topologically protected zero mode at the defect by a similar method. There is a correspondence between the zero mode of defect and the topological number in the 2D C3 class.

                  For the non-Hermitian continuous model described by
                  
    H(k,r)=   γ_1(-iv∂_x+iβ_1)+ γ_2 (-iv∂_y+iβ_2)+ bΔ_1γ_3 
       +bΔ_2γ_4+hσ_z-μτ_z ,

                  where Δ_1 and Δ_2 are functions of r, Δ=Δ_1+iΔ_2 has odd winding number on S^1, and S^1 is the loop around the defect. We can prove that there is topologically protected zero mode by a similar method.

                  

 §.§ Point defects in 2D PQC1 and QC5 class

             Consider the Hamiltonian Eq.(<ref>) with α_1=α_2=0, i.e., the Jackiw-Rossi model. This model has one more symmetry Q than Eq.(<ref>) with q=𝕀. According to the BL classification, the Jackiw-Rossi model belongs to the PQC1 class, and the Hermitian model has a correspondence between zero modes and ℤ topological number. Similarly, consider the Hamiltonian Eq.(<ref>) with α_1=α_2=0, i.e., the Fu-Kane model.  This model has one more symmetry Q than Eq.(<ref>) with q=𝕀.  According to the BL classification, the Fu-Kane model belongs to the  QC5 class, and the Hermitian model has a correspondence between zero modes and ℤ_2 topological number.



                   

§ SUMMARY

                     In summary, we have studied topological defects in non-Hermitian systems with point gap, real gap and imaginary gap and made a topological classification
                   for all the BL or GBL classes in all dimensions. While the BL class covers all 38  nonequivalent symmetry classes for the point-gap systems, we find that a full classification for non-Hermitian systems with line gap should include 54 nonequivalent GBL classes, which are a natural generalization of BL classes.
                   The periodical classification tables of point gap defects are summarized in table II for the AZ classes and in table III for the BL classes with point gap, respectively, and periodical classification tables of real and imaginary gap defects are summarized in table IV and V (GBL class), respectively.
                  By considering some concrete examples of point gap defects, we constructed explicitly the topological invariants.  Through the study of some concrete models, we calculated explicitly the topological invariants and discussed the correspondence between topological invariants and zero modes at the defect for some classes.





The work is supported by NSFC under Grants No. 11425419 and the National Key Research and Development Program of China (2016YFA0300600 and 2016YFA0302104).


                     

§ APPENDIX

                     
                     

 §.§ Real gap classification

                     According to the discussion in the main text, the classification  is equivalent to the classification of a Hermitian Hamiltonian with corresponding symmetries:
                     
    H=ϵ_k kH^*k^-1   , kk^*=η_k 𝕀   K  sym.
    
                          H=ϵ_q qH q^-1   , q^2=𝕀   Q  sym.
    
                          H=ϵ_c cH^* c^-1   , cc^*=η_c 𝕀   C  sym.
    
                          H=-pHp^-1   , p^2=𝕀   P  sym.

                         We can use Clifford algebra to represent the Hamiltonian:
                         
    H( k,r)=γ_0+k_1γ_1^k+...+k_dγ_d^k+r_1γ_1^r+...+r_Dγ_D^r .

                         Define K=k𝒦, Q=q, C=c𝒦 and P=p, where 𝒦 is a complex conjugate operator. We can get the space of mass term by constructing the Clifford algebra's extension. The correspondence between Clifford algebra's extension and space of mass term was listed in Table I and Table VI.
                          
                          Then we get the topological classification by calculating the homotopy group of the space of mass term. The following is the topological classification for some GBL classes.

                          Class Non: The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r }→{γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r}=Cl_d+D→ Cl_d+D+1. The space of mass term is C_d+D=C_-(d-D)=C_-δ. The topological classification is determined by π_0(C_-δ)=ℤ (0) for even (odd) δ.

                          Class P: The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P }→{γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P}=Cl_d+D+1→ Cl_d+D+2. The space of mass term is C_d+D+1=C_-(d-D)+1=C_-δ+1. The topological classification is determined by π_0(C_-δ+1)=0 (ℤ) for even (odd) δ.

                          Class Qa: The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,Q}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r}⊗{ Q }→{γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r}⊗{ Q }=Cl_d+D⊗ Cl_1 → Cl_d+D+1⊗ Cl_1. The space of mass term is C_d+D× C_d+D=C_-δ× C_-δ. The topological classification is determined by π_0(C_-δ× C_-δ)=ℤ⊕ℤ (0⊕ 0) for even (odd) δ.

                          Class C1: The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,C,J}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,C,JC}→{Jγ_0,γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,C,JC}=Cl_D,d+2→ Cl_D+1,d+2. The space of mass term is R_D-d=R_-δ.

                          Class PQ1: The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P,Q}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P}⊗{ Q }→{γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P}⊗{ Q }=Cl_d+D+1⊗ Cl_1 → Cl_d+D+2⊗ Cl_1. The space of mass term is C_d+D+1× C_d+D+1=C_-δ+1× C_-δ+1. The topological classification is determined by π_0(C_-δ+1× C_-δ+1)=ℤ⊕ℤ ( 0⊕ 0 ) for odd (even) δ.

                          Class PK1: The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P,K,J}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,JP,K,JK}→{Jγ_0,γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,JP,K,JK}=Cl_D+1,d+2→ Cl_D+2,d+2. The space of mass term is R_D-d+1=R_1-δ.

                          Class PC1: The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P,C,J}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,JP,C,JC}→{Jγ_0,γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,JP,C,JC}=Cl_D+1,d+2→ Cl_D+2,d+2. The space of mass term is R_D-d+1=R_1-δ.

                          Class QC1a: The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,Q,C,J}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,C,JC}⊗{Q }→{Jγ_0,γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,C,JC}⊗{Q }=Cl_D,d+2× Cl_0,1→ Cl_D+1,d+2× Cl_0,1. The space of mass term is R_D-d× R_D-d=R_-δ× R_-δ.

                          Class PQC1: The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P,Q,C,J}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,JP,C,JC}⊗{Q }→{Jγ_0,γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,JP,C,JC}⊗{Q }=Cl_D+1,d+2× Cl_0,1→ Cl_D+2,d+2× Cl_0,1. The space of mass term is R_D-d+1× R_D-d+1=R_1-δ× R_1-δ.



                     

 §.§ Imaginary gap classification

                      According to the discussion in the main text, the classification  is equivalent to the classification of a Hermitian Hamiltonian with corresponding symmetries:
                     
    H( -k,r)=-ϵ_k kH( k,r)^*k^-1   , kk^*=η_k 𝕀   K  sym.
    
                            H( k,r)=-ϵ_q qH( k,r) q^-1   , q^2=𝕀   Q  sym.
    
                            H( -k,r)=ϵ_c cH( k,r)^*c^-1   , cc^*=η_c 𝕀   C  sym.
    
                            H( k,r)=-pH( k,r)p^-1   , p^2=𝕀   P  sym.

                        We can use Clifford algebra to represent the Hamiltonian:
                        
    H(k,r)=γ_0+k_1γ_1^k+...+k_dγ_d^k+r_1γ_1^r+...+r_Dγ_D^r .

                        Define K=k𝒦, Q=q, C=c𝒦 and P=p, where 𝒦 is the complex conjugate operator. Then we can get the space of mass term by constructing the Clifford algebra's extension. And we get the topological classification by calculating the homotopy group of the space of mass term. The C and P give the same symmetry constraint for the topological equivalent Hermitian Hamiltonian in real-gap systems and imaginary-gap systems. Then the two systems have the same topological classification for GBL classes: Non, P, C1-4 and PC1-4. For one of  PQC1-8 classes, the constraint of topologically equivalent Hermitian Hamiltonian in imaginary-gap systems transforms to that in the corresponding real-gap systems after defining q̃=√(ϵ_pq)pq. Then real-gap and imaginary-gap classification of class PQC1-8 are also same. The following is the topological classification for some GBL classes.



                         Class Qa:  The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,Q}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,Q }→{γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,Q}=Cl_d+D+1→ Cl_d+D+2. The space of mass term is C_d+D+1=C_-(d-D)+1=C_-δ+1. The topological classification is determined by π_0(C_-δ+1)=0  (ℤ) for even (odd) δ.

                         Class K1a:  The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,K,J}. Clifford algebra's extension is {Jγ_1^k,Jγ_2^k,...,Jγ_d^k,γ_1^r,γ_2^r,...,γ_D^r,K,JK}→{γ_0,Jγ_1^k,Jγ_2^k,...,Jγ_d^k,γ_1^r,γ_2^r,...,γ_D^r,K,JK}=Cl_d,D+2→ Cl_d,D+3. The space of mass term is R_D-d+2=R_2-δ.

                         Class PQ1:  The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P,Q}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P}⊗{PQ }→{γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P}⊗{PQ}=Cl_d+D+1× Cl_1→ Cl_d+D+2× Cl_1. The space of mass term is C_d+D+1× C_d+D+1=C_1-δ× C_1-δ.

                         Class PK1:  The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P,K,J}. Clifford algebra's extension is {Jγ_1^k,Jγ_2^k,...,Jγ_d^k,γ_1^r,γ_2^r,...,γ_D^r,JP,K,JK}→{γ_0,Jγ_1^k,Jγ_2^k,...,Jγ_d^k,γ_1^r,γ_2^r,...,γ_D^r,JP,K,JK}=Cl_d+1,D+2→ Cl_d+1,D+3. The space of mass term is R_D-d+1=R_1-δ.

                         Class PC1:  The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,P,C,J}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,JP,C,JC}→{Jγ_0,γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,JP,C,JC}=Cl_D+1,d+2→ Cl_D+2,d+2. The space of mass term is R_D-d+1=R_1-δ.

                         Class QC1a:  The generators are {γ_0,γ_1^k,γ_2^k,...,γ_d^k,γ_1^r,γ_2^r,...,γ_D^r,Q,C,J}. Clifford algebra's extension is {γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,JQ,C,JC}→{Jγ_0,γ_1^k,γ_2^k,...,γ_d^k,Jγ_1^r,Jγ_2^r,...,Jγ_D^r,JQ,C,JC}=Cl_D+1,d+2→ Cl_D+2,d+2. The space of mass term is R_D-d+1=R_1-δ.



                          

 §.§ The correspondence between different notations of BL class and GBL class

                          There are three different notations of Bernard-LeClair class: Kawabata-Shiozaki-Ueda-Sato (KSUS) notations <cit.>, Zhou-Lee (ZL) notations <cit.> and our (LC) notations used in the present work. For the convenience of comparing with the existing results in references, we list the correspondence between different notations of Bernard-LeClair class in table VII. For the line gap systems, there are two different notations of generalized Bernard-LeClair class: Kawabata-Shiozaki-Ueda-Sato (KSUS) notations and our (LC) notations. We list the correspondence of different notations of generalized Bernard-LeClair class in table VIII.

 

    


    




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