A recent development in this field is to investigate non-Hermitian topological phases [20, 21, 30]. In non-Hermitian systems without the constraints of Hermiticity H 6= H†, the left eigenvectors are no longer the adjoint of the right eigenvectors. For a non-Hermitian Hamiltonian with discrete spectrum, the orthonormality can be achieved through the set of biorthonormal eigenbasis |uni, |unii which
satisfies
H|uni = En|uni, hhun|H = hhun|En, (1.109)
hhum|uni =hum|unii = δmn, (1.110) X n |unihhun| = X n |uniihun| =1. (1.111)
Here, the adjoints of a left eigenvector |unii and a right eigenvector |uni also satisfy hun|H†= hu n|En∗
and H†|unii = En∗|unii. As the choice of eigenbasis satisfying Eqs. (1.109) to (1.111) is in drastic
contrast to Hermitian systems, it is natural to ask whether the topological invariants can be defined similarly in non-Hermitian systems, and if they can, whether they can be used to characterize the non-Hermitian topological phases.
1.4 Non-Hermitian topological phases class d =0 1 2 3 4 5 6 7 A, DIII, CI 0 Z 0 Z 0 Z 0 Z AIII 0 Z ⊕ Z 0 Z ⊕ Z 0 Z ⊕ Z 0 Z ⊕ Z AI, D Z2 Z 0 0 0 2Z 0 Z2 BDI Z2⊕ Z2 Z ⊕ Z 0 0 0 2Z ⊕ 2Z 0 Z2⊕ Z2 AII, C 0 2Z 0 Z2 Z2 Z 0 0 CII 0 2Z ⊕ 2Z 0 Z2⊕ Z2 Z2⊕ Z2 Z ⊕ Z 0 0
Table 1.2: Classification of non-Hermitian topological phases [21]. A prominent feature is that Classes A, DIII and CI, classes AI and D, and classes AII and C are unified in the Altland-Zirnbauer ten-fold classes.
Berry phase We take the Berry phase in non-Hermitian systems as an example. The dynamics of the right eigenvector can be obtained similar to that of Eq. (1.9), which reads
En(k(t))|un(k(t))i = ~∂tθ(t)|un(k(t))i + i~∂t|un(k(t))i, (1.112)
with En(k(t)) the eigenvalue for the band n. Applying the left eigenvector hhun(k(t))| with the nor-
malization condition of hhun(k(t))|un(k(t))i = 1, after the integration, the phase is obtained as
θ(t) = 1 ~ Z t 0 En(k(t0))dt0− i Z t 0 hhun(k(t0))|∂t0|un(k(t0))idt0. (1.113) We emphasize that the second term at the RHS can still be treated as the geometrical phase. On a cyclic path of Γ in the Brillouin zone, it can be written as
γn= i Z t 0 hhun(k(t0))|∂t0|un(k(t0))idt0 = i I Γ dkhhun(k)|∇k|un(k)i. (1.114)
The form of the Berry connection can be derived accordingly γn= I Γ dk An= Z Z Γ dS ∇k× An, (1.115)
with the Berry connection and Berry curvature of
An= ihhun(k)|∇k|un(k)i, Ωn= ∇k× An(k) = i∇k× hhun(k)|∇k|un(k)i. (1.116)
From Eqs. (1.115) and (1.116), the Berry phase and Berry curvature can still be defined in terms of the biorthonormal eigenbasis.
It is important to notice that in Eq. (1.113), the first part which is called the dynamic phase is also different from the Hermitian case, because the energy eigenvalues are complex En(k) ∈ C. On a cyclic
path of Γ in the Brillouin zone, the winding number for the n-th band can be defined as [20] ωn=
I
Γ
dk
2π∇karg En(k), (1.117)
1.4.1 Classification of non-Hermitian topological phases
In this section, we extend the classification table of topological phases to non-Hermitian regime. An arbitrary invertible Hamiltonian H has a unique polar decomposition H = U P with U being unitary and P =
√
H†H being Hermitian and positive definite. It can be proved that H and U are homotopically
equivalent, H ' U [21]. Accordingly, the classification of Hamiltonians is equivalent to the classification of corresponding unitary matrices.
It should be noted that in the previous section, the classification of topological phases of gapped systems is achieved through Dirac Hamiltonians. A more fundamental treatment should start with a general gapped Hermitian Hamiltonian, which can be continuously deformed to a “spectrum flattened” Hamiltonian which satisfies H2 =1. The energy eigenvalues above and below the energy gap are de- formed to +1 and −1, respectively, and the wavefunctions are preserved during the spectrum flattening, so that any topological property is preserved. After this treatment, H can be considered as a generator of a Clifford algebra ClH. Notice that the symmetry operators already form a Clifford algebra ClS.
Thus the classifying space is formed by the extension of the Clifford algebra ClS → ClH. In this way, the classification of topological phases can be achieved from these classifying spaces.
For a non-Hermitian Hamiltonian H, after the unitarization by the polar decomposition H = U P , in general U2 6= ±1, which means U cannot be treated as a generator of a Clifford algebra. However, a corresponding Hermitian Hamiltonian can be constructed as [21]
HU = σ+⊗ U + σ−⊗ U†= U
U† !
. (1.118)
The constructed Hamiltonian satisfies HU2 =1 as U is unitary. It should be emphasized that after such a construction, the Hamiltonian automatically acquires a chiral symmetry with the symmetry operator Σ = σz⊗1,
{HU, Σ} = 0, (1.119)
with Σ2 =1. In this way, the topological property of U can be obtained from HU. Thus, the classi-
fication of non-Hermitian Hamiltonians H is equivalent to the classification of HU with an additional
chiral symmetry Σ.
Complex classes For class A, with the additional chiral symmetry Σ, the classification simply shifts to class AIII, since there are no other symmetry constraints.
For class AIII, because the additional chiral symmetry commutes with the original chiral symmetry σ0⊗ Γ, [Σ, σ0⊗ Γ] = 0, the topological number simply duplicates, that is, Z → Z ⊗ Z and 0 → 0 for
entries in class AIII.
Real classes AZ classes with a single anti-unitary symmetry This includes class AI (T2 = +1), class D (C2 = +1), class AII (T2 = −1) , and class C (C2 = −1). Denote the anti-unitary operator as A = UAK. Thus, from the relation AU = ηAU A and AU† = ηAU†A with ηA = ±1 from the single
anti-unitary symmetry, we can construct an anti-unitary operator AU = σ0⊗ A and obtain
AUHU = ηAHUAU, (1.120)
which means HU follows a similar anti-unitary symmetry as that of U . Since AU commutes with Σ, it can be inferred that there is another anti-unitary operator, the square of which is the same as A2.