A note on the topological insulator phase in non-Hermitian quantum systems

Topological ideas have resurfaced very often in analyzing physical problems, the recently discovered topological insulator [1–7] being one such example. The main characteristic of a topological insulator is the appearance of a boundary zero mode within the bulk gap, the stability of which is guaranteed due to the existence of some associated topological invariant. The classifications of topological insulators are based on the underlying discrete symmetries of the Hamiltonian admitting such phases, like the parity (), time-reversal () and particle–hole symmetries [6, 7]. Perturbations and/or deformations preserving these symmetries cannot destabilize a phase characterized by a topological invariant, which is usually an integer or a Z₂ quantity, as long as the bulk gap remains open. The decay of a particular phase within a given topological sector due to the time evolution of the states is also ruled out completely for Hermitian quantum systems.

The Hermiticity of an operator crucially depends on the metric of the Hilbert space on which it is defined. In the standard treatment of quantum mechanics, the metric is always taken to be an identity operator. An emerging view [8–13] in the context of the symmetric and/or pseudo-Hermitian quantum systems is that a consistent non-dissipative description of non-Hermitian quantum systems is admissible with a modified inner product in the Hilbert space. A few examples of non-dissipative non-Hermitian quantum systems with a complete and consistent description include the asymmetric XXZ spin chain, non-Hermitian transverse Ising model, non-Hermitian Dicke model, non-Hermitian quadratic form of bosonic (fermionic) operators and non-Hermitian many-particle rational Calogero model [13, 14]. Relativistic [11] and supersymmetric [12] non-Hermitian quantum systems have also been investigated within the same context. A proposal for optical realization of a non-Hermitian relativistic quantum system is described in [15]. Experimental results related to symmetric optical systems are also available [16].

The purpose of this paper is to present a general discussion on the construction of non-Hermitian Dirac Hamiltonians admitting a topological insulator phase, followed by a few explicit examples in one, two and three space dimensions. The method prescribed by the present author, in [11–13] and particularly in [11], is used to construct these models through suitable pseudo-Hermitian deformations of known Hermitian Hamiltonians admitting a topological insulator phase. All the model Hamiltonians presented in this paper have entirely real bulk eigenvalues and unitary time evolution. The topological invariant characterizing a particular topological insulator phase is identical for the non-Hermitian and the corresponding Hermitian Hamiltonian from which it is obtained by a pseudo-Hermitian deformation. The decay of a phase within a given topological sector is also ruled out, since the time evolution is unitary by construction.

A few attempts have been made in the recent past (see [17, 18]) to construct model non-Hermitian Hamiltonians admitting a topological insulator phase. The finding of [17] is that the topological insulator phase is absent in non-Hermitian symmetric Hamiltonians. It is worth recalling in this context that the reality of the entire spectra as well as the unitarity of a non-Hermitian Hamiltonian can generally be understood in terms of an unbroken anti-linear symmetry, which is not necessarily the standard symmetry [9, 11–14]. The anti-linear symmetry may be identified as the standard symmetry for some specific quantum systems. The topological insulator phase in a non-Hermitian Dirac Hamiltonian having an anti-linear symmetry is described in this paper. The second reference [18] contains examples of non-Hermitian Hamiltonians admitting a topological insulator phase in which the bulk eigenvalues of the models are in general complex. Consequently, the time evolution is not unitary and the states describing the topological insulator phase are expected to decay within a given topological sector. The examples presented in this paper are free from these shortcomings; all of the bulk eigenvalues are real and the time evolution is unitary.

The plan for presenting the results is the following. A brief review of pseudo-Hermitian quantum systems as applied to Dirac Hamiltonians is presented at the beginning of section 2. Results regarding the allowed forms of topological invariants for pseudo-Hermitian quantum systems are also included in this section. Examples of non-Hermitian Dirac Hamiltonians admitting topological insulator phases in one, two and three space dimensions are presented in sections 3.1, 3.2 and 3.3, respectively. Finally, the paper ends with a summary of the results obtained and relevant discussions in section 4. Suggestions for a possible classification scheme for topological insulators in non-Hermitian quantum systems are included.

The continuum description of topological insulators is given in terms of free particle Dirac Hamiltonians with translational invariance in D + 1 space–time dimensions. The main objective of this paper is to present Dirac Hamiltonians H, admitting a topological insulator phase, which are non-Hermitian in the Hilbert space _D that is endowed with the standard inner product 〈⋅|⋅〉. Any operator that is related to its adjoint in _D through a non-unitary similarity transformation is known as a pseudo-Hermitian operator [9], i.e.,

$$\begin{eqnarray} &&{H}^{\dagger }=\eta H{\eta }^{-1}. \end{eqnarray} \tag{ 1 }$$

A positive-definite similarity operator η: = η₊, if it exists, can be identified as a metric operator. A Hilbert space that is endowed with the metric η₊ and the inner product 〈〈⋅|⋅〉〉_η+: = 〈⋅|η₊⋅〉 is denoted as _η+. The Hilbert spaces _D and _η+ are identical in the limit of η₊ being an identity operator.

In general, Hermitian operators in _η+ are non-Hermitian in _D and vice versa. A Hermitian Hamiltonian H in _η+ is related to a Hermitian Hamiltonian h in _D through a non-unitary similarity transformation [9, 10],

$$\begin{eqnarray} &&h=\rho H{\rho }^{-1},\tqs \rho :=\sqrt{{\eta }_{+}}. \end{eqnarray} \tag{ 2 }$$

An operator O_η+ in _η+ may be introduced corresponding to each operator O_D in _D as O_D = ρO_η+ρ⁻¹ such that 〈O_D〉 = 〈〈O_η+〉〉_η+. This relation allows us to find the symmetry generator of H from that of h or vice versa, and defines physical observables in _η+. Both the operators O_D and O_η+ are Hermitian in _D as well as in _η+ for the special case for which these operators commute with the similarity operator ρ.

A comment is in order on the nature of the mapping described by equation (2). The non-Hermitian Hamiltonian H may be mapped to a Hermitian Hamiltonian $\tilde {h}=\tilde {\rho }H{\tilde {\rho }}^{-1}$ in _D that is different from h. Although the existence of such a similarity operator $\tilde {\rho }$ indicates non-uniqueness of the mapping (2), the Hermitian Hamiltonians thus obtained are known [9] to be unitarily equivalent to each other, i.e. $\tilde {h}={U}^{-1}h U$, $U:=\rho {\tilde {\rho }}^{-1}$. The unitary nature of U follows [9] from the identities ${\rho }^{-1}{\eta }_{+}{\rho }^{-1}=1={\tilde {\rho }}^{-1}{\eta }_{+}{\tilde {\rho }}^{-1}$. Without loss of any generality, the symmetry generators of H can thus be obtained by using the inverse similarity transformation of the corresponding generators of either h or $\tilde {h}$. For example, a symmetry generator T_D of h is related to the corresponding generator ${\tilde {T}}_{\mathrm{D}}$ of $\tilde {h}$ though the unitary transformation ${\tilde {T}}_{\mathrm{D}}={U}^{-1}{T}_{\mathrm{D}}U$. Consequently, the symmetry generators T_η+: = ρ⁻¹T_Dρ and ${\tilde {T}}_{{\eta }_{+}}:={\rho }^{-1}{\tilde {T}}_{\mathrm{D}}\rho$ of H are related to each other through a unitary transformation: ${T}_{{\eta }_{+}}={U}^{-1}{\tilde {T}}_{{\eta }_{+}}U$ and $\langle \langle {T}_{{\eta }_{+}}\rangle \rangle _{{\eta }_{+}}=\langle \langle {\tilde {T}}_{{\eta }_{+}}\rangle \rangle _{{\eta }_{+}}=\langle {T}_{\mathrm{D}}\rangle =\langle {\tilde {T}}_{\mathrm{D}}\rangle$. This implies that the expectation values of the physical observables and/or associated topological invariants of H can be computed either from that of h or from that of $\tilde {h}$. The different choices of the similarity operator $\tilde {\rho }$ correspond to different quantum canonical transformations in _D. Thus, without any loss of generality, the discussions below involve the similarity operator ρ.

The Hamiltonians H and h are isospectral. The Bloch eigenstates |ϕ_a(p)〉 of H(p) at each p with the eigenvalues E_a(p) are related to the corresponding states |ψ_a(p)〉 of h through the equation

$$\begin{eqnarray} &&\vert {\phi }_{a}(\mathbf{p})\rangle ={\rho }^{-1}\vert {\psi }_{a}(\mathbf{p})\rangle . \end{eqnarray} \tag{ 3 }$$

Further, the |ψ_a(p)〉 constitute a complete set of orthonormal states in _D, while the completeness and orthonormality of states |ϕ_a(p)〉 can be shown only in _η+. It is assumed throughout this paper that a bulk band gap exists for h that is centered around some fixed energy (which may be chosen to be zero without loss of any generality) and the quantum ground state is obtained by filling all the N states below this level. The isospectrality between h and H allows one to assume identical conditions for H. The Berry connection for the N number of filled Bloch states for h in the Hilbert space _D is defined as [6, 7]

$$\begin{eqnarray} \displaystyle \fl {a}_{i}^{\hat {a}\hat {b}}(\mathbf{p}){\mathrm{d}p}_{i}:=\langle {\psi }_{\hat {a}}(\mathbf{p})\vert \mathrm{d}{\psi }_{\hat {b}}(\mathbf{p})\rangle ,\tqs \hat {a},\hat {b}=1,2,\ldots ,N,&&\displaystyle \\ \displaystyle i=1,2,\ldots ,D.&&\displaystyle \end{eqnarray} \tag{ 4 }$$

The Berry connection for the non-Hermitian Hamiltonian H is introduced as

$$\begin{eqnarray} \displaystyle {\mathcal{A}}_{i}^{\hat {a}\hat {b}}(\mathbf{p})\hspace{0.167em} \mathrm{d}{p}_{i}&:=&\displaystyle \langle \langle {\phi }_{a}(\mathbf{p})\vert \mathrm{d}{\phi }_{b}(\mathbf{p})\rangle \rangle _{{\eta }_{+}},\\ \displaystyle &=&\displaystyle {a}_{i}^{\hat {a}\hat {b}}(\mathbf{p}){\mathrm{d}p}_{i}. \end{eqnarray} \tag{ 5 }$$

Besides an implicit assumption that ρ is independent of the momentum p, equation (3) and the modified inner product in _η+ have been used to obtain the expression in the second line of the above equation. The crucial observation at this point is that the gauge potentials ${\mathcal{A}}_{i}^{\hat {a}\hat {b}}$ in _η+ and ${a}_{i}^{\hat {a}\hat {b}}$ in _D are identical, leading to the same Berry curvature. Consequently, the Chern numbers and Chern–Simons invariants will have identical values in _D as well as in _η+. The readers are referred to [7] for explicit expressions for different topological invariants in terms of Berry curvature.

The topological invariants associated with chiral topological insulators are constructed in terms of the projectors onto the filled Bloch states. In this context, an idempotent operator Γ_D may be introduced in odd space dimensions that anti-commutes with the Dirac Hamiltonian h of a massive free particle:

$$\begin{eqnarray} &&{\Gamma }_{\mathrm{D}}^{2}=1,\tqs \{ h,{\Gamma }_{\mathrm{D}}\} =0. \end{eqnarray} \tag{ 6 }$$

The existence of such an operator implies that corresponding to each eigenstate |ψ_a(p)〉 of h with the eigenvalue E_a, $\vert {\tilde {\psi }}_{a}(\mathbf{p})\rangle ={\Gamma }_{\mathrm{D}}\vert {\psi }_{a}(\mathbf{p})\rangle$ is an eigenstate of h with the eigenvalue  − E_a. The operator that anti-commutes with H for a given Γ_D may be found as Γ_η+ = ρ⁻¹Γ_Dρ. Consequently, Γ_η+ relates the eigenstates $\vert {\tilde {\phi }}_{a}(\mathbf{p})\rangle$ and |ϕ_a(p)〉 of H corresponding to the eigenvalues  − E_a and E_a, respectively. In particular, $\vert {\tilde {\phi }}_{a}(\mathbf{p})\rangle ={\Gamma }_{{\eta }_{+}}\vert {\phi }_{a}(\mathbf{p})\rangle$.

The projector P_η+(p) onto the filled Bloch states of the Hamiltonian H at fixed p and the associated 'Q_η+ matrix' are defined as

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rc@{}} \displaystyle &\displaystyle {P}_{{\eta }_{+}}(\mathbf{p}):=\sum _{\hat {a}}\vert {\phi }_{\hat {a}}(\mathbf{p})\rangle \langle {\phi }_{\hat {a}}(\mathbf{p})\vert {\eta }_{+},\\ \displaystyle &\displaystyle {Q}_{{\eta }_{+}}(\mathbf{p}):=1-2{P}_{{\eta }_{+}}(\mathbf{p}), \end{array} &&\displaystyle \end{eqnarray} \tag{ 7 }$$

where $\hat {a}$ indicates a summation over all the filled Bloch states. The operators P_η+(p) and Q_η+(p) are Hermitian in the Hilbert space _η+ and satisfy the standard relations:

$$\begin{eqnarray} &&{P}_{{\eta }_{+}}^{2}(\mathbf{p})={P}_{{\eta }_{+}}(\mathbf{p}),\tqs {Q}_{{\eta }_{+}}^{2}(\mathbf{p})=1. \end{eqnarray} \tag{ 8 }$$

The projector P_D(p) onto the filled Bloch states of the Hamiltonian h at fixed p and the associated 'Q_D matrix' are related to P_η+(p) and Q_η+(p) through a non-unitary similarity transformation:

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rc@{}} \displaystyle &\displaystyle {P}_{\mathrm{D}}(\mathbf{p}):=\sum _{\hat {a}}\vert {\psi }_{\hat {a}}\rangle \langle {\psi }_{\hat {a}}\vert =\rho {P}_{{\eta }_{+}}(\mathbf{p}){\rho }^{-1},\\ \displaystyle &\displaystyle {Q}_{\mathrm{D}}(\mathbf{p}):=1-2{P}_{\mathrm{D}}(\mathbf{p})=\rho {Q}_{{\eta }_{+}}(\mathbf{p}){\rho }^{-1}, \end{array} &&\displaystyle \end{eqnarray} \tag{ 9 }$$

where $\hat {a}$ indicates a summation over all the filled Bloch states and ${P}_{\mathrm{D}}^{2}(\mathbf{p})={P}_{\mathrm{D}}(\mathbf{p})$, ${Q}_{\mathrm{D}}^{2}(\mathbf{p})=1$. The topological invariants for class AIII topological insulators in Hermitian quantum systems are constructed by using the projector Q_D(p) [6, 7], and equation (9) implies that the topological invariants for H and h are identical.

A comment is in order before the end of this section. The existence of Γ_D implies that the pseudo-Hermitian H is also pseudo-anti-Hermitian [18] with respect to an operator κ:

$$\begin{eqnarray} &&{H}^{\dagger }=-\kappa H{\kappa }^{-1},\tqs \kappa :=\rho {\Gamma }_{\mathrm{D}}\rho ={\eta }_{+}{\Gamma }_{{\eta }_{+}}. \end{eqnarray} \tag{ 10 }$$

The kinds of operators Γ_D and ρ that will be considered in this paper satisfy the identity

$$\begin{eqnarray} &&\rho {\Gamma }_{\mathrm{D}}\rho ={\Gamma }_{\mathrm{D}}. \end{eqnarray} \tag{ 11 }$$

The Hermitian Hamiltonian $\tilde {H}:=H+{H}^{\dagger }$ in _D anti-commutes with the operator Γ_D, when the validity of equation (11) is assumed, implying that $\tilde {H}$ can be transformed into a block off-diagonal form in a representation in which Γ_D is diagonal. The projector onto the filled Bloch states of the Hamiltonian $\tilde {H}$ can be used to construct topological invariants corresponding to H provided H is a normal operator, i.e. [H,H^†] = 0, so $\tilde {H}$ and H are simultaneously diagonalizable. The non-Hermitian Hamiltonians considered in this paper are not normal operators and, hence, this scheme of constructing topological invariants is not applicable. On the other hand, the projector $Q:=1-[{P}_{{\eta }_{+}}(\mathbf{p})+{P}_{{\eta }_{+}}^{\dagger }(\mathbf{p})]$ onto the filled Bloch states of H and H^† is not an idempotent operator. Consequently, a block in the block off-diagonal form of Q cannot necessarily be identified as an element of U(N). Thus, the use of Q to construct a topological invariant corresponding to non-Hermitian H seems problematic.

A few examples of non-Hermitian Hamiltonians admitting topological insulator phases are presented in this section. All the non-Hermitian Hamiltonians considered in this paper can be mapped to Hermitian Hamiltonians, through non-unitary similarity transformations, which are known to admit a topological insulator phase. It follows from the discussions in section 2 that the bulk eigenvalues and the topological invariants are identical for both the non-Hermitian Hamiltonian and its similarity transformed Hermitian version. The bulk eigenstates, zero modes and symmetry generators of the non-Hermitian Hamiltonian can be obtained from the corresponding quantities of the Hermitian Hamiltonian through the inverse similarity transformation. Thus, in general, only non-Hermitian Hamiltonians and associated non-unitary similarity transformations are mentioned explicitly in this note to avoid any repetition of known results. The 1 + 1-dimensional Dirac equation is treated in some detail in section 3.1 to give a general outline of the technique involved, that can be applied in similar situations.

3.1. The Dirac Hamiltonian in 1 + 1 dimensions

The first example considered in this paper is the 1 + 1-dimensional non-Hermitian Dirac Hamiltonian

$$\begin{eqnarray} \displaystyle \fl {H}^{(1)}={\sigma }^{2}{p}_{x}+m(x)\cosh \phi {\sigma }^{3}-\mathrm{i}m(x)\sinh \phi ~{\sigma }^{1},&&\displaystyle \\ \displaystyle \phi \in R,&&\displaystyle \end{eqnarray} \tag{ 12 }$$

where p_x is the linear momentum and m(x) is an arbitrary real function of its argument. The Hamiltonian H⁽¹⁾ was first introduced¹ in [11] in the context of a non-dissipative non-Hermitian relativistic quantum system in _D. It was shown [11] that H⁽¹⁾ is Hermitian in the Hilbert space ${\mathcal{H}}_{{\eta }_{+}^{(1)}}$ that is endowed with the metric ${\eta }_{+}^{(1)}$:

$$\begin{eqnarray} &&{\eta }_{+}^{(1)}:={\mathrm{e}}^{-\phi {\sigma }^{2}},\tqs {\rho }^{(1)}:={\mathrm{e}}^{-\frac{\phi }{2}{\sigma }^{2}}. \end{eqnarray} \tag{ 13 }$$

The non-unitary similarity operator ρ⁽¹⁾ maps H⁽¹⁾ to a Hermitian Hamiltonian h⁽¹⁾ in _D [11]:

$$\begin{eqnarray} &&{h}^{(1)}:={\rho }^{(1)}{H}^{(1)}({\rho }^{(1)})^{-1}={\sigma }^{2}{p}_{x}+m(x){\sigma }^{3}, \end{eqnarray} \tag{ 14 }$$

implying that H⁽¹⁾ and h⁽¹⁾ are isospectral. The Hamiltonian h⁽¹⁾ anti-commutes with σ¹. Correspondingly, the Hamiltonian H⁽¹⁾ anti-commutes with the operator $({\rho }^{1})^{-1}{\sigma }^{1}{\rho }^{(1)}={\sigma }^{1}{\eta }_{+}^{(1)}$.

The Dirac Hamiltonian h⁽¹⁾ with m(x) = m∈R provides an example of a chiral topological insulator in class AIII [7]. The relation equation (14) implies that the non-Hermitian Hamiltonian H⁽¹⁾ also admits a topological insulator phase. The bulk spectrum of H⁽¹⁾ contains a mass gap, ${E}_{\pm }=\pm \sqrt{{p}_{x}^{2}+{m}^{2}}\equiv \pm \lambda$, and the Bloch wavefunction corresponding to the negative eigenvalue state reads

$$\begin{eqnarray} &&\displaystyle \vert {\phi }^{-}({p}_{x})\rangle =\frac{({\rho }^{(1)})^{-1}}{2\sqrt{{p}_{x}^{2}+{m}^{2}}}\left(\begin{array}{@{}c@{}} \displaystyle \mathrm{i}{p}_{x}-m+\lambda \\ \displaystyle -(\mathrm{i}{p}_{x}-m)+\lambda \end{array}\right).&&\displaystyle \end{eqnarray} \tag{ 15 }$$

The Berry connection (p_x) and the associated Chern–Simons invariant CS₁ for the state |ϕ⁻(p_x)〉 may be computed as

$$\begin{eqnarray} &&\mathcal{A}({p}_{x})=-\frac{\mathrm{i}m}{2{\lambda }^{2}}\hspace{0.167em} \mathrm{d}{p}_{x},\tqs C{S}_{1}=\frac{m}{4\vert m\vert }, \end{eqnarray} \tag{ 16 }$$

which are identical with the corresponding expressions [7] for h⁽¹⁾ in _D. The projector Q_η+ can be expressed as follows:

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rc@{}} \displaystyle &\displaystyle {Q}_{{\eta }_{+}}=(U{\rho }^{(1)})^{-1}\left(\begin{array}{@{}cc@{}} \displaystyle 0&\displaystyle q\\ \displaystyle {q}^{\ast }&\displaystyle 0 \end{array}\right)(U{\rho }^{(1)}),\\ \displaystyle &\displaystyle q:=\frac{-\mathrm{i}{p}_{x}+m}{\lambda },\tqs U:={\mathrm{e}}^{-\frac{\mathrm{i}\pi }{4}{\sigma }^{2}}, \end{array} &&\displaystyle \end{eqnarray} \tag{ 17 }$$

where q* denotes the complex conjugate of q. It may be noted that the unitary operator U and the similarity operator ρ⁽¹⁾ are independent of p_x and m. The winding number is calculated as $\nu =\frac{\mathrm{i}}{2 \pi }\int {q}^{-1}\hspace{0.167em} \mathrm{d}q=\frac{m}{2\vert m\vert }$, which is twice the value of the Chern–Simons invariant CS₁.

The operator h⁽¹⁾ can be identified as the real supercharge of an  = 1 supersymmetric quantum system. The ground state of the supersymmetric Hamiltonian h_s: = [h⁽¹⁾]² in the supersymmetry-preserving phase is a zero mode of h⁽¹⁾. The zero mode of H can be obtained from the zero mode of h⁽¹⁾ by using equation (3). For example, assuming that Lt_x→±∞m(x) =± m,m∈R⁺, the zero mode of H reads

$$\begin{eqnarray} &&{\phi }_{0}=\frac{1}{\sqrt{2}}({\rho }^{(1)})^{-1}{\mathrm{e}}^{-\int \nolimits _{0}^{x}m(x)\hspace{0.167em} \mathrm{d}x}\left(\begin{array}{@{}c@{}} \displaystyle 1\\ \displaystyle -1 \end{array}\right). \end{eqnarray} \tag{ 18 }$$

The zero mode of H for different shapes of m(x) may be obtained in a similar way from the corresponding well-behaved zero-energy state of h⁽¹⁾.

3.2. The Dirac Hamiltonian in 2 + 1 dimensions

The second example is a four-component non-Hermitian Dirac equation in 2 + 1 dimensions. The relevant gamma matrices may be constructed in terms of the elements of the Clifford algebra:

$$\begin{eqnarray} &&\{ {\xi }^{p},{\xi }^{q}\} ={\delta }^{p q},\tqs p,q=1,2,\ldots ,5. \end{eqnarray} \tag{ 19 }$$

The three generators of the group O(3) are expressed as

$$\begin{eqnarray} &&{J}^{a}:=\frac{\mathrm{i}}{8}{\epsilon }^{a b c}[{\xi }^{b},{\xi }^{c}],\tqs a,b,c=1,2,3. \end{eqnarray} \tag{ 20 }$$

The Hilbert space ${\mathcal{H}}_{{\eta }_{+}^{(2)}}$ is endowed with the metric ${\eta }_{+}^{(2)}$:

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rc@{}} \displaystyle &\displaystyle {\eta }_{+}^{(2)}:={\mathrm{e}}^{-\phi ~\hat {n}\cdot \vec{J}},\tqs {\rho }^{(2)}:={\mathrm{e}}^{-\frac{\phi }{2}\hat {n}\cdot \vec{J}},\\ \displaystyle &\displaystyle \phi \in R,\tqs \hat {n}\cdot \hat {n}=1. \end{array} &&\displaystyle \end{eqnarray} \tag{ 21 }$$

The Hamiltonian H⁽²⁾ given by

$$\begin{eqnarray} &&\displaystyle \begin{array}{@{}l@{}} \displaystyle \fl {H}^{(2)}={\xi }^{4}{p}_{x}+{\xi }^{5}{p}_{y}+m\sum _{b=1}^{3}~{R}^{3 b}{\xi }^{b},\\ \displaystyle \fl {R}^{a b}\equiv {n}^{a}{n}^{b}(1-\cosh \phi )+{\delta }^{a b}\cosh \phi \\ \displaystyle +~\mathrm{i}{\epsilon }^{a b c}{n}^{c}\sinh \phi , \end{array}&&\displaystyle \end{eqnarray} \tag{ 22 }$$

is non-Hermitian in _D and Hermitian in ${\mathcal{H}}_{{\eta }_{+}^{(2)}}$. The non-Hermiticity in H⁽²⁾ arises due to the fact that the complex conjugate of R^ab is not equal to itself.

The non-unitary similarity operator ρ⁽²⁾ maps H⁽²⁾ to a Hermitian Hamiltonian h⁽²⁾ in _D:

$$\begin{eqnarray} \displaystyle {h}^{(2)}&:=&\displaystyle {\rho }^{(2)}{H}^{(2)}({\rho }^{(2)})^{-1}\\ \displaystyle &=&\displaystyle {\xi }^{4}{p}_{x}+{\xi }^{5}{p}_{y}+m{\xi }^{3}. \end{eqnarray} \tag{ 23 }$$

The elements of the Clifford algebra are realized in terms of the following matrices:

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rc@{}} \displaystyle &\displaystyle {\xi }_{1}:={\tau }^{2}\otimes I,\tqs {\xi }_{2}:={\tau }^{3}\otimes I,\\ \displaystyle &\displaystyle {\xi }_{3}:={\tau }^{1}\otimes {\sigma }^{3},\tqs {\xi }_{4}:={\tau }^{1}\otimes {\sigma }^{1},\\ \displaystyle &\displaystyle {\xi }_{5}:={\tau }^{1}\otimes {\sigma }^{2}, \end{array} &&\displaystyle \end{eqnarray} \tag{ 24 }$$

where τ^a,σ^a with a = 1,2,3 are two sets of Pauli matrices corresponding to two different sublattices and I is a 2 × 2 identity matrix. The Hamiltonian h⁽²⁾ is known to admit a topological insulator phase [7]. In fact, h⁽²⁾ describes two copies of the two-dimensional two-component Dirac Hamiltonian having a topological insulator phase. Following the general discussions in section 2, the conclusion is that H⁽²⁾ admits a topological insulator phase with identically the same bulk eigenvalues and topological invariant as Hamiltonian h⁽²⁾. The bulk eigenstates as well as the zero-mode state of H⁽²⁾ can be obtained from the corresponding states of h⁽²⁾ by using the relation (3) and making the identification ρ: = ρ⁽²⁾.

3.3. The Dirac Hamiltonian in 3 + 1 dimensions

A Hamiltonian in 3 + 1 dimensions may be introduced as follows:

$$\begin{eqnarray} \displaystyle \begin{array}{@{}rc@{}} \displaystyle &\displaystyle {H}^{(3)}=\vec{\alpha }\cdot \vec{p}+m{\mathrm{e}}^{{\gamma }_{5}\phi }\beta ,\tqs \phi \in R\\ \displaystyle &\displaystyle {\alpha }^{a}:={\tau }^{1}\otimes {\sigma }^{a},\tqs \beta :={\tau }^{3}\otimes I,\\ \displaystyle &\displaystyle {\gamma }^{5}:={\tau }^{1}\otimes I,\tqs a=1,2,3, \end{array} &&\displaystyle \end{eqnarray} \tag{ 25 }$$

where the Dirac representation of the γ matrices has been used and τ^a,σ^a correspond to two independent sets of Pauli matrices. The Hamiltonian is non-Hermitian in the Hilbert space _D for ϕ ≠ 0. The generator of the time-reversal transformation is defined as : = (I⊗iσ²)K, where K is the complex conjugation operator. The Hamiltonian H⁽³⁾ is invariant under the time-reversal transformation. The last term spoils the invariance of the Hamiltonian under the parity transformation with the standard form of the generator : = β. However, a non-standard generator of the parity transformation may be introduced as $\tilde {\mathcal{P}}:={\mathrm{e}}^{\phi {\gamma }_{5}}\beta$, which keeps H⁽³⁾ invariant.

The Hamiltonian H⁽³⁾ becomes Hermitian in _D in the limit ϕ → 0 and is Hermitian for any arbitrary ϕ in the Hilbert space _η+ that is endowed with a positive-definite metric ${\eta }_{+}^{(3)}$:

$$\begin{eqnarray} &&{\eta }_{+}^{(3)}:={\mathrm{e}}^{-\phi {\gamma }_{5}},\tqs {\rho }^{(3)}:={\mathrm{e}}^{-\frac{\phi }{2}{\gamma }_{5}}. \end{eqnarray} \tag{ 26 }$$

In fact, the non-unitary similarity operator ρ⁽³⁾ can be used to map H⁽³⁾ to a Hermitian Hamiltonian h⁽³⁾:

$$\begin{eqnarray} \displaystyle {h}^{(3)}&=&\displaystyle {\rho }^{(3)}{H}^{(3)}({\rho }^{(3)})^{-1}\\ \displaystyle &=&\displaystyle \vec{\alpha }\cdot \vec{p}+m \beta , \end{eqnarray} \tag{ 27 }$$

that is Hermitian in _D. Thus, H⁽³⁾ and h⁽³⁾ are isospectral. It is known that the Hamiltonian h⁽³⁾ admits a topological insulator phase [7]. Thus, the Hamiltonian H⁽³⁾ also admits a topological insulator phase with identically the same bulk eigenvalues and topological invariant as Hamiltonian h⁽³⁾. Other relevant quantities for H⁽³⁾ may be obtained from h⁽³⁾ by using equation (3) with the replacement of ρ by ρ⁽³⁾.

Examples of non-Hermitian Dirac Hamiltonians admitting a topological insulator phase have been presented in one, two and three spatial dimensions. All of these Hamiltonians are Hermitian in a Hilbert space that is endowed with a positive-definite metric and a modified inner product. All of the bulk eigenvalues of a given non-Hermitian Hamiltonian are thus real. It has also been shown that any non-Hermitian Hamiltonian belonging to this class can be mapped to a Hermitian Hamiltonian through a non-unitary similarity transformation, implying that these two quantum systems are isospectral to each other. Further, the time evolution of the bulk states as well as the zero modes of the non-Hermitian Hamiltonians are unitary in the Hilbert space H_η+. It appears that the topological insulator phase in non-Hermitian Dirac Hamiltonians admitting entirely real spectra and unitary time evolution have been discussed in this paper for the first time in the literature.

The topological insulator phases in Hermitian quantum systems have been classified previously [6, 7] according to certain discrete symmetries of Dirac Hamiltonians. Perturbations preserving the symmetries of the original Hamiltonian cannot destabilize the topological insulator phase as long as the gap is not closed. A direct application of this classification scheme to the non-Hermitian Hamiltonians presented in this paper may give conflicting results, since the non-Hermitian terms may or may not preserve the symmetry of the Hermitian part of the same Hamiltonian. However, the same classification scheme can be used for a non-Hermitian Hamiltonian provided that the identification of the topological class is based on the analysis of the Hermitian Hamiltonian to which it is related through a non-unitary similarity transformation. It may be worth emphasizing here that the topological index characterizing a particular type of insulator, as presented in this paper, is identical for the non-Hermitian and the corresponding Hermitian Hamiltonians. The non-Hermitian Hamiltonian and its Hermitian counterpart thus belong to the same topological class by construction. It may be emphasized here that the unitary equivalence between different Hermitian Hamiltonians h, $\tilde {h}$ in _D, which are obtained from the same non-Hermitian Hamiltonian H, allows a unique characterization of the topological class of H admitting topological insulator phases.

The approach taken in this paper in classifying topological insulator phases in non-Hermitian quantum systems may equally be extended to any generic non-Hermitian Hamiltonian that can be mapped to a Hermitian Hamiltonian. In general, finding the non-unitary similarity operator that maps the non-Hermitian Hamiltonian to a Hermitian one is a highly non-trivial task. However, the continuum descriptions of topological insulators are in terms of Dirac Hamiltonians which involve gamma matrices. The method described in [12] for a pseudo-Hermitian realization of the gamma matrices of arbitrary dimensions may be useful for finding the Hermitian Hamiltonian corresponding to a given non-Hermitian Hamiltonian. It is desirable that the method presented in this paper and in previous works [11–13] is utilized fully to construct at least one physically realizable non-Hermitian Dirac Hamiltonian admitting a topological insulator phase.