Hidden Chern number in one-dimensional non-Hermitian chiral-symmetric systems

TL;DR

This paper reveals a hidden Chern number in 1D non-Hermitian chiral-symmetric systems, linking topology to in-gap end states, and demonstrates the robustness of bulk-boundary correspondence despite non-Hermitian effects.

Contribution

It introduces a hidden Chern number in 1D non-Hermitian systems with chiral symmetry and shows its role in topologically protected end states.

Findings

Hidden Chern number determines topological phases.

End states are robust against non-Hermitian skin effect.

Derived phase diagram for a minimal model.

Abstract

We consider a class of one-dimensional non-Hermitian models with a special type of a chiral symmetry which is related to pseudo-Hermiticity. We show that the topology of a Hamiltonian belonging to this symmetry class is determined by a hidden Chern number described by an effective 2D Hermitian Hamiltonian $H^{\rm eff} (k, \eta)$, where $\eta$ is the imaginary part of the energy. This Chern number manifests itself as topologically protected in-gap end states at zero real part of the energy. We show that the bulk-boundary correspondence coming from the hidden Chern number is robust and immune to non-Hermitian skin effect. We introduce a minimal model Hamiltonian supporting topologically nontrivial phases in this symmetry class, derive its topological phase diagram and calculate the end states originating from the hidden Chern number.

Figures

Equations

$$\mathcal{S}{\mathcal{H}}_k\mathcal{S}=-{\mathcal{H}}_k^{†},$$

$$\mathcal{S}{\mathcal{H}}_k^h\mathcal{S}$$

$$\mathcal{S}{\mathcal{H}}_k^a\mathcal{S}$$

$$\mathcal{S}{\mathcal{H}}_k\mathcal{S}=-{\mathcal{H}}_k^{†}⟺{\mathcal{H}}_k=i\mathcal{S}{H}_k,H_k^{†}=H_k.$$

$${\mathcal{H}}_k=\left(\begin{array}{cc}i{P}_k& Q_k^{†}\\ Q_k& i{R}_k\end{array}\right),$$

$$\mathcal{S}=\left(\begin{array}{cc}\text{mathbbm}1& 0\\ 0& -\text{mathbbm}1\end{array}\right).$$

$${\mathcal{H}}_{\mathrm{obc}}|\psi ⟩=i\eta |\psi ⟩.$$

$$\left(H_{\mathrm{obc}}+\eta {\mathcal{S}}_{\mathrm{obc}}\right)|\psi ⟩=0.$$

$$H_{\mathrm{obc}}^{\mathrm{eff}}\left(\eta \right)\equiv H_{\mathrm{obc}}+\eta {\mathcal{S}}_{\mathrm{obc}}={\mathcal{S}}_{\mathrm{obc}}\left(\eta -i{\mathcal{H}}_{\mathrm{obc}}\right).$$

$$H^{\mathrm{eff}}\left(k,\eta \right)\equiv H_k+\eta \mathcal{S}=\mathcal{S}\left(\eta -i{\mathcal{H}}_k\right).$$

$$H^{\mathrm{eff}}\left(k,-\infty \right)=-H^{\mathrm{eff}}\left(k,+\infty \right).$$

$$H_{\mathrm{cp}}^{\mathrm{eff}}\left(k,\eta \right)={\mathcal{R}}_{\eta }{H}^{\mathrm{eff}}\left(k,\eta \right){\mathcal{R}}_{\eta }^{†}.$$

$${\mathcal{R}}_{\eta }=\exp \left[i\frac{\pi }{4}\left(1+\tanh \eta \right)\mathcal{G}\right],\mathcal{G}=\left(\begin{array}{cc}0& \text{mathbbm}1\\ \text{mathbbm}1& 0\end{array}\right).$$

$$H_{\mathrm{cp}}^{\mathrm{eff}}\left(k,\eta \to -\infty \right)=H_{\mathrm{cp}}^{\mathrm{eff}}\left(k,\eta \to +\infty \right)=-\left|\eta \right|\mathcal{S}.$$

$$C=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }d\eta {\int }_0^{2\pi }dk{\Omega }_{k,\eta },$$

$${\Omega }_{k,\eta }=\sum _{\genfrac{}{}{0ex}{}{n\le n_F}{m>n_F}}\mathfrak{Im}\frac{2⟨{\psi }_{k,\eta }^n|{\partial }_k{H}_{\mathrm{cp}}^{\mathrm{eff}}|{\psi }_{k,\eta }^m⟩⟨{\psi }_{k,\eta }^m|{\partial }_{\eta }{H}_{\mathrm{cp}}^{\mathrm{eff}}|{\psi }_{k,\eta }^n⟩}{{\left(E_{k,\eta }^{(n)}-E_{k,\eta }^{(m)}\right)}^2}.$$

$${\mathcal{H}}_k=\left(\begin{array}{cccc}i{g}_1& t& 0& t{e}^{-ik}\\ t& i{g}_2& t& 0\\ 0& t& i{g}_3& t\\ t{e}^{ik}& 0& t& i{g}_4\end{array}\right).$$

$$g_4=-g_1-g_2-g_3.$$

$$H^{\mathrm{eff}}\left(k,\eta \right)=\left(\begin{array}{cccc}\eta +g_1& -it& 0& -it{e}^{-ik}\\ it& -\eta -g_2& it& 0\\ 0& -it& \eta +g_3& -it\\ it{e}^{ik}& 0& it& -\eta -g_4\end{array}\right),$$

$$\mathcal{S}{\mathcal{H}}_k\mathcal{S}=-{\mathcal{H}}_k^{†}.$$

$${\mathcal{H}}_k=i\mathcal{S}{H}_k$$

$$H_k=-i\mathcal{S}{\mathcal{H}}_k.$$

$$H_k^{†}=i{\mathcal{H}}_k^{†}\mathcal{S}=-i\mathcal{S}{\mathcal{H}}_k=H_k.$$

$${\mathcal{H}}_k=i\mathcal{S}{H}_k,$$

$${\mathcal{H}}_k=\left(\begin{array}{cc}{a}_k& b_k\\ c_k& -a_k\end{array}\right),$$

$${\mathcal{H}}_k^{\prime }={\mathcal{U}}^{†}{\mathcal{H}}_k\mathcal{U}=\left(\begin{array}{cc}\frac{i}{2}\left(b_k-c_k\right)& a_k-\frac{i}{2}\left(b_k+c_k\right)\\ a_k+\frac{i}{2}\left(b_k+c_k\right)& -\frac{i}{2}\left(b_k-c_k\right)\end{array}\right).$$

Full text

Open quantum systems with loss (dissipation) and gain (coherent amplification) are described by non-Hermitian (NH) Hamiltonians Nature-Phys-Exp18 ; Konotop16RMP and have unexpected properties which often depend on the symmetries of the system Nature-Phys-Exp18 ; Konotop16RMP ; Ben07 ; Hatano-Nelson ; Efetov97 ; Bernard-LeClair ; Bernard-LeClair2 ; Magnea08 ; Sato12-review ; Deng12 ; Pikulin14 . Adding a non-Hermitian component to a Hamiltonian does not only broaden the resonances and allow the eigenstates to decay, but the eigenmodes can merge with each other at exceptional points, which are topological defects where not only the eigenvalues are degenerate but also the eigenvectors are parallel to each other Nature-Phys-Exp18 ; Miri19 ; Berry04 ; Heiss12 ; Lui19 . The flexibility to engineer gain and loss in a controllable manner for example in optics, (opto)mechanics, plasmonics, superconducting quantum circuits, dissipative Bose-Einstein condensates, exciton-polariton condensates and cold atom systems Nature-Phys-Exp18 ; Konotop16RMP ; Miri19 ; Muller ; Aspelmeyer14 ; Cao15 ; Gao15 ; Zeu15 ; Schomerus15 ; Xu16 ; Wei17 ; Xiao17 ; Fitzpatrick17 ; Zhou18 ; Wang18 ; Cer18 ; Silveri18 ; Silveri19a ; Silveri19b ; Yosh19 have naturally raised the interest to study the symmetry and topology in non-Hermitian physics systematically Gong18 ; Lieu18 ; Zhou19 ; Kaw19 ; Gha19 with potential applications for example in the design of topologically protected laser modes Bah17 ; Jean17 ; Par18 ; Zhao18 ; Har18 ; Ban18 .

Because in NH systems the energies are complex the Altland-Zirnbauer symmetry classes support new types of winding number and $\mathbb{Z}_{2}$ invariants determined by the complex spectra Gong18 – leading to NH topological phases with no Hermitian counterparts Gong18 ; Zhou19 ; Kaw19 . The classification is further enriched because for a NH Hamiltonian the transpose and complex conjugation are not equivalent so that the 10 Altland-Zirnbauer symmetry classes need to be extended Lieu18 to 38 non-Hermitian (nonspatial) symmetry classes Zhou19 ; Kaw19 . Furthermore, Hermitian Hamiltonians are gapped if the energy bands do not cross the Fermi energy, but non-Hermitian systems feature two different types of complex-energy gaps, so-called point (line) gaps where bands do not cross a point (line) in the complex-energy plane, giving rise to further ramification of the topological classification Kaw19 . Various models and realizations of the different NH topological phases have been proposed Nelson98 ; Rudner-Levitov ; Esa11 ; Lia13 ; Schomerus13 ; Rud16 ; Lee16 ; Ley17 ; Jin17 ; Lieu18B ; Shen18 ; Xu17 ; Tak18 ; Wang19 ; Hui19 . For Hermitian Hamiltonians the bulk-boundary correspondence guarantees that the topological invariants for periodic boundary conditions predict the presence of boundary states for open boundary conditions, but this is typically not the case for NH systems Xio18 ; Zirn19 , where the experimental consequences of topological invariants are less clear because the bulk-boundary correspondence can typically be established either on the level of singular value spectra Herv19 or on the level of biorthogonal density Kun18 ; Edv19 . One of the reasons for the breakdown of the bulk-boundary correspondence in NH systems is that the bulk states are often localized in the vicinity of the boundary (NH skin effect) so that the boundary effects are not described by the bulk Bloch Hamiltonian Yao18 ; Kaw18 ; Rui19 ; Bor19 ; Lee19 ; Hel19 ; Hof19 . Nevertheless, for particular NH symmetry classes, such as the pseudo-Hermitian Hamiltonians, the bulk-boundary correspondence can be established Kaw19 .

From the viewpoint of experiments, particularly relevant 1D NH Hamiltonians can be constructed by considering Hermitian hopping Hamiltonians with onsite gain and loss terms. Assuming chiral-symmetric hopping Hamiltonian this type of NH Hamiltonians ${\cal H}_{k}$ satisfy a special kind of NH chiral symmetry

[TABLE]

where ${\cal S}$ is a Hermitian unitary operator. Furthermore, in the following we assume that ${\cal S}$ is traceless so that the unit cell contains even number of lattice sites. In the topological classification discussed in Refs. Kaw19 , the NH chiral-symmetric Hamiltonians are discussed and the topological invariant and bulk-boundary correspondence has been established Zha19 ; Kaw19 . They can also be considered in the framework of pseudo-Hermitian Hamiltonians because $i{\cal H}_{k}$ is a pseudo-Hermitian matrix. In this paper we establish a novel perspective on the topological invariants and bulk-boundary correspondence for these Hamiltonians. Namely, we show that the topology of 1D NH chiral-symmetric Hamiltonian satisfying Eq. (1) is described by an effective 2D Hermitian Hamiltonian $H^{\rm eff}(k,\eta)$, where $\eta$ is the imaginary part of the energy. Moreover, we show that $H^{\rm eff}(k,\eta)$ supports Chern number as a topological invariant which determines the existence of boundary states also for the non-Hermitian Hamiltonian ${\cal H}_{\rm obc}$ with open boundary conditions via the bulk-boundary correspondence of Hermitian Hamiltonians (see Fig. 1). Finally, we introduce a minimal model Hamiltonian supporting topologically nontrivial phases in this symmetry class, derive its topological phase diagram and calculate the end states originating from the hidden Chern number. The hidden Chern numbers agree with the topological invariant considered in Ref. Kaw19 .

By inspecting the characteristic polynomial of ${\cal H}_{k}$ satisfying Eq. (1) we note that if $\lambda$ is the eigenvalue of ${\cal H}_{k}$ then also $-\lambda^{\star}$ is. Every square matrix can be easily decomposed in Hermitian and anti-Hermitian part, namely ${\cal H}_{k}={\cal H}_{k}^{h}+{\cal H}_{k}^{a}$ with ${\cal H}_{k}^{h/a}=\frac{1}{2}\left({\cal H}_{k}\pm{\cal H}_{k}^{\dagger}\right)$. The chiral relation (1) means that

[TABLE]

so ${\cal S}$ anticommutes with Hermitian and commutes with anti-Hermitian part of the Hamiltonian. Moreover, the NH Hamiltonian ${\cal H}_{k}$ satisfies the relation with a Hermitian Hamiltonian $H_{k}$ [see Appendix A]

[TABLE]

For traceless ${\cal S}$ we find that ${\cal H}_{k}$ satisfying Eq. (1) has a generic block structure in the eigebasis of ${\cal S}$. Namely

[TABLE]

where $P_{k}$ and $R_{k}$ are $N\times N$ Hermitian matrices and

[TABLE]

It is worth noticing that any $2\times 2$ real traceless Hamiltonian ${\cal H}_{k}$ can be put in the form of Eq. (4) so it satisfies the chiral symmetry (1), as shown in Appendix B.

Now consider the real space version of this Hamiltonian with open boundary conditions ${\cal H}_{{\rm obc}}$. If ${\cal H}_{k}$ satisfies (1) then also ${\cal H}_{{\rm obc}}$ satisfies it with ${\cal S}_{{\rm obc}}=\mathbbm{1}_{L}\otimes{\cal S}$, where $L$ is the number of unit cells stacked along the chain. We are interested about the end states of ${\cal H}_{{\rm obc}}$ with zero real part of the energy. Thus, we demand that there exists $\eta\in\mathbb{R}$ such that

[TABLE]

Using (3) we obtain ${\cal H}_{{\rm obc}}=i{\cal S}_{{\rm obc}}H_{{\rm obc}}$ and $H^{\dagger}_{{\rm obc}}=H_{{\rm obc}}$, so that we get from Eq. (6)

[TABLE]

Notice that this is a zero-energy eigenproblem for an effective Hermitian Hamiltonian

[TABLE]

We can use this to define a $k$-space effective Hermitian Hamiltonian

[TABLE]

The above Hamiltonian is defined in a two-dimensional $\left(k,\eta\right)$ space and its Chern number is quantized if it is gapped and it can be compactified in $\eta$. A non-trivial Chern number $C$ of $H^{{\rm eff}}\left(k,\eta\right)$ at half-filling implies that we have $C$ chiral boundary-modes of $H_{{\rm obc}}^{{\rm eff}}\left(\eta\right)$ crossing the energy gap and it means that we have $C$ solutions of the eigenproblem (7) or (6). Mapping the non-Hermitian Hamiltonians with open and periodic boundary conditions to the same Hermitian problem guarantees that if ${\cal H}_{k}$ has gapped real spectrum then ${\cal H}_{{\rm obc}}$ also has gapped real spectrum meaning that the non-Hermitian skin effect does not lead to a breakdown of the bulk-boundary correspondence. Moreover, if $H^{{\rm eff}}\left(k,\eta\right)$ is gapped and topological then $H_{{\rm obc}}^{{\rm eff}}\left(\eta\right)$ supports boundary states, and therefore also ${\cal H}_{k}$ is gapped and ${\cal H}_{{\rm obc}}$ supports end states with zero real part of the energy (see Fig. 1 for the schematic view of this induced bulk-boundary correspondence).

Now what remains is the question of quantization of the Chern number of $H^{{\rm eff}}\left(k,\eta\right)$. We know that it is quantized as long as $H^{{\rm eff}}\left(k,\eta\right)$ is periodic in $k$ and $\eta$. Periodicity in $k$ is obvious but in the canonical basis where Eqs. (4) and (5) are satisfied we obtain

[TABLE]

We can overcome this problem by defining a compactified version of Hamiltonian $H^{{\rm eff}}\left(k,\eta\right)$ given by

[TABLE]

where

[TABLE]

This way ${\cal R}_{-\infty}=\mathbbm{1}$ and ${\cal R}_{+\infty}=i{\cal G}$ so that $H_{{\rm cp}}^{{\rm eff}}\left(k,\eta\right)$ is compactified in $\eta$ as

[TABLE]

Note that the spectrum of $H_{{\rm cp}}^{{\rm eff}}\left(k,\eta\right)$ and $H^{{\rm eff}}\left(k,\eta\right)$ is the same. Now, the Chern number $C$ for $H_{{\rm cp}}^{{\rm eff}}\left(k,\eta\right)$ can be obtained using Kubo formula Thouless82 ; Nagaosa10

[TABLE]

where the Berry curvature $\Omega_{k,\eta}$ is given by

[TABLE]

Here $\left|\psi_{k,\eta}^{p}\right\rangle$ are eigenstates of $H_{{\rm cmp}}^{{\rm eff}}\left(k,\eta\right)$ (sorted in ascending order of eigenenergy) and $n_{F}$ is the number of occupied bands.

In what follows we focus on a minimal tight-binding model with uniform hopping $t$ supporting nontrivial phases. In this model we have a four-site unit cell with gain and loss terms $g_{1}$, $g_{2}$, $g_{3}$ and $g_{4}$, so that the NH Hamiltonian is

[TABLE]

Without loss of generality we can assume that

[TABLE]

This model satisfies chiral relation (1) with ${\cal S}=\mathbbm{1}\otimes\sigma_{z}$. From Eq. (9) we find that the effective Hermitian Hamiltonian for our model is

[TABLE]

and similarly from Eq. (8) we can get $H_{{\rm obc}}^{{\rm eff}}\left(\eta\right)$. The phase diagram of $H^{{\rm eff}}\left(k,\eta\right)$ is given in Fig. 2. The gapped phases are located mainly in the plane $g_{3}=-g_{1}$ and we find two non-trivial ones with hidden Chern numbers $C=-1$ along direction $g_{1}=-g_{2}$ and two trivial ones with $C=0$ along direction $g_{1}=g_{2}$. For better clarity Fig. 3 shows two-dimensionl phase diagrams in the planes $g_{3}=-g_{1}$ and $g_{3}=-g_{1}+0.2t$, see also Appendix C.

In Fig. 4 we show the spectra of the Hamiltonian $H_{{\rm obc}}^{{\rm eff}}\left(\eta\right)$ with open boundary conditions as a function of $\eta$ for the gapped phases shown in Fig. 3(a). In the non-trivial phase [Fig. 4(a)] we can see that the gap around zero energy is crossed by two states connecting valence and conduction band. Each of them cross the zero energy level once and values of $\eta$ at which this happen correspond to two solutions of Eq. (6), so that in the case of the original non-Hermitian Hamiltonian ${\cal H}_{{\rm obc}}$ the real part of the eigenenergy is zero. In the trivial case [Fig. 4(b)] these states are missing.

Fig. 5 shows energy spectra for a topologically nontrivial non-Hermitian system with open boundary conditions. We find two end states with zero real part of the energy and the imaginary parts of the energies stick out of the bulk spectrum. As one can see from Fig. 5(c) the states with zero real part of the energy are strongly localized at the left or right end of the chain. The rest of the states are delocalized in the bulk.

We have also computed the topological invariant for the model Hamiltonian [Eq. (16)] using the approach discussed in Ref. Kaw19 . We find that nontrivial invariants appear exactly in the same gapped regions where our hidden Chern number is non-zero. Therefore, we conclude that the hidden Chern number gives an alternative perspective on topological invariants and bulk-boundary correspondence of NH chiral-symmetric Hamiltonians.

In summary, we have shown that one-dimensional non-Hermitian chiral-symmetric models support a hidden Chern number as a topological invariant. The Chern number determines the number of end states with zero real part of the energy and the end states are immune to non-Hermitian skin effect. Moreover, we have calculated the topological phase diagram and end states for a minimal $4\times 4$ gain and loss model that supports a nontrivial topological phase. Our approach gives a new perspective on the topological invariants and bulk-boundary correspondence of non-Hermitian systems and the idea can be generalized to various dimensions and symmetries. Although there exists several proposals for realizing topological invariants of Hermitian systems in lower dimensional non-Hermitian systems Zen19 ; Vishwanath19 , the idea to utilize the imaginary part of the energy as a new dimension has not been explored so far. We point out that also gapless phases can support hidden Chern numbers, and in particular the gapless phase of the non-Hermitian Hamiltonian (16) carries hidden Chern number $C=-1$ in the vicinity of topologically nontrivial gapped phase – leading to localized topological end states, see Appendix D. The end states can be utilized for example in laser modes which have topologically protected frequency stability.

Appendix A Proof of equivalence (3)

Assume that there exist a Hermitian unitary operator ${\cal S}$ such that

[TABLE]

Then we can write

[TABLE]

where

[TABLE]

Now $H_{k}$ is Hermitian because

[TABLE]

On the other hand, if we assume that

[TABLE]

and ${\cal S}$ is Hermitian unitary and $H_{k}$ is Hermitian, then ${\cal S}{\cal H}_{k}{\cal S}=-{\cal H}_{k}^{\dagger}$.

Appendix B Real $2\times 2$ case

A real traceless $2\times 2$ Hamiltonian has a form of

[TABLE]

where $a_{k}$, $b_{k}$ and $c_{k}$ are real functions. By a unitary transformation ${\cal U}=\exp\left[i\frac{\pi}{3\sqrt{3}}(\sigma_{x}+\sigma_{y}+\sigma_{z})\right]$, we get

[TABLE]

The block structure of ${\cal H}^{\prime}_{k}$ is obviously the one of Eq. (4) of the main text and we find that $\sigma_{z}{\cal H}^{\prime}_{k}\sigma_{z}=-\left({\cal H}^{\prime}_{k}\right)^{\dagger}$.

Appendix C Phase diagrams

To establish a clear link between phase diagrams shown in Figs. 2 and 3 in the main text we present in Fig. 6 three dimensional phase diagrams cut by the planes of $g_{3}=-g_{1}$ and $g_{3}=-g_{1}+0.2t$. Curves along which they cut the branches of the phase diagram are the boundaries of the phases shown in Fig. 3 of the main text.

Appendix D Gapless phases

Gapless phases shown in Fig. 2 of the main text also carry hidden Chern numbers because the Chern number can change only in a band touching point. In a gapless phase close to $C=-1$ (or $C=0$) gapped phase we can still define a Chern number which takes the same value. The corresponding spectra for the Hermitian and non-Hermitian Hamiltonians with open boundary conditions are shown in Figs. 7 and 8. There is a continuum of bulk states crossing zero-energy level which is consistent with real part of ${\cal H}{}_{k}$ being gapless. Moreover, in the non-trivial case of we additionally see boundary states [see Fig. 7(a) and 8]. The end states are localized and the imaginary parts of the energies are in the gaps of the bulk spectrum [see 8(b)].