Edge states and topological invariants of non-Hermitian systems
For Hermitian systems, the creation or annihilation of topological edge modes is accompanied by the gap closing of Bloch Hamiltonian; for non-Hermitian systems, however, the edge-state transition points can differ from the gap closing points of Bloch Hamiltonian, which indicates breakdown of the usual bulk-boundary correspondence. We study this intriguing phenomenon via exactly solving a prototype model, namely the one-dimensional non-Hermitian Su-Schrieffer-Heeger model. The solution shows that the usual Bloch waves give way to eigenstates localized at the ends of an open chain, and the Bloch Hamiltonian is not the appropriate bulk side of bulk-boundary correspondence. It is shown that the standard Brillouin zone (a unit circle for one-dimensional systems) is replaced by a deformed one (a non-unit circle for the solved model), in which topological invariants can be precisely defined, embodying an unconventional bulk-boundary correspondence. This topological invariant correctly predicts the edge-state transition points and the number of topological edge modes. The theory is of general interest to topological aspects of non-Hermitian systems.
Introduction.–Topological materials are characterized by robust boundary states immune to perturbationsHasan and Kane (2010); Qi and Zhang (2011); Chiu et al. (2016); Bernevig and Hughes (2013); Bansil et al. (2016). According to the principle of bulk-boundary correspondence, the existence of boundary states is dictated by the bulk topological invariants, which, in the band-theory framework, are defined in terms of the Bloch Hamiltonian. The Hamiltonian is often assumed to be Hermitian. In many physical systems, however, non-Hermitian HamiltoniansBender and Boettcher (1998); Bender (2007) are more appropriate. For example, they are widely used in describing open systemsRotter (2009); Malzard et al. (2015); Carmichael (1993); Zhen et al. (2015); Diehl et al. (2011); Cao and Wiersig (2015); Choi et al. (2010); San-Jose et al. (2016); Lee and Chan (2014); Lee et al. (2014), wave systems with gain and lossMakris et al. (2008); Longhi (2009); Klaiman et al. (2008); Regensburger et al. (2012); Bittner et al. (2012); Rüter et al. (2010); Lin et al. (2011); Feng et al. (2013); Guo et al. (2009); Liertzer et al. (2012); Peng et al. (2014); Fleury et al. (2015); Chang et al. (2014); Hodaei et al. (2017, 2014); Feng et al. (2014); Gao et al. (2015); Xu et al. (2016); Ashida et al. (2017); Kawabata et al. (2017); Chen et al. (2017); Ding et al. (2016) (e.g. photonic and acoustic Ozawa et al. (2018); Lu et al. (2014); El-Ganainy et al. (2018); Longhi (2018)), and solid-state systems where electron-electron interactions or disorders introduce a non-Hermitian self energy into the effective Hamiltonian of quasiparticleKozii and Fu (2017); Papaj et al. (2018); Shen and Fu (2018). With these physical motivations, there have recently been growing efforts, both theoreticallyShen et al. (2017); Rudner and Levitov (2009); Esaki et al. (2011); Leykam et al. (2017); Lee (2016); Hu and Hughes (2011); Gong et al. (2017); Gong and Wang (2010); Rudner et al. (2016); Liang and Huang (2013); Kawabata et al. (2018); Ni et al. (2018); Zyuzin and Zyuzin (2018); Cerjan et al. (2018); Zhou et al. (2017a); González and Molina (2017); Klett et al. (2017, 2018); Menke and Hirschmann (2017); Yuce (2016, 2015); Xu et al. (2017); Hu et al. (2017); Li et al. (2017); Wang et al. (2015); Ke et al. (2017); Martinez Alvarez et al. (2017); Rivolta et al. (2017); Gong et al. (2018) and experimentallyZeuner et al. (2015); Zhan et al. (2017); Xiao et al. (2017); Weimann et al. (2017); Parto et al. (2017); Zhou et al. (2017b), to investigate topological phenomena of non-Hermitian Hamiltonians.
Among the key issues is the fate of bulk-boundary correspondence in non-Hermitian systems. Recently, numerical results indicate that the usual bulk-boundary correspondence may break downLee (2016); Xiong (2017). In particular, we observe a puzzling phenomenon in the data of Ref.Lee (2016) that the transition point, which divides phases with and without topological edge modes, does not seem to coincide with jump of any bulk topological invariant. Instead, the spectrum gap of Bloch Hamiltonian does not close at the transition point. The questions we address below are: How to determine the transition point? Is there a generalized bulk-boundary correspondence? What is the topological invariant responsible for the topological edge states?
We start from solving a one-dimensional(1D) lattice model with non-Hermiticity. Interestingly, the analytic solution shows that all the bulk eigenstates of an open chain become localized near the boundary (“non-Hermtian skin effect”), in sharp contrast to the extended Bloch waves in Hermitian cases. We will show that such a “non-Hermitian skin effect” is closely related to the breakdown of the usual bulk-boundary correspondence.
In essence, previous topological invariantsRudner and Levitov (2009); Esaki et al. (2011); Leykam et al. (2017); Shen et al. (2017); Lieu (2018) are formulated in terms of the Bloch Hamiltonian defined in the standard Brillouin zone, which, for 1D systems, is the unit circle parameterized by () on the complex plane. As we will explain, it is replaced by a “deformed Brillouin zone”, which is found to be a non-unit circle in the present model [Fig.3(b)]. In view of this non-Bloch-wave nature of bulk states, we introduce an unconventional topological invariant defined in the deformed Brillouin zone, which faithfully determines the topological edge modes. It embodies the unusual bulk-boundary correspondence of non-Hermitian systems.
Model.–The lattice model is pictorially shown in Fig.1. It is one of the non-Hermitian versions of Su-Schrieffer-Heeger(SSH) modelSu et al. (1980)111Related models has been studied for various purposes in Ref.Zhu et al. (2014); Yin et al. (2018); Lieu (2018)., which are relevant to quite a few experimentsWeimann et al. (2017); Zeuner et al. (2015); Poli et al. (2015). The Bloch Hamiltonian is222Compared to Ref. Lee (2016), a basis change is taken, bringing the physical interpretation closer to SSH.
where , , and are the Pauli matrices. The model has a chiral symmetryChiu et al. (2016) , which ensures that the eigenvalues appear in pairs: . The energy gap closes at the exceptional points , which requires () or ().
As found in Ref.Lee (2016), the open-boundary spectrum is fundamentally different from that of periodic boundary. We solve the real-space Hamiltonian for an open chain, taking as a parameter [Fig.2]. Zero modes exist for an interval of , and they are robust to perturbation [Fig.2(d)], which indicates their topological origin. A zero-mode transition point is located at (, as shown below), which is a unremarkable point viewed from whose spectrum is gapped there (). Although not explicitly mentioned333It was presumably due to insufficient numerical precision of Ref.Lee (2016). According to our analytic and improved numerical calculations, the zero-mode line in their Fig. 3(a) should span the entire interval, instead of two disconnected ones presented there. , this phenomenon are already appreciable in the spectrum in Ref. Lee (2016).
To gain insights, let us analytically solve an open chain. The wavefunction is written as . The real-space eigen-equation leads to and in the bulk of chain. We take the ansatz that , where each takes the exponential form (omitting the index temporarily)
Therefore, we have
which has two solutions, namely , where corresponds to . In the limit, we have
They can also be seen from Eq.(3). These two solutions correspond to and , respectively.
Restoring the index in , we have
which are equivalent because of Eq.(4). The general solution is written as a linear combination:
which should satisfy the boundary condition
Together with Eq.(6), they lead to
in which we have defined . We are concerned about the spectrum for a long chain, which necessitates for the bulk states. If not, suppose that , we would be able to discard the tiny term in Eq.(9), and the equation becomes essentially independent of . Eq.(4) tells us that , which leads to (via )
for bulk states. As an illustration, we plot the - curve for in Fig.3(a). As seen from Fig.2, the spectrum is real for this set of parameter, therefore, no imaginary part of is needed (the reality of spectrum in certain parameter intervals is due to the PT symmetryBender and Boettcher (1998); Bender (2007)). There is indeed an energy interval with (line in Fig.3(a))). As is increased from , moves towards left, and finally hits the axis, which apparently satisfies . Inserting Eq.(5) into this equation, we have
where the first two solutions are relevant for the parameters used in this paper (with ). At these locations of , the bulk spectrum touches zero energy, and transitions occur. This explains the numerical puzzle in Fig.2.
We emphasize that indicates that all the bulk eigenstates are localized at the left end of the chain. This phenomenon is dubbed the “non-Hermitian skin effect” (see Fig.4 for illustration). To see the bulk spectrum, let us parameterize the circle by (), then Eq.(4) leads to
which apparently recovers the spectrum of SSH model when .
Before proceeding, we comment on a standard method of finding zero modes, which has often been used in literature of non-Hermitian systems. Let us focus on our model for concreteness. Consider a semi-infinite system with a left end. One can see that appears as a zero-energy eigenstate. The normalizable condition is then imposed in the standard approach. Consequently, the transition points satisfy . However, it would predict , instead of , as a transition point. In contrast, our solution of SSH model shows that the zero mode merges into the bulk spectrum at
which correctly produces . This represents an unconventional bulk-boundary correspondence.
Shortcut.–Now we introduce a shortcut solution from hindsight (The longer solution is presented first because it is easier to generalize, e.g., to include longer-range hoppings). Let us take a similarity transformation of the real-space Hamiltonian: and
where is a diagonal matrix whose elements are , with given by Eq.(10). Now becomes a Hermitian SSH model for . In space:
with . The transition points via are identical to Eq.(11). Apparently, the case can also be solved this way, though is no longer Hermitian.
Non-Bloch topological invariant.–The curve reflects the non-Bloch nature of the bulk states of open chains. Notably, it remains a closed loop and can be intuitively viewed as a deformation of the standard Brillouin zone (unit circle). Departing from the usual framework, we define a topological invariant on as follows. First, a “non-Bloch” Hamiltonian is obtained from by the replacement :
where . One then find the right and left eigenvectors by
Chiral symmetry ensures that and are also right(left) eigenvectors, with eigenvalues and . To obtain these eigenvectors, one can diagonalize the matrix as , then each column of and is a right and left eigenvector, respectively. As a generalization of the usual “ matrix”Chiu et al. (2016), we define
which, due to the chiral symmetry , is off-diagonal: . Now we introduce a winding number
which differs essentially from the Hermitian cases in that is a general curve [e.g. Fig.3(b)] instead of the unit circle. It is useful to mention that the conventional formulations using may sometimes produce correct topological numbers, when happens to be a unit circle, even if the model is non-Hermitian444For example, it is the case for the model numerically studied in Ref. Lieu (2018)..
The numerical results for our model in Eq.(1) is shown in Fig.5, which is consistent with both the numerical and analytical spectrum obtained above. Quantitatively, counts the total number of robust zero modes at the left and right ends. For example, corresponding to Fig.2, there are two zero modes for , and none elsewhere. The analytic solution shows that, for , both modes live at the left end; for , one for each end; and for , both at the right end. Thus, the -gap closing points (called “half vortices” in Ref.Leykam et al. (2017)) are where zero modes migrate from one end to the other, conserving the total mode number. In fact, one can see at , indicating penetration into the bulk.
To provide a more generic example, we take , where introduces hopping between a pair separated by another pair. Now the curve is no longer a circle, yet correctly predicts the total zero-mode number [Fig.6].
Finally, we remarked that Eq.(19) can be generalized to multi-band systems. Each pair of bands (labeled by ) correspond to a curve, and the matrix [Eq.(18)] becomes , each one defining a winding number . The topological invariant is .
Conclusion.–Through the analytic solution of non-Hermitian SSH model (which itself will be quite useful as a benchmark for theories), we explained why the usual bulk-boundary correspondence breaks down, and how to generalize it to non-Hermitian systems in a precise manner. The bulk states lose the Bloch-wave nature and become localized near the boundary, which necessitates an unconventional correspondence not manifested in . We formulate this generalized correspondence by introducing a precise topological invariant. The physics presented is quite general and closely relevant to a vast variety of non-Hermitian systems, which will be left for future investigations.
Acknowledgements.–This work is supported by NSFC under Grant No. 11674189.
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