Anomalous edge state in a non-Hermitian lattice

Anomalous edge state in a non-Hermitian lattice

Abstract

We show that the bulk-boundary correspondence for topological insulators can be modified in the presence of non-Hermiticity. We consider a one-dimensional tight-binding model with gain and loss as well as long-range hopping. The system is described by a non-Hermitian Hamiltonian that encircles an exceptional point in momentum space. The winding number has a fractional value of 1/2. There is only one dynamically stable zero-energy edge state due to the defectiveness of the Hamiltonian. This edge state is robust to disorder due to protection by a chiral symmetry. We also discuss experimental realization with arrays of coupled resonator optical waveguides.

Introduction.— The bulk-boundary correspondence is a central principle that governs the band structure of tight-binding models Ryu and Hatsugai (2002); Asbóth et al. (2016). It says that the bulk of a lattice is characterized by a topological invariant, whose value determines the existence of gapless states that are localized on the edges of the sample. The bulk-boundary correspondence applies universally to all noninteracting tight-binding models, which are usually assumed to be Hermitian, i.e., closed systems. As such, it has been used to predict edge states in a variety of settings, including solid state Hasan and Kane (2010); Qi and Zhang (2011), cold atoms Aidelsburger et al. (2013); Miyake et al. (2013), photonics Kraus et al. (2012); Rechtsman et al. (2013); Zeuner et al. (2015), and even acoustics Yang et al. (2015); Süsstrunk and Huber (2015).

In a one-dimensional tight-binding model, the topological invariant is the winding number, which is always an integer Ryu and Hatsugai (2002); Asbóth et al. (2016). If the winding number is 0, there are no edge states. If the winding number is , there will be a pair of zero-energy edge states (one on the left and one on the right). As the Hamiltonian is modified, the edge states are “topologically protected” because the winding number changes only when the gap closes.

The bulk-boundary correspondence was developed for Hermitian systems, as motivated by solid state experiments. However, experiments with photonics are intrinsically non-Hermitian due to gain and loss, which raises the question of whether they can display physics beyond the bulk-boundary correspondence. The general conclusion so far is that the usual bulk-boundary correspondence still holds in the presence of non-Hermiticity, although the spectrum may be complex Rudner and Levitov (2009); Hu and Hughes (2011); Esaki et al. (2011); Diehl et al. (2011); Zhu et al. (2014); Malzard et al. (2015); Joglekar (); San-Jose et al. (2016).

In this Letter, we show that the bulk-boundary correspondence is modified in the presence of non-Hermiticity. We consider a one-dimensional tight-binding model with gain and loss, motivated by recent experiments with optical waveguides Guo et al. (2009); Rüter et al. (2010); Kraus et al. (2012); Rechtsman et al. (2013); Regensburger et al. (2012); Zeuner et al. (2015); Zhen et al. (2015). First, we show that the winding number can have a fractional value of 1/2, because the Brillouin zone is periodic when circling an exceptional point (a non-Hermitian degeneracy) Heiss (2012). An open chain exhibits a zero-energy eigenvalue, which is peculiar because it is defective Golub and van Loan (1996). Although there are two edge states, only one of them is dynamically stable. This edge state is protected by a chiral symmetry and is robust to disorder until the band gap closes. These new features are due to the fact that a non-Hermitian matrix can be defective, while a Hermitian matrix is always diagonalizable Golub and van Loan (1996). We also discuss experimental implementation with optical resonators and waveguides.

Figure 1: (a) In momentum space, the Hamiltonian encircles an exceptional point at . (b) The equivalent tight-binding model with gain on sublattice and loss on sublattice . The arrows denote the phase direction.

Model.— To motivate our model, we first consider a simple non-Hermitian Hamiltonian,

(1)

where are Pauli matrices. The eigenvalues are degenerate when . Such non-Hermitian degeneracies are called exceptional points Heiss (2012). We vary the parameters so as to encircle an exceptional point Milburn et al. (2015):

(2)

where is an external parameter for now. As increases, we make a circular trajectory in space [Fig. 1(a)]. When , encircles the exceptional point at . Without loss of generality, we assume that is positive, while can take any sign.

Figure 2: Chain with periodic boundaries, unit cells, and . (a) Real and (b) imaginary parts of the spectrum. (c) Trajectory of eigenvector in the Brillouin zone for when encircles 0 exceptional points (EPs) (blue, ), 1 exceptional point (red, ), and 2 exceptional points (black, ). Solid lines denote . Dashed line denotes .

Now, we find the tight-binding model whose momentum-space Hamiltonian is . The tight-binding model is given by a non-Hermitian Hamiltonian . As shown in Fig. 1(b), is a one-dimensional lattice with long-range hopping as well as gain and loss. To implement , the physical system we have in mind is a classical system of optical resonators with complex amplitudes that obey the equations of motion

for , where is the number of unit cells, so there are a total of sites. Equation (LABEL:eq:H) is mathematically equivalent to evolving a Schrödinger equation with Guo et al. (2009); Rüter et al. (2010); Regensburger et al. (2012); Zeuner et al. (2015); Zhen et al. (2015). is the non-Hermitian gain and loss on the and sublattices, respectively. denote the Hermitian hopping between sites.

Equation (LABEL:eq:H) is valid when the light fields are weak, such that the dynamics are linear. For strong light fields, the gain medium saturates, and the dynamics become nonlinear Dutta and Wang (2013). We are interested in the linear regime, when the system is equivalent to a tight-binding model. Note that an experiment would also have noise [not shown in Eq. (LABEL:eq:H)].

is similar to Hofstader’s model of a particle moving on a discrete lattice in a magnetic field, since the particle accumulates a phase of when it goes around a plaquette Hofstadter (1976). Later on, we discuss how to implement Eq. (LABEL:eq:H) experimentally in an array of coupled resonator optical waveguides Hafezi et al. (2011).

has two important symmetries. The first is chiral symmetry: letting , . So if has an eigenvector with eigenvalue , then is also an eigenvector with eigenvalue . There is also parity-time () symmetry: letting and , . This means that the eigenvalues of can be real, despite the non-Hermiticity Bender and Boettcher (1998).

Periodic boundary conditions.— For a periodic chain, the Hamiltonian is diagonalized in momentum space as , given in Eqs. (1)–(2), where . The eigenvalues of are

(4)

The real part of is gapped when . The imaginary part is gapped when . Figure 2 shows an example spectrum. The eigenvectors of are

(5)

where .

Figure 3: Chain with open boundaries, unit cells, and . (a) Real and (b) imaginary parts of the spectrum. Red lines follow the eigenvalue. (c) Zero-energy eigenvector for different values of .

The eigenvalues and eigenvectors of are periodic in when encircles one exceptional point Heiss (2012); Uzdin et al. (2011). This is a well-known feature of non-Hermitian systems and is due to the square root in Eq. (4) and the half angles in Eq. (5). After circling once around an exceptional point, the two eigenvectors exchange values; to come back to the initial value, one must circle twice.

Now, we calculate the winding number of the eigenvectors. We follow the trajectory of for an eigenvector as sweeps through the Brillouin zone and see whether it winds around the origin Rudner and Levitov (2009). As seen in Fig. 2(c), the winding number depends on whether encircles an exceptional point. When does not encircle an exceptional point, the winding number is 0. When encircles one exceptional point, the eigenvector does wind around the origin, but must sweep through in order to close the trajectory; thus, the winding number has a fractional value of 1/2 Viyuela et al. (). When encircles both exceptional points, the winding number is 1. (These results can be proven analytically.)

To experimentally observe the fractional winding number, one can modulate the hopping amplitudes in time to adiabatically sweep through the Brillouin zone (see Supplemental Material for details). An alternative approach may be to use Bloch oscillations Longhi (2009).

Since a periodic chain has nontrivial topology, we next investigate whether there are zero-energy edge states in a chain with open boundaries. However, there is a problem: Eq. (4) says that when one exceptional point is encircled, is gapless in both real and imaginary parts. This is worrisome because it precludes the existence of zero-energy edge states, which usually require a band gap.

Open boundary conditions.— Figure 3 shows the spectrum for an open chain. There are several remarkable features of this spectrum:

  • A gap opens up in the spectrum’s real part in the vicinity of , dividing most of the eigenvalues into two distinct bands [Fig. 3(a)].

  • Within the band gap, there is an eigenvalue, which is twofold degenerate but defective Golub and van Loan (1996). This eigenvalue is associated with an eigenvector and a generalized eigenvector. Under time evolution, the eigenvector dominates over the generalized eigenvector, so the latter is unimportant to the long-time dynamics.

  • The eigenvector for is localized either on the left edge (when ) or the right edge (when ) [see Fig. 3(c)]. The edge state is protected by chiral symmetry, and it appears when the gap opens and disappears when the gap closes.

  • For , the spectrum is purely real [Fig. 3(b)], i.e., symmetry is preserved, in contrast to a periodic chain Bender and Boettcher (1998).

Open chain: case of .— We discuss, in detail, the case of , where can be solved exactly. We seek the eigenvalues of , as well as their algebraic and geometric multiplicities Golub and van Loan (1996). It is more convenient to deal with , which is block upper triangular. It is easy to show that the characteristic polynomial of is , which implies that has eigenvalues with algebraic multiplicities , respectively. The Jordan normal form indicates that the geometric multiplicities of all three eigenvalues are 1. Thus, is highly defective at . Note that the eigenvalues are real.

The eigenvector for is the edge state

(6)

in the basis . So is localized on the left-most unit cell. It is its own chiral partner: . Physically, this state has zero energy because of destructive interference between the hopping and non-Hermiticity.

The generalized eigenvector for is given by Golub and van Loan (1996):

(7)

Since , population in is transferred to during the time evolution. Note that is also localized on the left edge.

For , can be similarly solved: and are similar to Eqs. (6)–(7) but localized on the right.

Open chain: case of .— As deviates from , the bands are no longer degenerate, and the band gap narrows and eventually closes. There is still a defective eigenvalue because it is protected by chiral symmetry [Fig. 3(c)]. However, when the band gap closes, the eigenvalue splits into two distinct eigenvalues that join the upper and lower bands [Fig. 3(a)].

Strictly speaking, for finite , the eigenvalue is defective only when . However, we find numerically that for a range of around , has one vanishingly small singular value Golub and van Loan (1996), which decreases as increases (see Supplemental Material). This indicates that for large , the eigenvalue remains defective for a range of .

In the Supplemental Material, we show that the eigenvalue is robust to disorder due to protection by chiral symmetry. The eigenvalue remains at until the disorder is strong enough to close the band gap.

Discussion.— The general time-dependent solution to Eq. (LABEL:eq:H) involves the eigenvectors and generalized eigenvector of . Consider the regime when the eigenvalue exists and is defective. In this case, the coefficient of increases linearly in time, while the coefficient of is independent of time Boyce and DiPrima (2012). Thus, as time increases, the population in the subspace is dominated by .

In a typical topological insulator, there are a pair of eigenvectors (one on the left and one on the right), and they are equally important to the dynamics Asbóth et al. (2016). In our non-Hermitian model, due to the defectiveness of the eigenvalue, it has one eigenvector and one generalized eigenvector (both on the same edge). However, only the eigenvector is present in the long-time-scale dynamics. Thus, the generalized eigenvector is dynamically unstable, and there is only one dynamically stable edge state. We note that in the Hermitian limit (), our model behaves entirely like a typical topological insulator.

Our non-Hermitian model is highly sensitive to boundary conditions. A periodic chain has a complex spectrum and a nonzero winding number, while an open chain can have a real spectrum but has no winding number. An open chain also has a band gap not present in a periodic chain. These differences are because an open chain is much more defective than a periodic chain.

We note that the Su-Schrieffer-Heeger model also has a single zero-energy state when the number of sites is odd Sirker et al. (2014). Because of the incompleteness of a unit cell, this state exists even when the gap closes. This state is not defective because the algebraic and geometric multiplicities are both 1. In contrast, our model’s zero-energy state is defective and disappears when the gap closes. Our zero-energy state originates from the non-Hermiticity instead of the peculiarity of an incomplete unit cell.

Experimental implementation.— Equation (LABEL:eq:H) can be implemented with an array of coupled resonator optical waveguides similar to Ref. Hafezi et al. (2011). In this setup, each site is a resonator, and waveguides between resonators allow photons to hop between sites. One would design an array of resonators with waveguide connections as in Fig. 1(b). The phase of a hopping amplitude can be tuned by making the corresponding waveguide asymmetric; in this way, one can engineer the imaginary hopping amplitudes in . To avoid cross talk between the diagonal waveguides that cross each other, one diagonal should be below the other.

Figure 4: Open chain of unit cells, starting with excitation on , (a,b) when zero-energy state is present () and (c,d) when it is absent (). (a,c) Population on each unit cell over time, plotted using color scale. (b,d) Fourier spectrum of .

In principle, the gain on the sublattice can be obtained by incorporating a pumped gain medium as in Ref. Rüter et al. (2010). In practice, it is easier to let both sublattices be lossy but with more loss on the sublattice Zeuner et al. (2015); Guo et al. (2009); such a passive setup avoids the complication of the gain medium. The physics is still described by Eq. (LABEL:eq:H) but on top of a background of decay.

When , has real eigenvalues, so the evolution is oscillatory (Fig. 4). It is easy to detect the state by exciting the left edge and measuring the frequency spectrum of the subsequent evolution. When the state is present, the spectrum has a peak at zero frequency [Fig. 4(b)]. When the state is absent, there is no peak at zero frequency [Fig. 4(d)].

Conclusion.— We have shown that non-Hermiticity breaks the usual bulk-boundary scenario. The new features are due to the fact that a non-Hermitian matrix can be defective, while a Hermitian matrix is always diagonalizable. In the future, it would be interesting to consider transport properties in the presence of a potential gradient Longhi (2009) or scattering properties, especially when the Brillouin zone is periodic. One should also extend the model to two dimensions to see whether the Chern number can be fractional and whether there is still only one dynamically stable edge state.

We thank P. Rabl and Y.N. Joglekar for useful discussions.

Supplemental Material

Appendix A Robustness to disorder

We now show that the zero-energy edge state is robust to disorder that preserves chiral symmetry. The eigenvalue remains in the spectrum until the disorder is large enough that the band gap closes. We allow to be different for each unit cell:

(8)

still has chiral symmetry: where . We note that the in the first line of Eq. (8) can be different from the in the second line and still preserve chiral symmetry.

Figure 5(a,b) shows an example spectrum with disorder in : , where is a uniform random variable on , and is the disorder strength. The eigenvalue is completely robust to disorder in this case, even when the band gap closes; its eigenvector remains localized on the left-most unit cell. The bands are affected by disorder, although all eigenvalues stay real.

Figure 5(c,d) shows an example spectrum with disorder in : . The eigenvalue is robust to disorder until the real gap narrows at , at which point it splits into two imaginary eigenvalues. Those eigenvalues undergo exceptional points with other eigenvalues at .

Figure 5(e,f) shows an example spectrum with disorder in : . The eigenvalue is robust to disorder until the real gap closes at , at which point the eigenvalue has an exceptional point with other eigenvalues.

Thus, the eigenvalue is robust to disorder as long as chiral symmetry is preserved and the gap does not close. (If the disorder did not preserve chiral symmetry, the eigenvalue would immediately deviate from for small .)

Figure 5: Open chain with (a,b) disorder in , (c,d) disorder in , and (e,f) disorder in . (a,c,e) Real and (b,d,f) imaginary parts of the spectrum. Red lines follow the eigenvalue. Parameters are , , , .

Appendix B Experimental detection of fractional winding number

The most direct way to see the periodicity of the eigenvectors is to modulate the hopping amplitudes in time. We multiply the hopping amplitudes by a phase factor :

(9)

This phase modulation can be implemented in an optical experiment like Ref. Hafezi et al. (2011) by adding a Kerr medium to one branch of the waveguides, allowing one to tune the index of refraction using an external source of light. In this way, the phase of a hopping amplitude can be changed in situ.

In momentum space, Eq. (9) becomes:

(10)
(11)

For a given , increasing effectively sweeps through the Brillouin zone. By changing adiabatically, one can directly observe the periodicity of the eigenvectors. Equations (10) and (11) have been studied in Refs. Milburn et al. (2015); Uzdin et al. (2011).

Figure 6: Singular value decomposition for an open chain with and various . (a) Smallest and (b) second smallest singular values. Numerical precision here is .

For example, suppose the system is initialized in the eigenstate with . Suppose also that , so one exceptional point is encircled. Starting from , we increase in time. When , the system does not return to the original eigenstate, but it will be in . In contrast, if the exceptional point is not encircled, then the system will return to the original eigenstate. In this way, one can clearly see the periodicity due to the exceptional point.

We note that the switching of the eigenvectors can depend on whether the trajectory around the exceptional point is clockwise or counter-clockwise Milburn et al. (2015); Uzdin et al. (2011).

Appendix C Range of defectiveness

Here, we provide numerical evidence that Eq. (3) of the main text has a defective eigenvalue for a range of . Figure 6 shows the smallest and second smallest singular values of . For a range of around , there is one vanishingly small singular value, and it decreases as increases. Thus, for large , the geometric multiplicity of the eigenvalue is 1 for a range of . Since its algebraic multiplicity is 2, the eigenvalue is defective in this range.

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