Topological phases and edge states in a non-Hermitian trimerized optical lattice

Topologically engineered optical materials support robust light transport. Herein, the investigated non-Hermitian lattice is trimerized and inhomogeneously coupled using uniform intracell coupling. The topological properties of the coupled waveguide lattice are evaluated, the PT-symmetric phase of a PT-symmetric lattice can have different topologies; the edge states depend on the lattice size, boundary configuration, and competition between the coupling and degree of non-Hermiticity. The topologically nontrivial region extends in the presence of periodic gain and loss. The nonzero geometric phases accumulated by the Bloch bands indicate the existence of topologically protected edge states between the band gaps. The unidirectional amplification and attenuation zero modes appear above a threshold degree of non-Hermiticity, which facilitate the development of a robust optical diode.


I. INTRODUCTION
Topological insulators are novel states of solid-state materials that have an insulating bulk band gap and a conducting edge or surface [1]. The edge or surface states are symmetry-protected against local disorder, and valuable for quantum transport and computation in quantum (anomalous) Hall insulators and quantum spin Hall insulators [2]. The optical realization of topological systems has stimulated the field of topological photonics, which enables the experimental studies of topological systems that are difficult to realize in condensed matter physics. The photons in coupled waveguides and optical lattices are manipulated in a manner similar to the electrons in solids, providing intriguing opportunities for novel optical devices [3][4][5]. The topologically protected unidirectional interface state propagates robustly against local impurities. This was experimentally demonstrated in coupled waveguide ring resonators [6]. Synthesizing artificial gauge fields for ultracold atoms in optical lattices enables the construction of a two-dimensional topological system [7]. The Su-Schrieffer-Heeger (SSH) model [8] is the simplest system that has topologically nontrivial edge states [9][10][11][12][13][14][15][16][17][18][19][20]; this model has been realized for a dimerized optical superlattice, wherein the topological properties of Bloch bands were experimentally measured [21].
Because extensive progress has been made in topological photonics and PT -symmetric optics, optical analysis of topological systems has been extended to non-Hermitian systems [58][59][60][61]. The topological interface states in non-Hermitian systems have been systematically discussed [62]. Topologically protected PTsymmetric interface states were demonstrated in coupled resonators [63]. Robust light interface states were discovered at the interface between SSH chains with distinct non-Hermiticity [64]. Topological properties are characterized using the generalized winding number in non-Hermitian systems [65]. PT -symmetric non-Hermitian Aubry-André systems [66,67] and Kitaev models [68,69] were theoretically investigated. Evidence reveals that universal non-Hermiticity may alter topological regions [70], but topological properties are robust against local non-Hermiticity.
In this paper, we investigate a coupled waveguide lattice that has a trimerized unit cell and in which gain and loss are balanced and separated by a passive waveguide. The lattice has universal non-Hermiticity. The intracell coupling, g 1 , is uniform and differs from the intercell coupling, g 2 . The nontrivial topology of the lattice implies that it has a conducting edge or surface. Under the periodical boundary condition, the geometric phases accumulated are π when the upper and lower bands circle one loop in the Brillouin zone in the topologically nontrivial region; corresponding edge states exist between the band gaps under an open boundary condition. In PT -symmetric configurations, the PT -symmetric regions possess different topologies depending on the coupling configuration and universal non-Hermiticity. In particular, we discover a non-Hermitian threshold above which there exist unidirectional amplified and damped edge states, which are located at opposite boundaries with identical resonant frequency. The topologically nontrivial regions with edge states are widened because of the non-Hermitian periodic gain and loss. The threshold corresponds to a bifurcation point in the imaginary part of the spectrum. The topological region expands when the lattice has substantial non-Hermiticity.
The reminder of the paper is organized as follows. In Sec. II, we discuss the topological properties of a Hermitian trimerized lattice. In Secs. III and IV, we investigate the lattice's PT -symmetric non-Hermitian extension based on coupled waveguides. The band structure, PT -symmetric phases, and edge states are investigated. The results are summarized in Sec. V.

II. THE TRIMERIZED LATTICE
We focus on the topological properties of a trimerized lattice, which has more edge states than a dimerized SSH chain [53][54][55]. The topologically protected edge states are related to the lattice structure and determined by the boundary [71,72]. The trimerized lattice investigated herein comprises three groups, which are indicated by lattice number N = 3n, 3n − 1, and 3n − 2; when considering the coupling configurations, the lattice is divided into three groups by their inhomogeneous couplings at the boundary: g 1 -g 1 -g 2 -, g 2 -g 1 -g 1 -, and g 1 -g 2g 1 -. The lattice has only six topologically distinct configurations because of its reflection symmetry. These configurations are illustrated above their spectra presented in Fig. 1, where each site represents a waveguide. The three configurations in Fig. 1(a)-1(c) are PT -symmetric, whereas those in Fig. 1(d)-1(f) are not. The differences in the configuration of the trimerized lattice result in more edge states in the topologically nontrivial phase.
To investigate the topological properties of the trimerized lattice, we set the intracell coupling g 1 to unity and force the intercell coupling to change periodically as The energy bands are plotted as a function of θ in Fig. 1.
The single line indicates the intracell coupling g 1 , and the double lines indicate the intercell coupling g 2 . Edge states with energies ±1 exist between the upper and lower band gaps and are symmetrically arranged about zero energy. In Fig. 1(a), four edge states (two-fold degenerate) are present in the regions −π < θ < −π/2 and π/2 < θ < π. The four edge states are displayed in Fig. 2 Fig. 1(b), the couplings at the left and right chain boundaries are the intercell coupling g 2 , and there is no edge state in any region of θ. In Fig. 1(c), four edge states (two-fold degenerate) exist in the region −π/2 < θ < π/2. In Fig. 1(d), there are two edge states localized on the left boundary in region −π/2 < θ < π/2, as shown in Fig. 2(b), but no edge states are localized on the right boundary. In Fig. 1(e), there are also two edge states, in the region −π/2 < θ < π/2, and they are both localized on the right boundary [ Fig. 2 (c)]. In other regions (−π < θ < −π/2 and π/2 < θ < π), the edge states are localized on the left boundary [ Fig. 2(b)]. In Fig. 1(f), two edge states exist in the regions −π < θ < −π/2 and π/2 < θ < π, localized on the right boundary. The existence of edge states is related to the lattice configuration. Edge states exist when the coupling g 1 is felt at the lattice boundary. The configuration with g 1g 2 -g 1 -at the boundary results in two edge states of energy ±1 in the region −π/2 < θ < π/2, whereas the configuration with g 1 -g 1 -g 2 -results in two edge states of energy ±1 in the regions −π < θ < −π/2 and π/2 < θ < π. When the lattice has reflection symmetry, there are four edge states (two-fold degenerate). The edge states disappear in the configuration with g 2 -g 1 -g 1 -at the boundary. The edges states appear at three cases. The edge states plotted in Fig. 2(a) are for the lattice with reflection symmetry [ Fig. 1(a)]. Two-fold degenerate edge states are located at the left and right boundaries, respectively. In Fig. 2(b) and 2(c), the edge states for the configuration presented in Fig. 1(e) are plotted; in this situation, one pair of edge states with energy ±1 is located on one boundary. The edge state amplitude is approximately {1, ±1, 0} for every three sites from the boundary. The amplitude decays according to (g 1 /g 2 ) p in the regions −π < θ < −π/2 and π/2 < θ < π and according to (g 2 /g 1 ) p in the region −π/2 < θ < π/2 from the lattice boundary (where p is the unit cell index). Inside the unit cell, the nonzero amplitudes of edge states are symmetric (upper panel in Fig. 2) and antisymmetric (lower panel in Fig. 2). For edge state with energy +1, the corresponding amplitude has overall phase difference e iπ between neighbor unit cells; for the edge state with energy −1, the amplitude has no such phase difference as illustrated in Fig. 2.

III. PT -SYMMETRIC NON-HERMITIAN LATTICE
In this section, we investigate a one-dimensional trimerized lattice of evanescently coupled optical waveguides. The lattice has three sublattices A, B, and C, and this spatial arrangement induces inhomogeneous couplings. The spacing between waveguides A and B is equal to that between B and C in a unit cell, which differs from the spacing between waveguides A and its nearest neighbor C. This spatial arrangement induces a periodic modulation in every the third coupling. The amplitudes of the waveguides in a unit cell are denoted ψ m,A , ψ m,B , and ψ m,C , where m labels the unit cell index. In coupled mode theory, the single-mode coupled waveguide lattice is modeled by a tight-binding system as follows: where g 1 is the intracell coupling and g 2 is the intercell coupling. The waveguides losses are denoted by γ A , γ B , and γ C for sublattices A, B, and C, respectively. Under the condition γ B − γ A = γ C − γ B ≡ γ, the system reduces to a PT -symmetric lattice with on-site po- tentials {iγ, 0, −iγ} in a three-site unit cell after the removal of a common loss rate γ B . The trimerized lattice chain is described by a PT -symmetric off-diagonal AAH Hamiltonian with non-Hermitian on-site potentials, i.e., The Hamiltonian H can be rewritten in a matrix form, examples of the SSH models are listed in Ref. [55]. Figure 3(a) schematically plots the PT -symmetric lattice under the periodical boundary condition. For N = 3n, take the Fourier transformation of Bloch waves, the waveguide lattice matrix is expressed as H = k H k , where the wave vector k = 2πm/n (m = 1, 2, ..., n) and Notably, H k is PT -symmetric [PT H k (PT ) −1 = H k ] and includes three energy bands. P is defined as the parity operator and satisfies PA k P −1 = C k , PB k P −1 = B k and PC k P −1 = A k (where A k , B k and C k are the corresponding sublattices in Bloch wavevector space). T is defined as the time reversal operator, which satisfies T iT −1 = − i. The band gaps are closed at the boundaries of the Brillouin zone: k = 0 and π. PT transition occurs at (γ 2 −2g 2 1 −g 2 2 ) 3 /3 3 +(2g 2 1 g 2 cos k) 2 /2 2 = 0. The phase diagram is depicted in Fig. 3(b). Region II (white) represents the region in which PT -symmetry is broken. Regions I and III are the regions of exact PT -symmetry. Inhomogenity is necessary for the existence of an exact PT symmetric phase and monotonously increases with the degree of non-Hermiticity.
The lattice topology is identified by the Zak phases of the energy bands [73,74], which is defined as θ = i π −π dk ψ R,k |dψ L,k /dk in a non-Hermitian system, where ψ L,k and ψ R,k are the left and right eigenstates of H k [15,64]. The geometric phases accumulated are π, 0, π for g 1 < g 2 when each Bloch band circles one loop in the Brillouin zone and are 0, 0, 0 for g 1 > g 2 . Bulkedge correspondence means that edge states exist in the band gaps under an open boundary condition. The exact PT -symmetric regions I and III have topologically distinct phases. In the PT -symmetric region I, the geometric phases accumulated for the three bands are all zero, which implies that region I is a topologically trivial phase; in the PT -symmetric region III, the geometric phases accumulated are 0 for the middle band and π for the upper and lower bands, which reflects that region III is a topologically nontrivial phase, and edge states exist simultaneously between the two band gaps under an open boundary condition similar to that shown in Fig. 1. As intercell coupling g 2 is varied from 0 to 2g 1 , the system transitions from an exact PT -symmetric phase with trivial topology, through the broken PT -symmetric phase, to an exact PT -symmetric phase with nontrivial topology.

IV. EDGE STATES
The existence of topologically protected edge state is an important feature of topological systems. In Fig. 4, the real and imaginary parts of the spectra for N = 3n, 3n − 1, and 3n − 2 at different γ are plotted. The lattice is the most sensitive to PT -symmetric gain and loss at g 1 = g 2 (θ = ±π/2). The broken energy levels are raised as γ is increased and the band gaps narrow. Edge states emerge in the bands but their frequency and amplifying or damping rate is independent of the intercell coupling g 2 for a lattice without defects at the boundary (N = 3n). For N = 3n − 1, the topology is not affected by weak non-Hermiticity, and there are no edge states [ Fig. 4(c)]. However, an edge state exists at large non-Hermiticity when γ > g 2 [ Fig. 4(d)], and its frequency is resonant with that of the waveguide; the amplifying (damping) rate is g 2 dependent and increases with γ. We calculate the edge states' resonant frequency and estimate the threshold gain and loss rate as follows.
Three PT -symmetric configurations are illustrated in Fig. 1(a)-1(c). Topologically protected edge states exist in the configurations in Fig. 1(a) and 1(c) in the absence of gain and loss when the system is Hermitian. Notably, the system topology is robust against the non-Hermiticity. In the presence of gain and loss, the Hamiltonian still commutes with the PT operator. Consequently, the eigenvalues become real or conjugate pairs. The edge state on one side switches to the other side of the lattice chain after the PT operation; therefore, the edge state is not the eigenstate of the PT operator and PT symmetry breaks. The edge states appear in conjugate pairs, which are robust amplification or attenuation modes. In the topological non-Hermitian optical lattice, the coexistence of topology and non-Hermiticity enables the lattice to function as a robust optical diode.
The Schrödinger equations for the edge states localized on the right side satisfy ψ N −2 = 0, Eψ N −1 = g 1 ψ N , and (E + iγ) ψ N = g 1 ψ N −1 , from which we obtain the edge states eigenvalues which are damped modes because the −iγ/2 term in E −,± . Indeed, the damping is because that the edge state probability is mainly distributed on the lossy sites. The amplitude of the edge states decays as (g 1 /g 2 ) p from the unit cell on the right boundary. The wave function amplitude is {0, σ +,± , 1} in each unit cell, where σ +,± = (iγ ± 4g 2 1 − γ 2 )/(2g 1 ) indicates the wave function distribution. The edge states can be expressed as ψ r = [0, (g 1 /g 2 ) m−1 σ +,± , (g 1 /g 2 ) m−1 , · · · , 0, σ +,± , 1] T , (7) for system N = 3m. For edge states localized on the left side, the eigenvalues are These two edge states are amplified modes. The wave function distribution is   Fig. 2(a) for γ = 0. When the gain and loss are introduced, the two-fold degenerate edge states in each band gap become one amplified and one damped pair, with the edge state damping or amplifying rate given by γ. The frequencies of the edge states are reduced from ±g 1 to ± g 2 1 − γ 2 /4 in the upper and lower band gaps separated by 4g 2 1 − γ 2 . The first ± in the subscript of E ±,± indicates that the edge states are damped or amplified; the second ± in the subscript of E ±,± indicates that the edge states are in the upper or lower band gap.
(11) The decay factor is g 2 /g 1 . E ±,+ (E ±,− ) corresponds to the edge states in the upper (lower) band gap. When g 2 is on the boundary, two edge states do not exist. If the lattice has a different boundary of g 1 -g 2 -g 1 -and g 1 -g 1g 2 -on the edge, the system has either right or left edge states, except when g 1 = g 2 .
Three types of PT -symmetric configurations are schematically illustrated in the PT -symmetric lattice in Fig. 5. In all configurations, the repeated unit cells are identical; the lattice numbers are distinct. The balanced gain and loss in the unit cell are {−iγ, 0, iγ}. In Fig. 5(b) and 5(c), the unit cells are defective at the edges. This leads to the existence of different edge states in the corresponding configurations. In particular, edge states exist in conjugate pairs as the degree of non-Hermiticity increases, and the real parts of their eigenvalues are zero.
The broken PT -symmetric edge states with real eigenvalues equal to zero have been demonstrated in honeycomb and square lattices [65]. In the configuration illustrated in Fig. 1(b), the chain number is N = 3n−1 and no edge state exists in the Hermitian system (at γ = 0). Affected by the universal gain and loss, the edge states exist when γ > g 2 . The Schrödinger equations for the edge state wave functions satisfy ψ N −2 = 0, (E + iγ) ψ N −1 = g 2 ψ N , and (E − iγ) ψ N = g 2 ψ N −1 . The eigenvalues of the edge states are E = ±i γ 2 − g 2 2 , implying that the large non-Hermiticity induces a pair of amplified and damped edge states with their frequency shifted to the waveguide resonant frequency. Correspondingly, the amplified edge state with energy E = i γ 2 − g 2 2 is this state is located on the right boundary with decay factor ρ − = −i(γ − γ 2 − g 2 2 )/g 2 . Conversely, the damped edge state with energy E = −i γ 2 − g 2 2 is this state is located on the left boundary with decay factor ρ + = +i(γ − γ 2 − g 2 2 )/g 2 . The amplitude of the unit cell decays as ρ p ± from the lattice boundary. The edge states are localized at the lattice boundary and decay exponentially; therefore, we analytically obtain the confinements for the three configurations as the system size approaches infinity (N → ∞). The detailed calculations of the edge states are presented in the Appendix. For a system constituted by integer unit cells (N = 3n), the edge states exist when g 2 > g 1 or the gain and loss satisfy γ > γ c,e = g 2 + g 2 1 /g 2 , when g 2 g 1 . The lattice with N = 3n − 1 has two defects in the unit cell at each boundary. Edge states exist when the gain and loss rates are larger than the coupling g 2 , γ > γ c,e = g 2 , and for a system with N = 3n − 2, which is a lattice with one missing site at each boundary of the unit cell, the edge states exist when g 1 > g 2 or the gain and loss satisfy γ > γ c,e = g 2 + g 2 1 /g 2 , when g 1 g 2 . The threshold γ c,e is depicted in Fig. 5 for the three PT -symmetric configurations. As shown in Fig. 5(a) and 5(c), γ c,e for finite systems of n = 30 (red circles) is slightly larger than that predicted for infinite systems (black lines), whereas γ c,e for a finite system of n = 300 (blue squares) approaches that predicted for infinite systems. Notably, γ c,e in Fig. 5(b) is system size independent; in this case, there is one pair of eigenstates E 2 = g 2 2 − γ 2 and the amplitudes at the passive sites (without gain or loss) vanish. Therefore, the wave functions are zero for every other two sites. The eigenvalues are real and the eigenstates are PT -symmetric when γ g 2 ; however, when γ > g 2 , this pair of eigenstates become a pair of edge states with purely imaginary eigenvalues. The edge states in the three PT -symmetric configurations for g 1 = 1, g 2 = 1.5, and γ = 2.5 are plotted in Fig. 6. The edge state located on the left boundary in Fig. 6(a) is the amplified mode, which is located on the right boundary in Fig. 6(b) and 6(c). These edge states are related to the edge state probability and gain (loss) distribution.
The spectra of PT -symmetric configurations are plotted as a function of γ for g 1 = 1 and g 2 = 1/2 and 3/2 in Fig. 7. As γ is increased, the real parts of the spectra narrow, the band gap widths are reduced and amplification (attenuation) zero mode edge states exist when the real parts are decreased to zero at γ c,e , which is where the edge state frequencies are resonant with the waveguide lattice and their imaginary parts bifurcate. The thresholds γ c,e in Fig. 7 are consistent with the red circles in Fig. 5. In Fig. 7(a), g 1 > g 2 , and thus the threshold is dependent on the intercell coupling g 2 , being γ c,e = 2.655; in Fig. 7(b), g 1 < g 2 , the threshold is 2g 1 , being γ c,e = 2.0. Both of these cases are for a lattice size of N = 3n = 90. The spectra presented in Fig. 7(c) and 7(d) are those for a lattice size N = 3n − 1 = 89, where the threshold equals g 2 , being γ c,e = 0.5 and 1.5, respectively. For a lattice size N = 3n − 1, the band gaps vanish when g 1 = g 2 . The bulk states at the top and bottom of the middle band become edge states when γ > γ c,e and g 1 > g 2 ; the edge states are converted from the bulk states at the bottom of the top band and the top of the bottom band when γ > γ c,e and g 1 < g 2 . Figure 7(e) and 7(f) displays the spectra obtained for a lattice size N = 3n − 2 = 88 when γ c,e = 2.0 and γ c,e = 2.226, respectively. In Fig. 7(b) and 7(e), the edge states exist at any γ, have broken PT symmetry, and have eigenvalues that linearly depend on γ until bifurcation occurs at γ = 2.0. Thereafter, zero mode edge states exist, and the amplification and attenuation of edge states depend on (γ, g 1 ) under a square root function, as indicated by E ±,± . In Fig. 7(a), 7(c), 7(d), 7(f), no edge state exists when γ < γ c,e , and a pair of bulk states becomes a conjugate pair of edge states when γ > γ c,e . This pair of edge states is localized on opposite edges with identi-cal frequency and an equal amount of amplification and attenuation.

V. SUMMARY
Topologically nontrivial non-Hermitian trimerized optical lattice is investigated. The PT -symmetric phases, band structure, and topologically protected edge states of a trimerized coupled waveguide lattice with universal non-Hermiticity are demonstrated. This lattice possesses topologies that are dependent on the degree of non-Hermiticity. In the topologically nontrivial region, the edge states are related to the configurations at the lattice boundary and depend on the coupling strengths. Two conjugate edge state pairs exist in the PT -symmetric configurations. These edge states are symmetrically arranged about the energy zero. Above a gain (loss) threshold, zero mode edge states exist, and these states are amplified and damped when propagating in opposite directions. Asymmetric transport through edge states in the photonic lattice represents a diode effect. The amplified edge state can be excited by dynamical creation, which is insensitive to the initial excitation or lattice imperfections. This robust one-way behavior has potential applications in optical manipulation, information processing, and unidirectional lasing.

ACKNOWLEDGMENTS
We acknowledge the support of NSFC (Grant No. 11605094) and the Tianjin Natural Science Foundation (Grant No. 16JCYBJC40800).

Appendix: Solution of the edge states
In the Appendix, we analyse the appearance of edge states in the three PT -symmetric configurations.