Topology of anti-parity-time-symmetric non-Hermitian Su-Schrieffer-Heeger model

We propose an anti-parity-time (anti-PT ) symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model, where the large non-Hermiticity constructively creates nontrivial topology and greatly expands the topological phase. In the anti-PT -symmetric SSH model, the gain and loss are alternatively arranged in pairs under the inversion symmetry. The appearance of degenerate point at the center of the Brillouin zone determines the topological phase transition, while the exceptional points unaffect the band topology. The large non-Hermiticity leads to unbalanced wavefunction distribution in the broken anti-PT -symmetric phase and induces the nontrivial topology. Our findings can be verified through introducing dissipations in every another two sites of the standard SSH model even in its trivial phase, where the nontrivial topology is solely induced by the dissipations.

The anti-PT symmetry can also protect the validity of BBC. In this work, we propose an anti-PT -symmetric non-Hermitian SSH model through alternatively incorporating the balanced gain and loss under the inversion symmetry in the standard SSH model. The band spectrum becomes partially complex in the presence of non-Hermiticity, indicating the thresholdless anti-PT symmetry breaking. The gain and loss help creating the nontrivial topology. The topological characterization and the geometric picture of the topological phases are elaborated. The topological phase transition occurs when the band gap closes and reopens. The degenerated topological edge states have zero-energy with net gain and localized at two lattice boundaries, respectively. Exciting the edge states enable topological lasing [117][118][119][120][121][122][123].
Model.-The schematic of the non-Hermitian SSH model is shown in Fig. 1(a), which describes a onedimensional coupled resonator array. All the resonators have identical resonant frequency. The staggered distance between the nearest neighbor resonators determines the lattice couplings t 1 and t 2 [115][116][117][118][119], which classify two sublattices in the SSH model where a † j (b † j ) and a j (b j ) are the creation and annihilation operators for the sublattice site indexed j. To create the anti-PT symmetry [124][125][126][127][128][129], the gain and loss are introduced in the resonators under the inversion symmetry in the form of {iγ, −iγ, −iγ, iγ} in the four-site unit cell (1) The Hamiltonian of the anti-PT -symmetric non-Hermitian SSH model reads As shown in Fig. 1(a), the anti-PT -symmetric non-Hermitian SSH model is invariant under a π rotation of the left (right) non-Hermitian dimer and glide half of the unit cell in the translational direction [130].
In comparison to the PT -symmetric non-Hermitian SSH model with the gain and loss {iγ, −iγ, iγ, −iγ} in the four-site unit cell, the only difference between the anti-PT -symmetric and PT -symmetric SSH models is that the arrangement of the even number of gain and loss pairs; the anti-PT -symmetric SSH model has the inversion symmetry, while the PT -symmetric SSH model does not. Interestingly, the role played by the non-Hermiticity γ is completely different in these two models. The non-Hermiticity γ in the anti-PT -symmetric SSH model constructively create nontrivial topology. The topologically trivial phase changes into the topologically nontrivial phase as the increasing of non-Hermiticity γ. The nontrivial topology of anti-PT -symmetric SSH model can be directly verified in many experimental platforms that used to demonstrate the PT -symmetric SSH model. The topological aspect of the anti-PT -symmetric SSH model completely differs from that of the nonsymmorphic RM model [131].
Applying the Fourier transformation, the Bloch Hamil-tonian of the lattice is obtained as where σ 0 and σ x,y,z are the two-by-two identical matrix and Pauli matrix. The system belongs to the BDI † class in the 38-fold topological classifications of non-Hermitian systems and the topological phase transition of the BDI † class is determined by the closure of the band gap of the real part of energy bands [2]. The interplay between the couplings and the non-Hermiticity alters the band topology and generates the nontrivial topology; furthermore, the loss can solely induce the nontrivial topology if we consider a common gain term iγ is removed from H k . This greatly simplifies the verification of the anti-PTsymmetric SSH model in experiments.
In contrast to the PT symmetry ensures the energy levels to be conjugate in pairs, the anti-PT symmetry ensures the energy levels in pairs with identical imaginary part and opposite real part. The four energy bands are In the absence of the gain and loss (γ = 0), the lattice is the Hermitian SSH model. At the topological phase transition point t 1 = t 2 , two bands ±2t 1 cos (k/2) of the SSH model are connected at the degenerate point (DP) k = ±π; the four-band spectrum E ±,± of H k can be regarded as the spectrum of the SSH model folded at k = ±π/2 and stretched to the entire Brillouin zone (BZ). Thus, the central band gap closes at the DP at the center of the BZ k = 0 and the band folding generates another DP at the edge of the BZ k = ±π. At t 1 = t 2 , the SSH model is gapped and the central gap is open as shown in Fig. 1(b); however, the spectrum of H k still has a DP at the edge of BZ protected by the nonsymmorphic symmetry in the four-site unit cell of the SSH model [130].
In the presence of the gain and loss (γ = 0), the non-Hermiticity splits the edge DP into two exceptional points (EPs) associated with the anti-PT symmetry breaking [ Fig. 1(c)]. As the increase of the non-Hermiticity, the two EPs gradually move and the complex energy region expands from the edge to the center of the BZ as shown in Figs. 1(d) and 1(e). When γ 2 = t 2 1 , two EPs merge to one EP at the center of the BZ [ Fig. 1(f)] and disappear for γ 2 > t 2 1 [ Fig. 1(g)]. The band gap of the central two bands closes at E = 0 as presented in Figs. 1(d), which requires γ 2 + t 2 2 − t 2 1 2 + 4t 2 1 t 2 2 sin 2 (k/2) = 0. The central two bands touch at the DP k = 0 associated with the topological phase transition at the critical non-Hermiticity Phase diagram.-The anti-PT -symmetric non-Hermitian SSH model reduces to the SSH model for γ = 0, which has the topologically nontrivial phase for t 2 2 > t 2 1 and the topologically trivial phase for t 2 2 < t 2 1 . However, the situation changes in the presence of non-Hermiticity as shown in the phase diagram Fig. 2(a). The non-Hermiticity creates the nontrivial topology and the topological region expands in the anti-PT -symmetric non-Hermitian SSH model, where γ 2 + t 2 2 < t 2 1 is the topologically trivial phase and γ 2 + t 2 2 > t 2 1 is the topologically nontrivial phase. The nontrivial topology of the anti-PT -symmetric SSH model can be solely created by the non-Hermiticity because large non-Hermiticity induces unbalanced distributions of the wavefunction probability. The non-Hermiticity generates nontrivial topology in the uniform chain at t 2 1 = t 2 2 [132] and even in the trivial phase of the Hermitian SSH model at t 2 2 < t 2 1 . The real part and imaginary part of the energy bands under OBC as a function of the non-Hermiticity are shown in Figs. 2(b) and 2(c), respectively. The anti-PT -symmetric non-Hermitian SSH lattice under OBC has one pair of edge states in the topologically nontrivial phase. To further elucidate the band structure, the energy bands in the complex energy plane for the non-Hermitian SSH lattice under OBC are plotted as shown in Figs. 3(a)-3(f); the corresponding PBC spectra are shown in Figs. 1(b)-1(g). For γ 2 t 2 1 and γ 2 + t 2 2 = t 2 1 , the real part of energy bands is gapped (the central gap is open). However, the energy bands E 1 and E 2 (E 3 and E 4 ) are inseparable [4] because of the existence of edge DP or EPs (blue crosses) as shown in Figs. 1(b)-1(c) and 1(e)-1(f); in this sense, the four energy bands can be regarded as two energy bands E r (cyan) and E l (magenta) according to the real part of energy bands as shown in Figs. 3(a)-3(b) and 3(d)-3(e). For γ 2 + t 2 2 = t 2 1 (topological phase transition), the real gap is closed (the central gap is closed) and the two energy bands E r and E l become single band [ Fig. 3(c)]. For γ 2 > t 2 1 , the EP disappears between E 1 and E 2 (E 3 and E 4 ) and the four where λ = 3 α + √ β with α = (18t 2 2 + 8γ 2 − 9t 2 1 )γ and β = 27{ 4(2t 2 2 + 4γ 2 − 5t 2 1 )t 2 2 − t 4 1 γ 2 + t 2 1 + t 2 2 3 }. The left edge state localizes at the left boundary of the lattice. Without loss of generality, the wave functions in the j-th unit cell of the left edge state |ψ L can be expressed as χ j−1 {1, (ε − iγ) /t 1 , −χt 2 /t 1 , 0} with χ = (iγ − ε)/ (iγ + ε). For γ = 0, the edge states reduce to zero modes with ε = 0 and the wave functions in the jth unit cell as (t 1 /t 2 ) j−1 {1, 0, −t 1 /t 2 , 0}. The right edge state |ψ R is the mirror reflection of the left edge state |ψ L . The edge states have net gain rate and are useful for topological lasing.
Geometric picture of band topology.-The topology of the anti-PT -symmetric non-Hermitian SSH model relates to the geometry of the Bloch Hamiltonian winding around the degeneracy points in a two dimensional parameter space, we show how the non-Hermiticity creates the nontrivial topology. For the Bloch Hamiltonian Eq.
(3), we replace e ik by h x +ih y to create a two dimensional parameter space (h x , h y ) where the k-dependent Bloch Hamiltonian Eq. (3) corresponds to a unit circle h 2 x +h 2 y = 1 in the two dimensional parameter space (h x , h y ). Figure 4 depicts the complex energy bands extended to the two dimensional parameter space (h x , h y ) for the anti-PT -symmetric non-Hermitian SSH model at fixed parameters t 1 = 1 and t 2 = 1/2 for different non-Hermiticity γ. The edge DP (blue cross) on the unit circle at γ = 0 splits into two EPs on the unit circle at nonzero non-Hermiticity γ < 1; and the two EPs are symmetrically distributed about h y = 0 because the EPs are symmetrically distributed about k = 0 in the BZ [see Figs. 1(c)-1(e)]. As the non-Hermiticity increases, the two EPs move on the unit circle from (h x , h y ) = (−1, 0) to (h x , h y ) = (1, 0); at γ = 1, the two EPs merge into single EP at (h x , h y ) = (1, 0); and the EP vanishes for γ > 1. The edge DP or the EPs remain on the unit circle, the topology is fully determined by the central DP (black cross). From the unit circle winding around the central DP, we can observe how the nontrivial topology is created by the non-Hermiticity γ. For γ = 0 in Fig. 4(a), the central DP is outside the unit circle; thus, the system is in the topologically trivial phase. The nonzero non-Hermiticity γ moves the central DP along h y = 0 towards the negative h x direction in the parameter space. For γ = 1/2 in Fig. 4(b), the central DP moves to (h x , h y ) = (3.284, 0), and the system enters the white region of the phase diagram as shown in Fig. 2(a). Fig. 4(c), the central DP moves to (h x , h y ) = (1, 0) and locates on the unit circle; the system is at the boundary of the white and cyan regions. For γ = 9/10 in Fig. 4(d), the central DP is enclosed in the unit circle, the topology of the system changes and the system enters the cyan region. For γ = 1 in Fig. 4(e), the central DP keeps inside the unit circle; the system is in the nontrivial phase for the nonzero t 2 , but the energy bands are not completely separated. For γ = 3/2 in Fig.  4(f), the central DP is still inside the unit circle; in this situation, the four bands are completely separated and the system is in the nontrivial phase.
Zak phase and partial global Zak phase.-When the four complex energy bands are separated at γ 2 > t 2 1 , each energy band is associated with a Zak phase In the definition of Θ n , |ϕ n is the left eigenstate and |ψ n is the right eigenstate, H k |ψ n = E n |ψ n and H † k |ϕ n = E * n |ϕ n , where the subscript n is the band index. E 1 denotes the band with positive real and imaginary energy, E 2 denotes the band with positive real and negative imaginary energy, E 3 denotes the band with negative real and imaginary energy, and E 4 denotes the band with negative real and positive imaginary energy as shown in Figs. 3(f) and Fig. 1(g). Their wavefunctions are |ψ 1 , |ψ 2 , |ψ 3 , and |ψ 4 , respectively. The system has the inversion symmetry, which ensures that the Zak phase for each separated energy band is an integer of π. Thus, the Zak phase is used for topological characterization. In Fig. 5(c), the Zak phases for the bands E 1 and E 4 are π and the Zak phases for the bands E 2 and E 3 are 0; which are consistent with the geometric picture in Fig.  4(f), the central DP belongs to energy bands E 1 and E 4 , and predicts the existence of one pair of the topological zero modes with gain for the system under OBC.
For the energy bands embedded with EPs (γ 2 t 2 1 and γ 2 + t 2 2 = t 2 1 ), there are only two energy bands E r and E l . The two-state coalescence EP2 12 exists only in the energy bands E 1 and E 2 , and the two-state coalescence EP2 34 exists only in the energy bands E 3 and E 4 . In this sense, we define two partial global Zak phase In the calculation of the partial global Zak phase, the momentum ranges [k EP −∆k, k EP +∆k] are removed because that the coalesced wavefunctions are self-orthogonal at the EPs [132], where ∆k is an infinite small positive real number. The partial global Zak phase is valid for the topological characterization.
For γ 2 t 2 1 and γ 2 + t 2 2 < t 2 1 , both the partial global Zak phase Θ r and Θ l are 0 as shown in Fig. 5(a). This indicates the phase is topologically trivial without any edge state under OBC. However, for γ 2 t 2 1 and γ 2 + t 2 2 > t 2 1 , both the partial global Zak phase Θ r and Θ l are π as shown in Fig. 5(b). This indicates the phase is topologically nontrivial and one pair of topological edge states appear under OBC.
Conclusion.-We propose the anti-PT -symmetric non-Hermitian SSH model as a prototypical anti-PTsymmetric topological lattice. The gain and loss are alternatively introduced in pairs in the standard SSH model through holding the inversion symmetry. The inversion symmetric gain and loss result in the thresholdless breaking of anti-PT symmetry and the energy spectrum is partially or fully complex. We provide novel insights on the roles played by the anti-PT -symmetry and non-Hermiticity in the topological phases. The large non-Hermiticity constructively creates the nontrivial topology and greatly expands the topologically nontrivial region of the SSH model. The topological edge states localized at two boundaries of the lattice are degenerate and suitable for topological lasing. Besides, the dissipation can solely induce the nontrivial topology. In comparison to the PT -symmetric non-Hermitian SSH model, only the arrangement of gain and loss in the anti-PT -symmetric non-Hermitian SSH model is different; the proposed anti-PT -symmetric non-Hermitian SSH model can be easily implemented in the microring resonator arrays, coupled optical waveguides, photonic crystals, electronic circuits, and acoustic lattices .