Su-Schrieffer-Heeger chain with one pair of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯-symmetric defects

The topologically nontrivial edge states induce \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯 transition in Su-Schrieffer-Heeger (SSH) chain with one pair of gain and loss at boundaries. In this study, we investigated a pair of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯-symmetric defects located inside the SSH chain, in particular, the defects locations are at the chain centre. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯 symmetry breaking of the bound states leads to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯 transition, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯-symmetric phases and the localized states were studied. In the broken \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯-symmetric phase, all energy levels break simultaneously in topologically trivial phase; however, two edge states in topologically nontrivial phase are free from the influence of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯-symmetric defects. We discovered \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯-symmetric bound states induced by the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯-symmetric local defects at the SSH chain centre. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯-symmetric bound states significantly increase the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{P}\mathcal{T}$$\end{document}𝒫𝒯 transition threshold and coalesce to the topologically protected zero mode with vanishing probabilities on every other site of the left-half chain and the right-half chain, respectively.

Introduction. The parity-time (PT ) symmetric non-Hermitian Hamiltonians can possess real spectra [1][2][3][4][5][6][7][8] but nonunitary dynamics, such as faster evolution [9,10] and power oscillation [11]. The PT system experiences a phase transition when its spectrum changes between real and complex. The PT transition point is the exceptional point associated with eigenstates coalescence [8]. In onedimensional system, the exceptional point varies as system size and structure [12,13]. The critical gain/loss rate approximately equals to the coupling strength of a uniform chain for a pair of PT -symmetric gain and loss defects at boundary [14]. For defects locations at center, the critical gain/loss rate is the coupling strength between the two defects [15]. Topological properties were extensively investigated in condensed matter physics [16][17][18][19][20][21][22][23][24][25][26][27] and in photonic systems [28][29][30]; furthermore, the PTsymmetric topological insulator was proposed in twodimensional (2D) coupled optical resonators. Different with traditional Hermitian topological insulator, the edge states are unidirectional amplified and damped [31]. Topological insulator states are PT symmetry breaking in a PT -symmetric non-Hermitian extension of Dirac Hamiltonians, because the PT operator switches the edge states locations at boundary [32]. The Chern number was generalized for non-Hermitian systems, the tightbinding model on the honeycomb and square lattices under different symmetry classes were examined, broken PT -symmetric edge states with real part eigen energies being zero were found [33]. The topologically chiral edge modes found in 2D non-Hermitian system were related to the exceptional point of the bulk Hamiltonian that characterized by two half-integer charges of the exceptional point [34].
The Su-Schrieffer-Heeger (SSH) chain [35] with a pair of PT -symmetric defects at boundary was studied, the edge states found in topologically nontrivial phase are * jinliang@nankai.edu.cn sensitive to the non-Hermiticity [36] and the critical non-Hermitian gain and loss approach zero [37]. The non-Hermitian Kitaev and extended Kitaev models were investigated similar as the SSH model [38,39]. Optical systems are fruitful platforms for the investigation of PT symmetry [40][41][42][43][44][45][46]. The robust light interface states were discovered at the interface of a combined two SSH chains with different quantum phases of PT symmetry [47]. Recently, non-Hermitian SSH chains were experimentally realized by coupled dielectric microwave resonators [48,49] and photonic lattices [50,51]. In passive SSH chains with periodical losses, single coupling disorder induces asymmetric topological zero mode [49] and PT -symmetric topological zero mode interface states, respectively [51]. PT symmetry breaking and topological properties were theoretically investigated in other PT -symmetric systems [52][53][54], the competition between two lattice defects can induce PT symmetry breaking and restoration as non-Hermiticity increasing in Aubry-André-Harper model [55].
In this work, we study an open SSH chain with one pair of PT -symmetric gain and loss. The PT -symmetric thresholds, the topologically nontrivial edge states, and the local defects induced PT -symmetric bound states are investigated. The PT -symmetry breaking is closely related to the appearance of localized states. When considering the defects located near the chain boundary, the edge states in topologically nontrivial region break the PT symmetry if the defects are at the sites with nonzero states distribution probabilities; otherwise, the edge states are free from the influence of the on-site defects and the PT symmetry phase transition is induced by the bulk states: The extended (bound) state induces the PT symmetry phase transition at weak (strong) non-Hermiticity for defects near the chain boundary (at the chain center).
The PT transition threshold is the largest when the defects located at the chain center, being the weak inhomogeneous couplings of the SSH chain. Two edge states and four bound states exist at large non-Hermiticity, the number of breaking energy levels increases as defects moving from the chain boundary to the center. For defects near the chain center, when PT transition happens, all energy levels break simultaneously in topologically trivial phase; by contrast, two topologically nontrivial edge states are not PT symmetry breaking although in broken PT symmetry phase. The PT transition is associated with the PT symmetry breaking of the eigenstate. The edge states and bound states probabilities localize around the PT -symmetric defects; therefore, they are PT symmetry breaking states except when the defects are the nearest neighbours. We discovered a pair of PT -symmetric bound states for the defects at the chain centre, the PT -symmetric bound states significantly increase the PT transition threshold, at which, the bound states coalesce to the topologically protected zero mode but their probabilities are not only confined to either the loss or the gain sublattice; the probabilities vanish on every other site of the left-half chain and the right-half chain, respectively.
PT -symmetric non-Hermitian SSH chain. In this section, we introduce a one dimensional N -site SSH chain with one pair of PT -symmetric imaginary defects, the system is schematically illustrated in Fig. 1. The couplings between neighbor sites are staggered 1 ± ∆ cos θ, which are modulated by parameter ∆. The coupled chain can be realized by optical resonators [56][57][58]. The defects pair includes a loss (in red) and a balanced gain (in green) [5,6,11,40,42,45]. We define P as the parity operator, which equals to a reflection symmetry with respect to the chain center, satisfying PjP −1 = N + 1 − j. The time-reversal operator satisfies T iT −1 = −i. Under these definitions, the balanced gain and loss as onsite defects pair satisfies the PT symmetry. The pri- is the creation (annihilation) operator on site j for fermionic particles. The chiral symmetry protects the topological properties of H 0 in topologically nontrivial region. In this work, we confine our discussion within system with even N , the non-Hermitian gain and loss defects are located at reflection symmetric positions m and N + 1 − m; the non-Hermitian extended SSH chain is Note that H satisfies the PT symmetry, and is expected to have purely real spectrum. The analysis and conclusions are applicable to corresponding bosonic particles [59]. Topologically nontrivial edge states disappear in system with universal non-Hermiticity [32,33], but remain in system with localized non-Hermiticity [36,53]. The topology is changed in the presence of universal non-Hermiticity, but is robust to several impurities although the impurities are non-Hermitian. The traditional Hermitian SSH model has inhomogeneous staggered hoppings between neighbor sites, the SSH Hamiltonian is a two-band model. Under periodical boundary condition, the Berry phases of the two bands can be calculated, both being π in topologically nontrivial phase when −π/2 < θ < π/2 and both being 0 in topologically trivial phase when −π θ −π/2 and π/2 θ π.
Under open boundary condition, the bulk-edge correspondence indicates the existence of edges states. For even N , two zero edge states appear in region −π/2 < θ < π/2; by contrast, for odd N , single edge state exists when θ = ±π/2 [59]. The edge states probabilities are localized on the chain boundary, thus edge states are immune to non-Hermitian on-site defects at the SSH chain center.
PT -symmetric phases. We first consider PTsymmetric SSH chain with even n as follows (Fig. 1a). The defects are at the chain center (m = n). The SSH chain with odd n has similar results (Fig. 1b). The Hamiltonian H is always in the exact PT -symmetric phase as θ varies when γ < 1 − ∆; the exact PTsymmetric region shrinks for 1 − ∆ < γ < 1 + ∆. At γ > 1 + ∆, H is in the broken PT -symmetric phase for arbitrary θ. In PT symmetry broken phase, all energy levels are PT symmetry breaking in topologically trivial region; however, two edge states are robust to gain and loss defects in topologically nontrivial region, and can be composed into a pair of PT -symmetric states. As an illustration, we numerically calculate the SSH chain spectra and depict them as a function of θ for N = 100 and ∆ = 1/2 in Fig. 2. The real and imaginary parts of the spectra are plotted in blue and red lines, respectively. Two topologically nontrivial edge states appear in the region −π/2 < θ < π/2. Figures 2a and 2b show the entirely real spectrum as a function of θ at γ = 1/4 and 1/2; however, the PT - Other system parameters are N = 100, ∆ = 1/2; the gain and loss are at sites 50 and 51.
symmetric SSH chain is nondiagonalizable for γ = 1/2 at θ = ±π, where the coupling inhomogeneity is at maximum. The Hamiltonian H become n 2 × 2 Jordan blocks after diagonalization, which indicates n pairs of two states coalescence. For γ > 1/2, the PT symmetry breaking appears at θ = ±π, the PT symmetry of all eigen states breaks simultaneously. In Fig. 2c, we depict the SSH chain spectrum for γ = 3/4. The exact PT -symmetric region is determined by γ = 1 + ∆ cos θ. Thus, the system is in exact PT -symmetric phase in region −2π/3 θ 2π/3; in other regions, all the N eigenvalues form n conjugate pairs. The eigen states with the largest imaginary part (we refer to the absolute values in the comparison) have highest probabilities localized near the chain center. The inverse participation ratio (sum of fourth power of the wave function amplitude j |ψ j | 4 ) for the extended state scales as system size in order of N −1 , but the IPR for the localized state approaches constant at large system size. As γ increasing, the probabilities are more localized and form bound states at approximately γ 1. The bound states are attributed to the PT -symmetric non-Hermitian impurities; and the bound states probabilities are localized near the chain center.
In Fig. 2d, γ = 1, exact PT -symmetric phase shrinks to −π/2 θ π/2. In broken PT -symmetric phase, the two red lines (two folders) with the maximum imaginary parts in regions −π θ −0.7π and 0.7π θ π correspond to four bound states. Notably, the bound states eigenvalues change more rapidly than the extended states and enter the extended states band around θ ≈ ±0.7π. As γ increasing to γ = 5/4 as shown in Fig. 2e, exact PTsymmetric phase shrinks to −π/3 θ π/3 (indicated by yellow line in the inset), where the SSH chain has two degenerate zero edge states and two bound states with real eigenvalues. The PT symmetry breaks out of region −π/3 θ π/3. In the regions −π/2 θ < −π/3 and π/3 < θ π/2, the SSH chain is in broken PTsymmetric phase with two topologically nontrivial edge states. There exist two bound states and two edge states, the two bound states have pure imaginary eigenvalues with the largest imaginary part, exiting the extended states band at θ ≈ ±π/2 and become real for |θ| π/3. Notably, the two edge states are also pure imaginary but with the smallest imaginary parts, approaching zero as |θ| close to 0. This is because weak inhomogeneity for |θ| close to π/2 induces more spreading of edges states; by contrast, strong inhomogeneity for |θ| close to 0 induces more localization of the edge states. In the numerical results, the imaginary parts of edge states are negligible in the region −0.4π θ 0.4π (indicated by green line in the inset), determined by the coupling inhomogeneity ∆. The two zero edge states are free from the influence of non-Hermitian defects at (or close to) the chain center. In the regions −π θ −π/2 and π/2 θ π, there are four bound states with the largest imaginary eigenvalues. In Fig. 2f, the spectrum for γ = 3/2 is plotted. The bound states always have the largest imagi-nary eigenvalues, the bound states are two folders out of region −π/2 < θ < π/2, and one folder in region −π/2 < θ < π/2. For γ > 3/2, the SSH chain spectrum is in the broken PT -symmetric phase at arbitrary θ. The only real energy states are the two topologically nontrivial edge states in the region −0.4π θ 0.4π (indicated by green line in the inset).
In Figs. 2g and 2f, 3/2 < γ < 2.96, the imaginary parts of the bound states experience a bifurcation in topological trivial regions −π θ −π/2 and π/2 θ π. After bifurcation, the eigenvalues of the bound states become pure imaginary, which is reflected from the real parts being zero. In the real part of the energy spectrum, the zero is four folders, and corresponds to the four bound states in the topologically trivial phase but corresponds to two bound states and two edge states in the topologically nontrivial phase. Notably, the four folders zero always exists in the real part of energy spectrum for γ > 2.96. The bound states have pure imaginary eigenvalues. The bifurcation behavior disappears in the imaginary parts of the bound states. In this situation, one pair of bound states have larger imaginary eigenvalues than the other pair at arbitrary θ.
In the situation of odd n (Fig. 1b), the coupling between the gain and loss is 1 − ∆ cos θ; by contrast, this coupling is 1 + ∆ cos θ for even n case. The spectrum structures are approximately the same as the even n case shown in Fig. 2, but shifted by π in parameter θ, but two zero edge states still exist in topologically nontrivial phase −π/2 < θ < π/2. In our discussion of odd n case, the coupling at the boundaries are unchanged as 1 − ∆ cos θ.
In the following, we discuss a general situation that the PT -symmetric defects are inside the SSH chain (0 < m < n) rather than at the SSH chain center. The configuration is illustrated in Figs. 1c and 1d. The topological properties are robust to the one pair of gain and loss defects; however, the PT -symmetric properties change significantly. The edge states with probabilities localized at the chain boundary are free from the influence of non-Hermitian defects when they are close to the chain center (m ∼ n) at strong coupling inhomogeneity. By contrast, for the gain and loss defects at the chain boundary (m = 1), the PT symmetry of the SSH chain is fragile to the non-Hermitian defects in the presence of topologically nontrivial edge states. This is because the probabilities of edge states are the highest at the chain boundary and decay exponentially. Thus, the influence of defects is the greatest for edge states; any small gain and loss rate breaks the PT symmetry of the SSH chain in the topologically nontrivial region −π/2 < θ < π/2; in topologically nontrivial region, the two edge states, forming a conjugation pair with pure imaginary eigenvalues, are the only PT symmetry breaking states [36].
As parameter θ varies, the number of breaking energy levels at maximum appears in the topologically trivial phase. The number of breaking energy levels at maximum is larger for larger m, which equals to 2m + 2 for odd m n − 2 in the topologically trivial phase but equals to 2m in the topologically nontrivial phase. By contrast, the number of breaking energy levels at maximum is 2m for m = n−1 and n in both topologically trivial and nontrivial phases. When m = n, all energy levels break simultaneously as shown in Fig. 2c-e. The number of breaking levels at maximum in topologically nontrivial region is two less than that in topologically trivial region for m ≤ n − 2 at odd m.
For N = 100, in topologically nontrivial phase of θ = 0, the coupling inhomogeneity is the strongest. The number of PT -symmetric breaking energy levels at maximum is 2m for even m < n − 2; which changes to 2m − 2 (2m) for odd 25 m < n−2 (m < 25). The two edges states have pure imaginary eigenvalue |E ES | < 10 −10 for m 25 along with defects moving from the chain boundary to the center, the edge states can be considered as unaffected and the eigenvalues are zero for odd m 25. The critical gain/loss rate is γ c ≈ 0.07 at m = 25. At weak non-Hermiticity (γ 1), bound states disappear and two pairs (four) of extended states break first with equal amount of energy imaginary parts. For n − 2 m n, the number of PT -symmetric breaking energy levels at maximum is 2m − 2. The topologically nontrivial zero edge states are real valued.
Edge states and bound states. The defects support localized modes, which can induce PT symmetry breaking [60]. In the SSH chain, the edge states break the PT symmetry if the gain and loss are at the sites with nonzero distribution probabilities in topologically nontrivial phase; otherwise, the edge states are free from the influence of gain and loss and the PT symmetry phase transition is induced by the bulk states, including the extended states induced PT symmetry phase transition at weak non-Hermiticity for defects near the chain boundary and the bound states induced PT symmetry phase transition at strong non-Hermiticity for defects at the chain center.
The probabilities of two edge states for γ = 0 are staggered decreasing from the chain boundary [59], the probability approaches zero for every other site. The probabilities of two edge states on site m and its P-symmetric position N + 1 − m are both zeros for even m. Thus, the edge states are unaffected (Figs. 3a and 3c). By contrast, the influence of defects pair on the edge states is remarkable for odd m, in particular, for the defects close to the chain boundary (Fig. 3e). The topologically nontrivial edge states are robust to non-Hermitian defects, the nonvanishing distribution probabilities at the chain boundary lead to PT symmetry breaking, and the energies of two edge states become conjugate imaginary pair for small gain and loss [37].
At large non-Hermiticity, six localized states at maximum are found, which include four bound states and two edge states as depicted in Fig. 3c. The number of localized states depends on the non-Hermiticity and the defects locations. For large enough gain and loss, all localized states eigenvalues have zero real parts. In topo- logically trivial phase, the localized states are four bound states; however, the number differs in topologically nontrivial phase. Figure 3 depicts the edge states and bound states probabilities for different defects locations. For m = 1, the defects at the chain boundary, two edge states are the only localized states; for 2 m n − 1, two edge states and four bound states are found; for m = n, there are four localized states, including two edge states and two bound states.
In Fig. 3a, the localized states in topologically nontrivial phase with gain and loss at the center are depicted. The localized states include one conjugate pair of bound states and two edge states. The eigenvalues are +3.6494i, −3.6494i, and two degenerate 0 from top to bottom. In Fig. 3b, the SSH chain is in topologically trivial phase, the eigenvalues of two pairs of bound states are +3.2995i, −3.2995i, +0.6062i, and −0.6062i from the top to the bottom.
In Fig. 3c, the eigenvalues of the four bound states are +3.2594i, −3.2594i, +0.6136i, and −0.6136i from top to middle. The other two states at bottom are the zero edge states. In Fig. 3d, the eigenvalues of four bound states are the same as those shown in Fig. 3c, but with probabilities distributions slightly different. This is because θ = 0 in Fig. 3c but θ = π in Fig. 3d, the two couplings 1 ± ∆ between each site and its nearest neighbors switch in the two structures. The bound states probabilities are localized near the defects and decay to zero at the chain boundary. Therefore, the SSH chain structures at the boundary are not important for the four bound states and they are approximately identical in the two cases of θ = 0 and θ = π. This conclusion is invalid when the gain and loss defects locations are close, where bound states probabilities decay from defects iγ and −iγ affected each other.
In case of defects at the chain boundary, the bound states disappear in topologically nontrivial phase. In Fig. 3e, the two edge states eigenvalues are +3.9445i, −3.9445i, and being ±iγ for n → ∞, both two edge states are fragile to impurities at the chain boundary. The staggered decay of edge states disappears when γ is large. In Fig. 3f, the eigenvalues are +3.3384i, −3.3384i, +0.5991i, and −0.5991i. The staggered decay of state probabilities is clearly seen for states with small imaginary eigenvalues (absolute values).
The bound states with positive (negative) imaginary eigenvalues are centered at the gain (loss) site. In the two pairs of bound states, the probabilities decay faster for the one with larger imaginary eigenvalues. The probabilities maxima of these two bound states are at the gain and loss sites. The other pair of bound states has smaller imaginary eigenvalues, which decay in a staggered way instead of monotonously and the decay is slower. The probabilities maxima of this pair of bound states are at the nearest neighbor site of the impurity, the site which has stronger coupling strength between the impurity and its neighbors. As shown in Figs. 3c and 3d, the stronger couplings are 1+∆ between 20 (81) and 21 (80) as shown in Fig. 3c and 19 (82) and 20 (81) as shown in Fig. 3d. The pair of bound states with smaller positive (negative) imaginary parts are localized at 21 (80) and 19 (82), respectively. By contrast, for the gain and loss at the chain center, bound states with smaller imaginary eigenvalues vanish (Fig. 3a). The situations are different for defects at the boundary (m = 1) and the center (m = 50), the dimerized unit with stronger couplings is incomplete at FIG. 4. The numerically calculated γc as a function of location m. (a) γc minimum in region θ ∈ [−π, π] is depicted, which indicates the PT -symmetric SSH chain spectrum is entirely real for γ < γc at arbitrary θ. The blue bars are for odd m, which approaches zero for m < 30; the green bars are for even m. (b) γc depicted for full θ ∈ [−π, π], the dark blue from left to dark red on the right represent θ from −π to π, the green area in the center corresponds to θ = 0, indicated by arrows in the upper right corner. Other parameters are N = 100, ∆ = 1/2. the boundary and the center in comparison with other cases, and the localized states partially vanish accordingly.
In Fig. 4a, we depict γ c as a function of location m. γ c is maximal at m = n, being 1 − ∆; the minimum γ c approaches zero for small odd m (gain and loss defects close to the chain boundary). The localized states are non-PT -symmetric except for the two degenerate zero edge states, which can be composed into PT -symmetric form; and the real valued bound states appear when defects are at the chain center. For even m, the edge states are unaffected; for odd m, the edge states break the PT symmetry when m is small (defects near the chain boundary). The PT symmetry is fragile to nonzero non-Hermiticity; when m is large (defects near the chain center), the edge states are still unaffected because the probabilities of edge states decayed to zero at the locations of defects pair. In Fig. 4a, γ c is no longer approaching 0 for odd m > 30 and monotonously increases as location m when m > 40. These all reflect that the influence of defects pair on the edge states is negligible and the two edge states energies are real and still being zero. The bound states appear in conjugation pairs, being non-degenerate; the bound states probabilities localize around each impurity. PT symmetry is thus fragile to the bound states. An exception is that when the defects are at the chain center (m = 50), PT -symmetric bound states can appear in topologically nontrivial phase for γ < γ c (in exact PTsymmetric phase). Figure 4b depicts the contours of γ c at different location m as function of θ in full region of θ ∈ [−π, π]. At m = 1, γ c maximum equals to 1 around |θ| = π/2 and shapely changes to zero in |θ| < π/2, where topologically nontrivial edge states appear. This is because that the edge states are fragile to the on-site non-Hermitian gain and loss. Affected by the edge states, the shape change of γ c occurs near |θ| = π/2 at odd m for defects near the chain boundary. The influence of edge states vanishes for defects near the chain center and the bulk states induce PT symmetry phase transition. Notably, γ c increases dramatically at m = 50 in comparison with other cases. γ c increases from 1 − ∆ = 1/2 at |θ| = π to 1 + ∆ = 3/2 at |θ| = 0. This large PT transition threshold implies the PT -symmetric bound states may appear.
The PT -symmetric bound states. In a large N system, the bound states located in the chain center have amplitude decayed to zero at the chain boundaries. At m = n, the bound states are analytically calculated. The eigenenergy is E = (t 1 χ + t 2 ) cos φ, where sin φ = −γ/t 2 , χ is the decay factor that depends on the chain configuration and the inhomogeneous couplings. The eigenenergy is real for |γ/t 2 | 1. For even n, t 1 = 1 − ∆ cos θ, t 2 = 1 + ∆ cos θ; for odd n, t 1 = 1 + ∆ cos θ, t 2 = 1 − ∆ cos θ.
Previously, the topologically protected PT -symmetric zero modes were demonstrated in non-Hermitian SSH chains, the SSH chains globally possess balanced gain and loss in each dimerized unit cell; the topologically protected zero modes are interface states that induced by the coupling disorder in the SSH chain centre and the global loss [48,49,51]. The interface states are confined to the passive sites with vanishing probability distributions on the lossy sites [49], and spread on both sublattices [51]. In our SSH chain with one pair of balanced gain and loss at the centre, the existed PT -symmetric bound states are different. The PT -symmetric bound states are induced by the non-Hermitian local defects and significantly enhance the PT transition threshold. The two bound states energies are symmetric about zero energy and approach zero as the non-Hermiticity increases. At the PT transition threshold, the bound states coalesce to the PT -symmetric zero mode, the zero mode is defective and topologically protected by the band gap [34]. The PT -symmetric zero mode still differs with that found at the interface between two topologically distinct PT -symmetric lattices induced by the coupling disorder [48,49]. The coalesced PT -symmetric zero mode probability vanishes on every other site of the left-half chain and the right-half chain, respectively. The two PTsymmetric bound states are composed by the edge state localized on the right edge n of the left half chain and the edge state localized on the left edge n+1 of the right half chain. When the coupling strength between the neighbours at chain center (sites n and n + 1) is stronger in the inhomogeneous couplings, the topologically protected zero mode appears at γ = max(t 1 ,t 2 ). The wave function contributions of the on-site defects ±iγ and the coupling between the neighbour sites n and n+1 cancel each other, they mimic a free-like boundary except for ψ n = iψ n+1 (ψ j represents the wave function amplitude for site j). The wave function amplitude stepped decays in form of χ l , where l is the dimerized unit cell index. The decay factor χ = −t 1 /t 2 < 1 for even n (χ = −t 2 /t 1 < 1 for odd n). In Fig. 5, the PT -symmetric bound states are depicted in topologically nontrivial phase at θ = 0. The real parts (upper panels) of the bound states are even functions of position while the imaginary parts (middle panels) of the bound states are odd functions of position. In this case (t 1 = 1/2, t 2 = 3/2), the PT transition threshold is at γ = 3/2 = t 2 , the PT -symmetric bound states coalesce and turn to the topologically protected zero mode; we depicted it in Fig. 5a, the probability distribution vanishes for every other site of the left-half chain and the right-half chain, respectively. This topological zero mode differs with that found in the SSH chain with loss in each unit cell due to the distinct interface at the chain centre. [48,49].
For |γ/t 2 | = 1, the decay factor χ is We choose the amplitude in the chain center |ψ n+1 | = 1 instead of renormalized the wave function for convenience.
The wave function of the PT -symmetric bound states at even n is where n c = (N +1)/2, the power exponent [σ j (j − n c )] of (−1) in equation 3 represents the integer part of σ j (j−n c ) and σ j is a sign function defined as σ j = sgn(j − n c ). At odd n case, the expression of ψ j is still valid; however, the bound states are not real valued and PT -symmetric bound states vanish. The values φ for the bound states with complex eigenvalues are φ 1 = sin −1 (−γ/t 2 ), φ 2 = − sin −1 (−γ/t 2 ), φ 3 = π − sin −1 (−γ/t 2 ), and φ 4 = −π + sin −1 (−γ/t 2 ); the corresponding decay factors χ and the wave function of bound states can be obtained from equations 2 and 3.

Conclusion.
We have studied a pair of balance gain and loss defects in a non-Hermitian SSH chain, the influence differs significantly as the PT -symmetric defects locations. The PT transition threshold has been investigated, the number of broken energy levels at maximum increases as the defects close to the chain center in the broken PT -symmetric phase; for the defects at the chain center, all energy levels break the PT symmetry simultaneously in topologically trivial phase, but two edge states are free from PT symmetry breaking in topolog-ically nontrivial phase. When the defects are near the chain boundaries, the edge states in topologically nontrivial phase break the PT symmetry if defects are at the sites with nonzero edge states distribution probabilities; the PT symmetry breaking is caused by the extended states at weak non-Hermiticity or by the bound states at strong non-Hermiticity. The bound states probabilities are localized at the defects and decay exponentially, thus are PT -symmetric breaking; however, the PT -symmetric bound states can be formed when the defects are at the SSH chain center, where the gain and loss are the nearest neighbors. Therefore, the PT transition threshold in this situation increases significantly, which is the largest and equals to the weak inhomogeneous coupling. The PTsymmetric bound states are the topologically protected coalesced zero mode at the PT transition threshold.
We acknowledge the support of National Natural Science Foundation of China (Grant Nos. 11605094 and 11374163) and the Tianjin Natural Science Foundation (Grant No. 16JCYBJC40800).