Symmetry-protected localized states at defects in non-Hermitian systems

Understanding how local potentials affect system eigenmodes is crucial for experimental studies of nontrivial bulk topology. Recent studies have discovered many exotic and highly nontrivial topological states in non-Hermitian systems. As such, it would be interesting to see how non-Hermitian systems respond to local perturbations. In this work we consider chiral and particle-hole-symmetric non-Hermitian systems on a bipartite lattice, including the Su-Schrieer-Heeger model and photonic graphene, and find that a disordered local potential could induce bound states evolving from the bulk. When the local potential on a single site becomes infinite, which renders a lattice vacancy, chiral-symmetry-protected zero-energy mode and particle-hole-symmetry-protected bound states with purely imaginary eigenvalues emerge near the vacancy. These modes are robust against any symmetry-preserved perturbations. Our work generalizes the symmetry-protected localized states to non-Hermitian systems.

Figure 1

Illustration of the non-Hermitian SSH model. The dotted rectangle denotes the unit cell. (a) Here $t_a\pm \lambda$, $t_{LR}$, and $t_b$ are tunneling strengths and $\pm i\gamma$ denote balanced gain and loss. (b) The red dashed circle at $i_0$ denotes the site under a local potential $V_d=V_0(1-\beta )/\beta$. The hopping amplitude related to this site is proportional to $\beta$.

Figure 2

Spectrum of the SSH model with respect to varying disordered strength $\beta$, showing the (a) real and (b) imaginary parts of the eigenvalues. The top (bottom) inset shows the particle-density distribution of localized mode with $\beta =0$ ($\beta =0.8$) in real space. The parameters are $t_a=1.5$, $\lambda =0.1$, $t_{LR}=1.0$, $t_b=0.1$, $\gamma =0$, $V_0=10.0$, and $N_u=100$.

Figure 3

(a) Real and (b) imaginary spectra of the SSH model with chiral symmetry. The orange and red dots indicate two zero-energy states localized near the vacancy in two topologically distinct phases. The insets (L1) and (L2) in (b) show the real part of energies $\mathrm{Re}\left(E\right)$ versus the indices of states $n$ and the particle density distribution $\rho$ versus lattice site indices $i$ of the vacancy-induced zero-energy mode in the topological phase (Top.) [$\beta =0.2$, indicated by the orange dot in (a) and (b)], respectively. The insets (R1) and (R2) present $\mathrm{Re}\left(E\right)$ and $\rho$ of the zero-energy mode in trivial phase (Tri.) [$\beta =1.5$, indicated by the red dot in (a) and (b)], respectively. The parameters are $\lambda =0.1$, $t_{LR}=1.0$, $t_b=0.1$, $\gamma =0$, $V_0=10.0$, and $N_u=100$.

Figure 4

Similar to Fig. 3 but plotted with different non-Hermitian parameters $\lambda =0$ and $\gamma =0.1$.

Figure 5

Illustration of a honeycomb lattice. The parameters $t_a$ and $t_b\pm \lambda$ are tunneling strengths and $\pm i\gamma$ denote balanced gain and loss. The lattice spacing is set as $a=1$.

Figure 6

(a) Real and (b) complex spectra of the graphene model with chiral symmetry versus the parameter $t_b/t_a$. The orange disk indicates the zero-energy state localized near the vacancy in the gapped phase. The top inset $\left(\mathrm{L}1\right)$ shows the real part of energies when $t_b/t_a=4.0$ and the bottom inset $\left(\mathrm{L}2\right)$ shows the density distribution of the localized zero mode. The density is proportional to the radius of the pink spots. The parameters are $\lambda =0.2$, $t_a=1.0$, and $\gamma =0$.

Figure 7

Similar to Fig. 6 but with modified non-Hermitian parameters $\lambda =0$ and $\gamma =0.2$.

Figure 8

(a) Real part $\text{Re}\left(E\right)$ and (b) imaginary part $\text{Im}\left(E\right)$ of the eigenenergies of the SSH model with a single lattice vacancy versus the gain or loss strength $\gamma$. The other parameters are fixed as $t_a=1.8$, $t_{LR}=1$, $t_b=0.1$, and $\lambda =0$.

Figure 9

(a) The SSH model with multivacancies. (b) Eigenenergies of the SSH model versus the the number of vacancies. Here $N_{\mathrm{v}}=m$ corresponds to vacancies existing at sites $1,2,...,m$, as shown in (a). Also shown is the particle density distribution of localized modes in the presence of (c) $N_{\mathrm{v}}=2$ and (d) $N_{\mathrm{v}}=5$ vacancies. The density is proportional to the radius of the pink spots. The parameters are $t_a=1.4$, $t_{LR}=0.5$, $t_b=0.1$, $\lambda =0.1$, $\gamma =0$, and $N_u=100$.