Figure 1

(a) The left-hand panel shows energy spectra for the robust $\mathcal{P}\mathcal{T}$-symmetric chain, Eq. (1), with $N=500$ sites and a position-dependent hopping amplitude $t_k=t_0{k}^{\alpha }$ with $\alpha =\{0,1,2,-1\}$. The energy is normalized by its maximum value. When $\alpha =0$ (black solid line), we recover the well-known tight-binding chain dispersion $E_n/(2t_0)=-\cos (k_n)$. When $\alpha =1$ (green dotted line), we get a linear spectrum. When $\alpha >1$ (red dashed line), the energy spectrum develops an inflection point at zero energy. In contrast, when $\alpha =-1$ (blue dotted-dashed line), the energy spectrum is linear at the origin, has a steep slope near the band extrema, and develops two symmetrical inflection points. (b) The right-hand panel shows corresponding (un-normalized) densities of states ${\rho }_{\alpha }(E)$. When $\alpha =2>1$ (red dashed line), ${\rho }_{\alpha =2}(E)$ develops a maximum at zero energy and it monotonically decreases to a finite value at the band edges. When $\alpha =-1<0$ (blue dotted-dashed line), the density of states shows a two-peak structure. When $\alpha =0$ (black solid line), we recover the well-known result ${\rho }_0(E)$ that diverges at the band edges. These results show that the energy spectrum and density of states are widely tunable through the exponent $\alpha$.Reuse & Permissions

Figure 2

(a) The left-hand panel shows the minimum and maximum values of inverse-participation ratio ($R$) for an $N$-site chain with Hamiltonian $H$, Eq. (3), as a function of position-dependent hopping $t_k=t_0{k}^{\alpha }$. When $\alpha =0$ (black solid circles), we obtain the analytical result $R(N)\propto 1/N$. The $\alpha =1$ (blue dotted-dashed line) and $\alpha =2$ (red dashed line) results show that the $R$s decrease monotonically with increasing chain size. These results strongly suggest that all eigenstates are extended when $\alpha ⩾0$. (b) The right-hand panel shows the $R$ results for $\alpha ⩽0$. The $\alpha =-1/2$ (blue solid circles) and $\alpha =-1$ (red open squares) results show that the minimum $R$ is essentially independent of $\alpha$. The maximum $R$ saturates to a nonzero value and indicates the presence of localized eigenstates with energies $\pm E$. These results show that when $\alpha <0$, the system has both extended and localized states.Reuse & Permissions

Figure 3

The evolution of the inverse-participation-ratio ($R$) distribution with the chain size $N$. Note the logarithmic scale. The top left-hand and bottom left-hand panels show that when $\alpha =1$, as $N$ increases, the entire distribution of $R$s shifts to lower values. It suggests that all eigenstates of the Hamiltonian $H$, Eq. (3), are extended in the absence of disorder. The top right-hand and bottom right-hand panels show that when $\alpha =-1$, as $N$ is increased, although most of the $R$s shift to lower values, they saturate to a nonzero value for the states shown in the red oval. These eigenstates are localized at the two ends of the chain, and each finite value of the $R$ is fourfold degenerate. Thus, when $\alpha <0$, the system has both extended and localized eigenstates in the absence of disorder.Reuse & Permissions