Anderson localization transition in a robust $\mathcal{PT}$-symmetric phase of a generalized Aubry-André model

We study a generalized Aubry-André model that obeys $\mathcal{PT}$ symmetry. We observe a robust $\mathcal{PT}$-symmetric phase with respect to system size and disorder strength, where all eigenvalues are real despite the Hamiltonian being non-Hermitian. This robust $\mathcal{PT}$-symmetric phase can support an Anderson localization transition, giving a rich phase diagram as a result of the interplay between disorder and $\mathcal{PT}$ symmetry. Our model provides a perfect platform to study disorder-driven localization phenomena in a $\mathcal{PT}$-symmetric system.

Figure 1

(a) Maximum violation of the $\mathcal{PT}$ symmetry $max(S_{\mathcal{PT}}/V_0)$ for even and odd chains at ${\gamma }_0=2$, $t=0$, and $t=1$. (b) $\langle I_{\mathcal{PT}}\rangle /V_0$ reveals the robust $\mathcal{PT}$-symmetric phase existing for ${\gamma }_0<1$ for arbitrary $t/V_0$.

Figure 2

$\langle \mathrm{MIPR}\rangle$ and $\langle r\rangle$ as a function of $R=\sqrt{t^2+{\gamma }_0^2}$. (a) The $\langle \mathrm{MIPR}\rangle$ at several different $\theta ={tan}^{-1}({\gamma }_0/t)$, indicating that the localization transition occurs at $R=V_0$ for all $\theta$. (b) The gap statistics $\langle r\rangle \approx 0.38$, the Poisson distribution value, in the strong disorder limit $t/V_0\to 0$, and a rapid decay at the localization transition boundary.

Figure 3

Phase diagrams of the system for $N=233$. (a) The $\langle \mathrm{MIPR}\rangle$ and (b) gap statistics $\langle r\rangle$, both of which identify a localized phase within the quarter circle $\sqrt{t^2+{\gamma }_0^2}\le V_0$. The localization-transition and $\mathcal{PT}$-transition boundaries are also indicated in (a) by the thin dashed curve and the dotted line, respectively. A thick dashed line illustrates the “special ray” $\{t=0,\gamma >1\}$ detailed in the main text. We also mark several specific points ${\mathrm{P}}_1$, ${\mathrm{P}}_2$, ${\mathrm{P}}_3$, and ${\mathrm{B}}_{60}$ in different phase regimes, which correspond to $\left\{t/V_0,{\gamma }_0/V_0\right\}\approx \{0.24,0.42\}$, $\{1.2,0.69\}$, $\{1.0,1.74\}$, and $\{cos\left({60}^{\circ }\right),sin\left({60}^{\circ }\right)\}$. We exemplify properties of different phases on these points as detailed in the main text.

Figure 4

Energy spectra $\mathrm{Im}\left(E_k\right)$ as a function of $\mathrm{Re}\left(E_k\right)$ and $\sqrt{|p_0^{LR}\left(j\right)|}$ of the state with $E_0\approx 0$ for $N=1597$ for the different sets of parameters marked in Fig. 3. The spectra are shown in (a), (c), (e), and (g) and the wave functions are shown in (b), (d), (f), and (h) for ${\mathrm{P}}_1$, ${\mathrm{P}}_2$, ${\mathrm{P}}_3$, and ${\mathrm{B}}_{60}$, respectively.

Figure 5

(a) $\langle {\alpha }_{\text{min}}\rangle$ for different chain length $N=F_n$ with $n=13–17$ for ${\mathrm{P}}_1$, ${\mathrm{P}}_2$, ${\mathrm{P}}_3$, and ${\mathrm{B}}_{60}$ defined in Fig. 3. Extrapolation of $\langle {\alpha }_{\text{min}}\rangle$ to the $1/n\to 0$ limit can distinguish extended, localized, and critical phases. (b) The values of $\langle {\alpha }_{\text{min}}\rangle$ for $1/n\to 0$ obtained from extrapolation for different $\theta$, illustrating the localization transition at $R=V_0$. The inset illustrates a zoom-in near $R=V_0$, emphasizing that the critical indexes $\langle {\alpha }_{\text{min}}\rangle$ all collapse approximately on 0.36 for different $\theta$, except $\theta ={90}^{\circ }$.