Reentrant localization and mobility edges in a spinful Aubry-André-Harper model with a non-Abelian potential

We study localization properties and mobility edges of a generalized spinful Aubry-André-Harper (AAH) model, which is the dimensional reduction of the two-dimensional Hofstadter model with a non-Abelian SU(2) gauge potential. Depending on whether the quasiperiod is comparable with the lattice size, the model has different localization properties. In the noncomparable case, the generalized AAH model still retains duality properties. Tuning the non-Abelian gauge can make the system undergo an unconventional reentrant localization phase transition as the strength of quasiperiodic potential increases. Furthermore, mobility edges exist in the mixed phase where the localized states sit at the center of spectra. Nevertheless, the non-Abelian gauge potential results in more mobility edges than that the Abelian gauge potential does, when the model is in the semiclassical limit where the quasiperiod is comparable with the lattice size. Moreover, exact expressions of the mobility edges and localization phase diagrams are analytically obtained by a semiclassical method.

Figure 1

(a) FDs of single-particle states, as functions of the corresponding energies $E$ and quasiperiodic potential strength $J_y$, for systems with $\alpha =1$ and $\beta =\pi /2$. (b) Semilogarithmic plots of the distributions of IPRs for systems with the same $\alpha =1$ and $\beta =\pi /2$ but different strengths of quasiperiodic potential. $k$ is the index of states, which are arranged in the ascending order of energies. (c) Typical spatial distributions of states, for the system with $J_y=2$. The irrational number $\mathrm{\Omega }=(\sqrt{5}-1)/2\simeq 610/987$, and the total number of lattice sites $L=987$.

Figure 2

(a) MIPR and MNPR vs $J_y$ for the system with $\alpha =0.4\pi$ and $\beta =0.34\pi$. (b) Semilogarithmic plots of the distributions of IPRs for the same system but with different strengths of quasiperiodic potential. $\mathrm{\Omega }\simeq 610/987$ and $L=987$.

Figure 3

Contour plot of the quantity $\kappa$ in Eq. (5), which represents localization phase diagram, for the system with $\beta =0.5\pi$ (a) or $\alpha =0.5\pi$ (b). $\mathrm{\Omega }\approx 610/987$ and $L=987$.

Figure 4

Contour plot of the quantity $\kappa$ in the ($\beta ,J_y$) plane for systems with different $\alpha$. $\mathrm{\Omega }\simeq 610/987$ and $L=987$.

Figure 5

(a) ${log}_{10}$(IPRs) of single-particle states, as function of the corresponding energies $E$ and quasiperiodic potential strength $J_y$, for systems with $\alpha =0$ and $\beta =0.3\pi$. The irrational $\mathrm{\Omega }=\sqrt{2}/618\pi \ll 1$, and the total number of lattice sites $L=1000$. (b) Typical spatial distributions of states, whose positions in spectra are shown in (a) (symbols).

Figure 6

(a) ${log}_{10}$(IPRs) of single-particle states for systems with $\alpha =0.4\pi , \beta =0.2\pi , \mathrm{\Omega }=\sqrt{2}/618\pi$, and $L=1000$. (b) Semilogarithmic plots of spatial distributions of states, whose positions in spectra are shown in (a) (symbols).