Transition from metal to higher-order topological insulator driven by random flux

Random flux is commonly believed to be incapable of driving full metal-insulator transitions in noninteracting systems. Here we show that random flux can after all induce a full metal–band insulator transition in the two-dimensional Su-Schrieffer-Heeger model. Remarkably, we find that the resulting insulator can be an extrinsic higher-order topological insulator with zero-energy corner modes in proper regimes, rather than a conventional Anderson insulator. Employing both level statistics and finite-size scaling analysis, we characterize the metal–band insulator transition and numerically extract its critical exponent as $\nu =2.48\pm 0.08$. To reveal the physical mechanism underlying the transition, we present an effective band structure picture based on the random-flux averaged Green's function.

Figure 1

(a) Schematic of the 2D SSH model with random flux. Blue (red) thick and thin bonds mark dimerized hopping amplitudes in the $x\left(y\right)$ direction. The round arrow (with different sizes and opacities) in each plaquette indicates the random flux. (b) Energy spectrum of the model without random flux for $(t_x,t_y)=(0.2t,0.6t)$. (c) Disorder-averaged spectrum as a function of $U$ for $(t_x,t_y)=(0.2t,0.6t)$. For large $U$, the system acquires a bulk gap that protects four zero-energy modes (red). (d) Density plot of the directly disorder-averaged gap as a function of $t_x$ and $t_y$ at $U=2\pi$. The dimension of the system is $L\equiv L_x=L_y=30$ with open (periodic) boundaries in (c) [(d)]. Here, 200 random-flux configurations are considered.

Figure 2

(a) Averaged IPR $\langle I\rangle$ as a function of $U$ for $L=20$, 30, and 40, respectively. (b) Averaged LSR $\langle r\rangle$ as a function of $U$ for $L=20$, 30, and 40, respectively. (c) Distribution of LSR $P\left(r\right)$ in different limits. (d) Averaged LSR $\langle r\rangle$ near the critical point as a function of $U$ for various $L$. (e) Scaling behavior of IPR in the metallic (blue) and insulating (green) phases, and at the critical point $U_c\approx 0.75\pi$ (red). (f) Number variance ${\mathrm{\Sigma }}_2$ as a function of $\sqrt{\langle N\rangle }$ at the critical point. We consider $N_E=16$ and 4000 random-flux configurations in (a), (b), (c), and (e). The parameters $t_x=0.2t, t_y=0.6t$ and periodic boundary conditions are chosen for all plots.

Figure 3

(a) Phase diagram of $q_{xy}$ against $t_x$ and $t_y$. The dimension of the system is $L=30$ with 30 random-flux configurations. (b) Electron charge density in the extrinsic HOTI phase at half filling. (c) Disorder-averaged energy spectrum as a function of $t_x$ for $t_y=0.3t$ under open boundary conditions. (d) Disorder-averaged edge polarization $P_x$ as a function of $t_x$ for $t_y=0.3t$. $U=2\pi$ for all plots.