Fate of topological states and mobility edges in one-dimensional slowly varying incommensurate potentials

We investigate the interplay between disorder and superconducting pairing for a one-dimensional $p$-wave superconductor subject to slowly varying incommensurate potentials with mobility edges. With amplitude increments of the incommensurate potentials, the system can undergo a transition from a topological phase to a topologically trivial localized phase. Interestingly, we find that there are four mobility edges in the spectrum when the strength of the incommensurate potential is below a critical threshold, and a novel topologically nontrivial localized phase emerges in a certain region. We reveal this energy-dependent metal-insulator transition by applying several numerical diagnostic techniques, including the inverse participation ratio, the density of states, and the Lyapunov exponent. Since the precise control of the background potential and the $p$-wave superfluid can be realized in the ultracold atomic systems, we believe that these novel mobility edges can be observed experimentally.

Figure 1

The spectrum (only the lowest four eigenenergies close to zero are shown) of the Hamiltonian (1) with $\mathrm{\Delta }=0.3$ as a function of $V$ under the open boundary condition. Here the total number of sites is set as $L=10000$. Surprisingly, the lowest excitation, i.e., the 10 000th and 10 001th eigenenergies stay at zero as $V$ increases. The inset shows the blow up of these two eigenenergies. We also carry numerical calculations by choosing other $\mathrm{\Delta }$'s, and find that the zero energy modes are independent of $V$ too. The spatial distributions of $\varphi$ and $\psi$ for the lowest excitation with various $V$'s are shown in the lower figures. The lower left picture corresponds to the wave functions of the Majorana zero energy mode, and the lower right picture corresponds to the wave functions of the other zero energy mode.

Figure 2

${\mathrm{\Delta }}_g$ as a function of $V$ with various $\mathrm{\Delta }$'s under the twisted periodic boundary condition. The total number of sites is set to $L=1000$. Here ${\mathrm{\Delta }}_g$ is chosen to be twice of the lowest excitation energy. The inset shows the finite size analysis of $\mathrm{\Lambda }=\frac{{\mathrm{\Delta }}_g}{2}$ near $V=2$. The red symbols correspond to $\mathrm{\Delta }=0.5$ and the blue symbols correspond to $\mathrm{\Delta }=1$. We can clearly see that when $V=1.9, \mathrm{\Lambda }$ is finite as $L\to \infty$. When $V=2.1, \mathrm{\Lambda }$ vanishes as $L\to \infty$. When $V=5, \mathrm{\Lambda }$ also vanishes as $L\to \infty$, which is consistent with the result in Fig. 1 that there exists a widely gapless range. Interestingly, when $V=2, \mathrm{\Lambda }\to L^{-a}$ with $a>0$, hence $\mathrm{\Lambda }\to 0$ as $L\to \infty$, the energy gap vanishes at $V_T=2$.

Figure 3

Topological quantum numbers $Q_L$ and $Q_R$ versus $V$ for systems with various $\mathrm{\Delta }$'s and $L=10000$.

Figure 4

The potential landscape of $V_i=Vcos\left(2\pi \beta i^v\right)$. The site number $i$ ranges from 1 to 10 000.

Figure 5

The distribution of IPR as a function of eigenenergy for various $(\mathrm{\Delta },V)$. “Black dotted lines” correspond to two turning points of IPR located at the mobility edges $E_{c1}=\pm (2-V)$ and $E_{c2}=\pm 2\mathrm{\Delta }$, respectively. (a) When $(\mathrm{\Delta },V)=(0.5,0.5)$ and $(0.5,0.8), E_{c1}=\pm 1.2,\pm 1.5$ and $E_{c2}=\pm 2\mathrm{\Delta }=\pm 1$ are located at the spectrum due to $V<2-2\mathrm{\Delta }=1$, while when $(\mathrm{\Delta },V)=(0.5,1)$, the mobility edges disappear at the spectrum due to $V=2-2\mathrm{\Delta }=1$. (b) When $(\mathrm{\Delta },V)=(0.6,0.5), E_{c1}=\pm 1.5$ and $E_{c2}=\pm 1.2$ are located at the spectrum due to $V<2-2\mathrm{\Delta }=0.8$, while when $(\mathrm{\Delta },V)=(0.6,1)$ and $(0.6,1.5)$, there are no mobility edges and all wave functions are localized due to $V>2-2\mathrm{\Delta }=0.8$, however, the zero energy modes still exist. Therefore, when the strength of the incommensurate potentials is less than the threshold $V_L=2-2\mathrm{\Delta }$, there exist four mobility edges located at $E_{c1}=\pm (2-V)$ and $E_{c2}=\pm 2\mathrm{\Delta }$ in the spectrum. The number of sites is set as $L=5000$.

Figure 6

The finite size analysis of IPR corresponding to four typical eigenenergies $E=1.1,1.3,1.4,1.6$, when $(\mathrm{\Delta },V)=(0.6,0.5)$ (blue)/$(0.6,1.5)$ (red), $\beta =F_{m-1}/F_m$, and $L=F_m$.

Figure 7

The probability density $P_i=u_i^2+v_i^2$ corresponding to four typical eigenenergies $E=1.1,1.3,1.4,1.6$, when $(\mathrm{\Delta },V)=(0.6,0.5)$ (blue)/$(0.6,1.5)$ (red), and $L=6765$. (a) E = 1.1; (b) E = 1.3; (c) E = 1.4; (d) E = 1.6; (e) E = 1.1; (f) E = 1.3; (g) E = 1.4; and (h) E = 1.6.

Figure 8

Eigenstates $u$ and $v$ near the mobility edge $E_{c1}=1.5$, when $\mathrm{\Delta }=0.5$ and $V=0.5$. Here we choose three typical eigenenergies (with four significant digits): high energy localized state above $E_{c1}$ (a) and (b), critical state near $E_{c1}$ (c) and (d), and low energy extended state below $E_{c1}$ (e) and (f). (a) E = 1.5026 above edge; (b) E = 1.5026 above edge; (c) E = 1.5015 near edge; (d) E = 1.5015 near edge; (e) E = 1.4975 below edge; and (f) E = 1.4975 below edge.

Figure 9

Eigenstates $u$ and $v$ near the mobility edge $E_{c2}=1.0$, when $\mathrm{\Delta }=0.5$ and $V=0.5$. Here we choose three typical eigenenergies (with four significant digits): high energy extended state above $E_{c2}$ (a) and (b), critical state near $E_{c2}$ (c) and (d), and low energy localized state below $E_{c2}$ (e) and (f). (a) E = 1.0012 above edge; (b) E = 1.0012 above edge; (c) E = 1.0000 near edge; (d) E = 1.0000 near edge; (e) E = 0.9991 below edge; and (f) E = 0.9991 below edge.

Figure 10

DOS as a function of eigenenergy with three different sets of parameters $(\mathrm{\Delta },V)=(0.5,0.4), (0.5,0.6)$, and $(0.6,0.4)$. Obviously a dramatic change occurs when the eigenenergy passes through the mobility edges $E_{c1}=\pm (2-V)$ and $E_{c2}=\pm 2\mathrm{\Delta }$, which are in accordance with the IPR predictions.

Figure 11

The Lyapunov exponent $\gamma \left(E\right)$ vs eigenenergy with three different sets of parameters $(\mathrm{\Delta },V)=(0.5,0.4), (0.5,0.6)$, and $(0.6,0.4)$. When the eigenenergy is located in the intervals $[V-2,-2\mathrm{\Delta }]$ and $[2\mathrm{\Delta },2-V], \gamma \left(E\right)\to 0$, indicating that the corresponding state is extended. Otherwise $\gamma \left(E\right)$ is finite, indicating that the corresponding state is localized.