Quantum walks in the commensurate off-diagonal Aubry-André-Harper model

Due to the topological nature of the Aubry-André-Harper (AAH) model, interesting edge states have been found existing in one-dimensional periodic and quasiperiodic lattices. In this article, we investigate continuous-time quantum walks of identical particles initially located on either edge of commensurate AAH lattices in detail. It is shown that the quantum walker is delocalized among the whole lattice until the strength of periodic modulation is strong enough. The inverse participation ratios (IPRs) for all of the eigenstates are calculated. It is found that the localization properties of the quantum walker is mainly determined by the IPRs of the topologically protected edge states. More interestingly, the edge states are shown to have an evident “repulsion” effect on quantum walkers initiated from the lattice sites inside the bulk. Furthermore, we examine the role of nearest-neighbor interaction on the quantum walks of two identical fermions. Clear enhancement of the repulsion effect by strong interaction has been shown.

Figure 1

Sketch of a one-dimensional period-2 off-diagonal AAH lattice under different boundary conditions. (a) Open boundary condition. (b) Periodic boundary condition.

Figure 2

Single-particle quantum walks on a one-dimensional period-2 off-diagonal AAH lattice with $L=100$. The quantum walker is initially positioned on the left boundary site. (a)–(c) $\varphi =0$, open boundary condition. (d)–(f) $\varphi =0.6\pi$, open boundary condition. (g)–(i) $\varphi =0$, periodic boundary condition. The first column is corresponding to ${\lambda }_{od}=0.1$, the second to ${\lambda }_{od}=0.3$, and the third to ${\lambda }_{od}=0.9$.

Figure 3

Upper panel shows the edge states of the one-dimensional period-2 off-diagonal AAH lattice with $L=100$ under the open boundary condition. The lower panel is the energy spectrum plotted as a function of $\varphi$. (a),(d) ${\lambda }_{od}=0.1$. (b),(c),(e) ${\lambda }_{od}=0.9$.

Figure 4

Phase diagram of the one-dimensional period-2 off-diagonal AAH model. $\gamma$ is the Zak phase plotted as a function of the phase $\varphi$ and the strength ${\lambda }_{od}$ of the off-diagonal modulations.

Figure 5

(a) Inverse participation ratio of all eigenstates plotted as a function of the strength of the off-diagonal modulations ${\lambda }_{od}$ for the one-dimensional period-2 off-diagonal AAH model under the open boundary condition with $\varphi =0$ and $L=100$. (b) Inverse participation ratio of the edge state as a function of ${\lambda }_{od}$.

Figure 6

Single-particle quantum walks on a one-dimensional period-2 off-diagonal AAH lattice with $L=30$. The quantum walker is initially positioned on the center site 15. (a)–(c) $\varphi =0$, open boundary condition. (d)–(f) $\varphi =0.6\pi$, open boundary condition. (g)–(i) $\varphi =0$, periodic boundary condition. The first column is corresponding to ${\lambda }_{od}=0.1$, the second to ${\lambda }_{od}=0.3$, and the third to ${\lambda }_{od}=0.9$.

Figure 7

The time-dependent distribution of the quantum walker on the left boundary site for a relatively long time. The quantum walker is initially positioned on the center site 15. (a) $\varphi =0$, open boundary condition. (b) $\varphi =0.6\pi$, open boundary condition. (c) $\varphi =0$, periodic boundary condition.

Figure 8

Quantum walks of two identical fermionic particles with nearest-neighbor interaction on a one-dimensional period-2 off-diagonal AAH lattice with $L=30$ under the open boundary condition. The phase $\varphi =0$ and ${\lambda }_{od}=0.9$. One quantum walker is initially positioned on the left boundary site, and the other is located on the center site 15. (a) $V=0$. (b) $V=1$.

Figure 9

Quantum walks of two identical fermionic particles with nearest-neighbor interaction on a one-dimensional period-2 off-diagonal AAH lattice with $L=30$ under the open boundary condition. The phase $\varphi =0$ and ${\lambda }_{od}=0.9$. The initial state is prepared on $\left|1,2\right.〉$. (a) $V=0$. (b) $V=1$.

Figure 10

(a),(b) Density distributions of eigenstates corresponding to the eigenenergies in (c) and (d) denoted by star symbols. (c),(d) Eigenenergies in ascending order for a period-2 off-diagonal AAH lattice with ${\lambda }_{od}=0.9, \varphi =0$, and $L=30$ under the open boundary condition. The nearest-neighbor interaction is (c) $V=0$; (d) $V=1$.

Figure 11

Quantum walks of two identical fermionic particles with attractive nearest-neighbor interaction on a one-dimensional period-2 off-diagonal AAH lattice with $L=30$ under the open boundary condition. The phase $\varphi =0, {\lambda }_{od}=0.9$, and $V=-1$. (a) The initial state is prepared on $\left|1,15\right.〉$. (b) The initial state is prepared on $\left|1,2\right.〉$.