Quantum walks of two interacting anyons in one-dimensional optical lattices

We investigate continuous-time quantum walks of two indistinguishable anyons in one-dimensional lattices with both on-site and nearest-neighbor interactions based on the fractional Jordan-Wigner transformation. It is shown that the two-body correlations in position space are symmetric about the initial sites of two quantum walkers in the Bose limit ($\chi =0$) and Fermi limit ($\chi =1$), while in momentum space this happens only in the Bose limit. An interesting asymmetry arises in the correlation for most cases with the statistical parameter $\chi$ varying in between. It turns out that the origin of this asymmetry comes from the fractional statistics that anyons obey. On the other hand, the two-body correlations of hard-core anyons in position space show uniform behaviors from antibunching to cowalking regardless of the statistical parameter. The momentum correlations in the case of strong interaction undergo a smooth process of two stripes smoothly merging into a single one, i.e. the evolution of fermions into hard-core bosons.

Figure 1

Two-body correlations of anyonic quantum walkers in position space. The nearest-neighbor interaction strength $|V/J|=0,1,4$, and 80 from top to bottom and the on-site interaction strength $U/J=0$. Here we only show the instantaneous correlations before colliding with the boundaries $L=\pm 10$. The corresponding evolution times are $Jt=4,4.5,7.5$, and 110 from top to bottom. For each row, $\chi =0,0.25,0.5,0.75,1$ (from left to right).

Figure 2

Two-body correlations of anyonic quantum walkers in momentum space with the same parameters as in Fig. 1. Note that spatial asymmetry in the bottom panel for $\chi =1$ persists for very strong interaction.

Figure 3

Two-body correlations of anyonic quantum walkers in position space. The on-site interactions are $U/J=0,1,4$, and 80 (from top to bottom) with fixed $|V/J|=1$ and the corresponding evolution time is given by $Jt=4.5$.

Figure 4

Two-body correlation fluctuations and the density distributions in position space. The on-site and nearest-neighbor interaction strengths are $(U/J,|V/J\left|\right)=(0,0),(0,1)$ and $(1,0)$ from top to bottom. The evolution time is given by $Jt=4$.

Figure 5

Two-body correlations in position space for HCAs. The nearest-neighbor interaction strength $|V/J|=0,1,$ and 4 from top to bottom and the corresponding evolution times are given by $Jt=4,4.5,$ and $7.5$, respectively. In each row, $\kappa =0,0.25,0.5,0.75,1$ (from left to right).

Figure 6

Two-body correlations in momentum space for HCAs. The nearest-neighbor interactions are $|V/J|=0,1,4$, and 80 from top to bottom. And the corresponding evolution times are given by $Jt=4,4.5,7.5,$ and $110,$ respectively. The figures in the first and last column are exactly the same as those in Ref. [23].