Effects of long-range hopping and interactions on quantum walks in ordered and disordered lattices

We study the effects of long-range hopping and long-range interparticle interactions on the quantum walk of hard-core bosons in ideal and disordered one-dimensional lattices. We find that the range of hopping has a much more significant effect on the particle correlation dynamics than the range of interactions. We illustrate that long-range hopping makes the correlation diagrams asymmetric with respect to the sign of the interaction. We examine the relative role of repulsive and attractive interactions on the dynamics of scattering by isolated impurities and Anderson localization in disordered lattices. We show that weakly repulsive interactions increase the probability of tunneling through isolated impurities and decrease the localization.

Figure 1

Pair correlations for two hard-core bosons with $\beta =1$ in Eq. (3) computed at time $2\pi /t$ in a 1D lattice with 50 sites. The particles are initially in sites $i=25$ and $j=26$. The values along the diagonal are the probabilities of cowalking. The labels $n_1$ and $n_2$ denote the site indexes of particles 1 and 2, respectively.

Figure 2

Pair correlations for two hard-core bosons with $\beta =3$ in Eq. (3) computed at time $2\pi /t$ in a 1D lattice with 50 sites. The particles are initially in sites $i=25$ and $j=26$. The labels $n_1$ and $n_2$ denote the site indexes of particles 1 and 2, respectively.

Figure 3

Eigenstates of two noninteracting hard-core bosons in a 1D lattice. The different panels correspond to the different range of hopping $t_{ij}$: (a) nearest neighbor hopping, (b) $\alpha =3$, (c) $\alpha =2$, (d) $\alpha =1$. The energy is in the units of $t$ and the wave number is in the units of $1/a$, where $a$ is the lattice constant.

Figure 4

Particle density $\mathcal{P}$ integrated over the part of the lattice between two impurities at sites 10 and 31 at time $20/t$ for two particles with long-range hopping and long-range interactions initially placed in a 1D lattice at sites 20 and 21. The part of the wave packet tunneling to $i<10$ and $i>31$ is absorbed to eliminate reflection from the lattice boundaries, thus leading to irreversible decay of $\mathcal{P}$ as a function of time. The values of $\alpha$ and $\beta$ specify the parameters of hopping (2) and interactions (3).

Figure 5

Particle density $\mathcal{P}$ integrated over the part of the lattice between two impurities at sites 10 and 31 at time $20/t$ for two particles with long-range hopping and long-range interactions initially placed in a 1D lattice at sites 20 and 21. The part of the wave packet tunneling to $i<10$ and $i>31$ is absorbed to eliminate reflection from the lattice boundaries, thus leading to irreversible decay of $\mathcal{P}$ as a function of time. The values of $\alpha$ and $\beta$ specify the parameters of hopping (2) and interactions (3).

Figure 6

Pair correlations in the limit of long time $\tau ={10}^4/t$ for two hard-core bosons with long-range hopping and dipolar interactions $\beta =3$ in a 1D lattice disordered by a random distribution of impurities with the concentration $p=0.1$. The results are averaged over 5000 realizations of disorder. The labels $n_1$ and $n_2$ denote the site indexes of particles 1 and 2, respectively.

Figure 7

The inverse of the inverse participation ratio calculated at $\tau ={10}^4/t$ from localized density distributions for two particles initially placed in adjacent sites of a disordered 1D lattice with $N=50$ and $p=0.9$. The results are averaged over 5000 realizations of disorder.