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.
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.
Two interacting particles in lattices, in the absence of dissipation, can not distin- guish between attractive or repulsive interaction when the range of their tunnelling is limited to nearest neighbor sites. However, we find that, in the case of long-range tunnelling, the particles exhibit different dynamics for different types of interactions of the same strength. The nature of dynamical correlations between particles also becomes significantly different. For weak interactions, particles develop a character in correlation which is in between that of antiwalking and cowalking when the tunnelling is long-range. For strong interactions, particles cowalk independently of their statistics. A few recent experiments have demonstrated such effects of interactions on quantum walk of photons, atoms and spin excitations on various lattice platforms. In disordered lattices the effect of coherent backscattering makes particles localize to their initial position. We find that a weak repulsive interaction reduces localization and a strong interaction enhances localization. We also calculate the correlations between the particles in the disordered 1D and 2D systems. The effect of long-range tunnelling on localization of particles in disordered 1D systems has been explored. For large ordered or disordered lattices, computation of localization parameters becomes difficult. In these cases, an efficient recursive algorithm is used to calculate Green’s functions exactly. We extend such algorithm to disordered systems in both one and two dimensions. We also illustrate that this recursive algorithm maps directly to some graph structures like binary trees. We perform calculations for quantum walk of interacting particles on such graphs. The method is also used to calculate the properties of interacting particles on lattices with gauge fields. For disordered 2D lattices, we introduce and test approximations which produce accurate results and make the calculations more efficient. We examine the localization parameters for a broad range of interaction and disorder strengths and try to find differences among parameters within the range.
We study the effects of interparticle interactions and power-law tunneling couplings on quantum walks executed by both a single one and a pair of hard-core bosons moving in clean and disordered one-dimensional lattices. For this purpose, we perform exact diagonalization to explicitly evaluate the short and long time probabilities of finding the walkers within a surveillance area. Our main conclusions, summarized in phase diagrams in the disorder-power-law and interaction-disorder spaces, allowed us to discern two different scenarios for the single and two quantum walkers dynamics. While in the single particle case the transition to localized and extended regimes is identified for well defined values of the disorder amplitude and power law hopping, those frontiers are replaced by diffuse contours in the interacting two particle case. In fact, counterintuitive transport regimes as diffusion enhanced by disorder, and space constrained dynamics assisted by both interactions and short tunneling range are found. Our results are of direct relevance for quantum systems with long-range interactions that are currently realized in the laboratory.
The quantum walk of two hard-core bosons in one-dimensional lattice under the effect of long-range inter-particle interaction is studied in detail. We also simulate the influence of an isolated defect that may exist in the lattice on the quantum walk of two particles by adding an additional potential energy to a certain lattice site. Using exact diagonalization method, the continuous-time quantum walk is directly simulated. The numerical simulations show that the range of interaction (long-range or short-range), the strength of the inter-particle interaction, the initial state of the two particles and the presence of the isolated defect have great influences on the quantum walk. Under the effect of strong long-range interaction, the particles initially located on the non-adjacent lattice sites have a co-walking behavior, while under the short-range interactions (nearest-neighbor interactions) only two particles initially located on the neighboring lattice sites can exhibit co-walking. After introducing the isolated defect into the system with strong interaction, two particles residing on the same side of the isolated defect keep co-walking, while two particles located on either sides of the isolated defect or one particle located on the isolated defect and the other particle staying on the side of the isolated defect, the two particles keep stationary or co-walking near the defect, displaying the characteristics of localization. By using the second-order perturbation theory of degenerate quantum system, a comprehensive theoretical analysis of the above numerical results is given. The theoretical analysis reveals the underlying physical law of quantum walks of two particles in one-dimensional lattice under the effects of strong long-range interaction and isolated defect in the lattice.
Non-interacting particles with long-range hopping are known to be delocalized in disordered systems of infinite size. It is thus natural to assume that such particles can traverse any finite-size lattice. Here, we demonstrate that this is not generally true. The delocalization mechanism is induced by resonances between distant lattice sites. The number of such resonances diverges with the system size. For a finite-size lattice the number of resonances is finite and may not be sufficient to result in delocalization. We consider particles with long-range (dipolar) hopping in three-dimensional lattices with diagonal disorder and random dilution. We compute the wavepacket dynamics of particles placed in an individual lattice site, the inverse participation ratios and the fluctuations of the wavefunctions with disorder. We show that, for certain finite-size disordered lattices, particles remain localized within a finite volume much smaller than the lattice size, and that the wavefunctions exhibit the log-normal fluctuations characteristic of Anderson localization. We characterize the localization properties as functions of dilution and diagonal disorder. We combine our results with scaling theory to obtain the size dependence of the localization--diffusion crossover. Our results indicate that particles with long-range hopping undergo exponential localization in lattices of finite size, even macroscopically finite. Our phase diagrams illustrate a rather unusual phenomenon: quantum particles can diffuse through a lattice of size 10A, but not through a lattice of size A.
It has become widely accepted that particles with long-range hopping do not undergo Anderson localization. However, several recent studies demonstrated localization of particles with long-range hopping. In particular, it was recently shown that the effect of long-range hopping in one-dimensional (1D) lattices can be mitigated by cooperative shielding, which makes the system behave effectively as one with short-range hopping. Here, we show that cooperative shielding, demonstrated previously for 1D lattices, extends to 3D lattices with isotropic long-range ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ hopping, but not to 3D cubic lattices with anisotropic long-range hopping. The specific anisotropy we consider corresponds to the interaction between dipoles aligned along one of the principal axes of the lattice. We demonstrate the presence of localization in 3D lattices with uniform ($\ensuremath{\alpha}=0$) isotropic long-range hopping and the absence of localization with uniform anisotropic long-range hopping by using the scaling behavior of eigenstate participation ratios. We use the scaling behavior of participation ratios and energy-level statistics to show that the existence of delocalized, extended nonergodic, or localized states in the presence of disorder depends on both the exponents $\ensuremath{\alpha}$ and the anisotropy of the long-range hopping amplitudes.
Correlation of interacting particles is studied in their dynamics and localization in ideal and disordered lattice systems with the help of numerical tools. Both 1D and 2D systems are considered. In 1D lattices with long-range hopping, differences between dynamics of attractively and repulsively interacting particles are noted. For calculations of 2D systems, a recursion based numerical algorithm for two-particle Greens functions is implemented which provides accurate results. Using this algorithm, spectral properties of 2D Hubbard and Hofstadter models is computed, along with localization parameters of 2D disordered systems.