Effect of the anisotropy of long-range hopping on localization in three-dimensional lattices

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 $r^{-\alpha }$ 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 ($\alpha =0$) 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 $\alpha$ and the anisotropy of the long-range hopping amplitudes.

Figure 1

The energy gap at the top of the spectrum as a function of the hopping range exponent $\alpha$. Panel (a) is for long-range hopping in a 1D lattice with periodic boundary conditions. Panel (b) is for a 3D lattice with isotropic hopping, determined analytically under periodic boundary conditions; see Eq. (9). Panels (c) and (d) show the energy gap for the Hamiltonian with anisotropic hopping, determined numerically and under open boundary conditions. Panel (d) is an expanded view of panel (c). $N$ is the system side length (total of $N^3$ sites in the 3D case). In all plots, $p=1$ and $W=0$. Note that we examine the top of the spectrum as the effective mass is negative.

Figure 2

Energy of several top eigenstates as a function of $\alpha$ for isotropic hopping and anisotropic hopping with open boundary conditions (top and bottom panels, respectively). For the top panel, the red dashed line is triply degenerate; the other is nondegenerate. For the bottom panel, the orange line is doubly degenerate for $\alpha \lesssim 1$; the rest are nondegenerate. In both plots, $N^3={30}^3$, $p=1$, and $W=0$.

Figure 3

Participation ratio (PR) as a function of energy for the eigenstate of a 3D system with isotropic hopping, $W=16.5$, $p=1$, and $\alpha =0$. The number of disorders included ranges from 148 to 2100, as required to obtain sufficiently small error bars. Top panel: Plot of the PR of each eigenstate vs its eigenvalue for several disorder realizations for $N=11,21,\text{and}31$ (green $\blacksquare$, orange $\blacksquare$, and blue $\blacksquare$, respectively). Note the symmetric logarithmic scaling of the abscissa, where the linear portion is within $\pm 10\tilde{t}$. Large black circles merely highlight the location of the states at the top of the band. Lower panel: PR averaged within energy bins and over disorder realizations. The plot is focused on the band near zero energy. Note the linear scale of the abscissa. Error bars for the average PR are 95% confidence intervals and smaller than the marker size where not seen.

Figure 4

Participation ratio (PR) as a function of energy for the eigenstate of a 3D system with anisotropic hopping; $W=1\text{and}16.5$; $p=1$; $\alpha =0$; and $N=11,21,\text{and}31$ (green $\blacksquare$, orange $\blacksquare$, and blue $\blacksquare$, respectively). The number of disorders included ranges from 50 to 2350, as required to obtain sufficiently small error bars. Note the symmetric logarithmic scaling of the abscissa, where the linear portion is within $\pm 1\tilde{t}$ or within $\pm 16\tilde{t}$. Top panels: Plot of the PR of each eigenstate vs its eigenvalue for several disorder realizations. Lower panels: PR averaged within energy bins and over disorder realizations for $W=1\text{and}16.5$. Error bars for the average PR are 95% confidence intervals and smaller than the marker size where not seen.

Figure 5

Top panels: Plot of the participation ratio (PR) of each eigenstate vs its eigenvalue for several disorder realizations. Lower panels: Plot of the $\mathrm{PR}/N^3$ of each eigenstate vs its eigenvalue for several disorder realizations. Plots are for a 3D system with isotropic and anisotropic hopping; $W=16.5$; $p=1$; $\alpha =1$; and $N=11,21,\text{and}31$ (green $\blacksquare$, orange $\blacksquare$, and blue $\blacksquare$, respectively). The number of disorders included ranges from 100 to 350, as required to obtain well-formed distributions. Note the symmetric logarithmic scaling of the abscissa, where the linear portion is within $\pm 10\tilde{t}$ or within $\pm 16\tilde{t}$.

Figure 6

Top panels: Plot of the participation ratio (PR) of each eigenstate vs its eigenvalue for several disorder realizations. Lower panels: Plot of the $\mathrm{PR}/N^3$ of each eigenstate vs its eigenvalue for several disorder realizations. Plots are for a 3D system with isotropic and anisotropic hopping; $W=16.5$; $p=1$; $\alpha =3$; and $N=11,21,\text{and}31$ (green $\blacksquare$, orange $\blacksquare$, and blue $\blacksquare$, respectively). The number of disorders included ranges from 99 to 350, as required to obtain well-formed distributions. Note the linear scaling of the abscissa.

Figure 7

Mean energy-level spacing ratio $\langle r\rangle$ as a function of disorder strength $W=\frac{\omega }{t_{\max }}$ for the isotropic and anisotropic hopping Hamiltonians with open boundary conditions. The hopping range exponent is $\alpha =3$ and the filling fraction is $p=1$ in both cases. $N$ denotes the lattice side length ($N^3$ sites). The horizontal dashed lines denote the values of $\langle r\rangle$ for the Poisson distribution $(\sim 0.38629)$ and the GOE $(\sim 0.53590)$. The error bars are 95% confidence intervals based on 96 disorders and are smaller than the marker size where not seen.

Figure 8

Upper panel: Same as Fig. 7 (upper panel), but with the mean energy-level spacing ratio $\langle r\rangle$ including only the eigenvalues in the energy range $-\tilde{t}\le E\le \tilde{t}$. The results are averaged over 1071 disorders for $N^3={11}^3$ and ${21}^3$, and 96 disorders for $N^3={31}^3$. Lower panel: Same as upper panel, but for $\alpha =1$. The results are averaged over 1100 disorders for $N^3={11}^3$ and ${21}^3$, and 100 disorders for $N^3={31}^3$.

Figure 9

Mean energy-level spacing ratio $\langle r\rangle$ as a function of disorder strength $W=\frac{\omega }{t_{\max }}$ and filling fraction $p$ for the isotropic and anisotropic hopping Hamiltonians with open boundary conditions. The hopping range exponent is $\alpha =3$ and the lattice size is $N^3={31}^3$ in both cases. The color varies from the value of $\langle r\rangle$ for the Poisson distribution $(\sim 0.38629)$ to that for the GOE $(\sim 0.53590)$. The white circles denote the values obtained via exact diagonalization of the Hamiltonian; piecewise cubic interpolation is used for the rest of the plot. The black lines indicate where $\langle r\rangle =0.49$. The results are averaged over 96 realizations of disorder, giving 95% confidence intervals that are at most $\pm 0.002$. The green data points mark the location of the Anderson transition for the 3D tight-binding model in the infinite-size limit, as determined by Root and Skinner [37].