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

Abstract

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 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 −α hopping, but not to 3D lattices with dipolar-like anisotropic long-range hopping. We demonstrate the presence of localization in 3D lattices with uniform (α = 0) isotropic long-range hopping and the absence of localization with uniform anisotropic long-range hopping by using the scaling behaviour of eigenstate participation ratios. We use the scaling behaviour of participation ratios and energy level statistics to show that the existence of delocalized, non-ergodic extended, or localized states in the presence of disorder depends on both the exponents α and the isotropy of the long-range hopping amplitudes.

Renormalization to localization without a small parameter

“…These eigenstates are polynomially localized ψ n,typ (r m ) 2 ∼ |r n − r m | −2µ with the exponent µ = max{a, 2d − a} for all positive values of the parameter a. Although this model is very similar to the one of Burin and Maksimov (BM) [53] considered also in [36,52,55], the above wavefunction duality in PLE model has not yet been explained theoretically as the matrix inversion trick invented by one of us in [36] breaks down in the absence of diagonal disorder. The RG approach developed in the present paper provides the desired explanation.…”

“…1. ) 12 This estimate is given by solution of the equation R d 0 P (rmin) = 1, with the distribution of distances between adjacent sites, homogeneously distributed in d-dimensional space with unit density, given by Poisson formula, P (r) ∼ re −r it is not the ground (or anti-ground) state with the energy diverging with the system size which matters for the localization of the bulk spectrum (like in the cooperative shielding approach [35,52]), but the presence of high-energy delocalized states (with high effective charges) do this job. The latter high energy states do not need to be at the very spectral edge, see the right panel of Fig.…”

“…Indeed, unlike the models with the diagonal disorder, there is no way to treat the ERMs without the diagonal potential with the locator expansion approximation even for the infinitesimally small hopping term, due to the ideal resonance of all bare diagonal levels. This fact can be understood on the example of low-dimensional models with translation-invariant polynomial hopping which show localization of all bulk wavefunctions for any finite disorder either in the diagonal potential [32,35,36,52] or in the position of the lattice sites [32], whereas in the complete absence of the disorder these models are translation-invariant and, hence, delocalized. To overcome the above mentioned principal difficulty we generalized the known renormalization group (RG) approach (developed by Levitov [33,34] and extended by Burin and Maksimov (BM) in [53] and by Mirlin and Evers in [54]) to the case of absence of a small parameter and to generic smooth Euclidean hopping term.…”

Robustness of delocalization to the inclusion of soft constraints in long-range random models

“…[18] and to generalize both Anderson localization [3,49,58] and Mott ergodicity [59] principles for the models with correlated hopping. However recently there have been found several many-body [60,61] and higher-dimensional, d > 1, single-particle models [62,63], applied to which this method easily uncovers their phase diagrams and the wave-function structure by the extension the locator expansion validity range.…”

Emergent fractal phase in energy stratified random models

“…Indeed, in the case of completely correlated hopping amplitudes (see, e.g., disordered Richardson's model [56,57] and Burin-Maksimov model [58,59]) even the long-range character of hopping terms cannot delocalize the majority of the states. These effects called in the literature the cooperative shielding [60,61] and the correlation-induced localization [21,[62][63][64][65][66] are based on the observation that in the disorder-free versions of such systems due to the correlated nature of the kinetic long-range terms there is measure zero of high-energy states that take the most spectral weight of the hopping and effectively screen the bulk states from the off-diagonal matrix elements. The coexistence of few high-energy states with the nearly degenerate bulk states forms a kind of energy stratification, when measure zero of states are separated from each other and from the rest of the spectrum.…”