Shielding and localization in the presence of long-range hopping

Abstract

We investigate a paradigmatic model for quantum transport with both nearest-neighbor and infinite range hopping coupling (independent of the position). Due to long range homogeneous hopping, a gap between the ground state and the excited states can be induced, which is mathematically equivalent to the superconducting gap. In the gapped regime, the dynamics within the excited states subspace is shielded from long range hopping, namely it occurs as if long range hopping would be absent. This is a cooperative phenomenon since shielding is effective over a time scale which diverges with the system size. We named this effect Cooperative Shielding. We also discuss the consequences of our findings on Anderson localization. Long range hopping is usually thought to destroy localization due to the fact that it induces an infinite number of resonances. Contrary to this common lore we show that the excited states display strong localized features when shielding is effective even in the regime of strong long range coupling. A brief discussion on the extension of our results to generic power-law decaying long range hopping is also given. Our preliminary results confirms that the effects found for the infinite range case are generic.

Renormalization to localization without a small parameter

“…First of all, Eq. (35) in case of PLE model give R i+1 R 2 i , which is compatible with the approximation of effective charges provided R i 1 as R 2 i R i should be valid. It means that the zero-cutoff radius R 0 is finite and large compared to the unity.…”

“…Here we used the definition χ ε (R) = |q ε (R)| 2 ν R (ε), the asymptotic expression for the probability of single resonances p i ε (r) 2χ ε (R i )f (r) for R i , R i+1 1, and the expression (19). Equation (35) together with the condition (21) and Eq. (28) form a complete set of requirements for the RG to be applicable.…”

“…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

“…Indeed, the new paradigm of the ALT suggested there states that hopping correlations j mn j m n − j mn j m n = 0 shrink in general an ergodic phase towards smaller disorder strengths extending both localized and multifractal phases. In the case when all hopping integrals are fully-correlated (with unit Pearson's coefficient) the localization at any disorder strength is restored [15,18,46] similar to the case of the short-range Anderson model in d = 1, 2 [3,47]. An example of such random matrix models with fully-correlated hopping elements j m =n = C|m − n| −a , decaying with the distance |m − n| as a power-law like in PLRBM, has been suggested in a seminal paper by Burin and Maksimov (BM) [15].…”