Renormalization to localization without a small parameter

Abstract

We study the wave function localization properties in a d-dimensional model of randomly spaced particles with isotropic hopping potential depending solely on Euclidean interparticle distances. Due to generality of this model usually called Euclidean random matrix model, it arises naturally in various physical contexts such as studies of vibrational modes, artificial atomic systems, liquids and glasses, ultracold gases and photon localization phenomena. We generalize the known Burin-Levitov renormalization group approach, formulate universal conditions sufficient for localization in such models and inspect a striking equivalence of the wave function spatial decay between Euclidean random matrices and translationinvariant long-range lattice models with a diagonal disorder.

Single-particle localization in a two-dimensional Rydberg spin system

“…For example, one could try to extend the duality found for one-dimensional systems in [33] to the two-dimensional case. One could apply the analytical treatment based on a renormalization procedure introduced in [27] to the present problem by including the blockade constraint. Self consistent perturbative methods such as the locator expansion employed in [41] could be applied to predict spectral and eigenstate properties in the two-dimensional case.…”

“…The problem of localization in systems with power-law hopping (∝ r −a ) has been studied theoretically by Levitov [8] and others [25][26][27] finding full localization for a > d, where a is the power-law exponent of the hopping and d the system dimension. In this case the eigenstates show power-law tails which means that they are not Anderson localized in the strict sense, which is signaled by exponential localization of the eigenstates.…”

Emergent fractal phase in energy stratified random models

“…and shows the immediate emergence of eigenstate fractality of mid-spectrum states as soon as the number N β of energy stratified states becomes extensive, β > 0 (unlike the case of the Burin-Maksimov model [62][63][64]). This is the main result of our paper.…”

“…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.…”

Non-ergodic delocalized phase with Poisson level statistics

“…However, the latter delocalization property at small powers a < d of power-law longrange systems might fail in the dipolar systems, where all the dipoles are not randomly oriented [44,45], but aligned, e.g., by an electric field [28,[47][48][49][50][51]. In the literature this is called the Burin-Maksimov model by the names of the authors of [47] first suggested it.…”