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

Abstract

Motivated by the constrained many-body dynamics, the stability of the localization-delocalization properties to the inclusion of the soft constraints is addressed in random matrix models. These constraints are modeled by correlations in long-ranged hopping with Pearson correlation coefficient different from zero or unity. Counterintuitive robustness of delocalized phases, both ergodic and (multi)fractal, in these models is numerically observed and confirmed by the analytical calculations. First, matrix inversion trick is used to uncover the origin of such robustness. Next, to characterize delocalized phases a method of eigenstate calculation, sensitive to correlations in long-ranged hopping terms, is developed for random matrix models and approved by numerical calculations and previous analytical ansatz. The effect of the robustness of states in the bulk of the spectrum the inclusion of to soft constraints is generally discussed for single-particle and many-body systems.

Renormalization to localization without a small parameter

“…An actual necessary and sufficient condition has to deal with meaning of relative fluctuations of f (r) in the Ri−vicinity of the typical distance between resonances on the ith step. This fact actually allows the same RG treatment not only for the ERM models with smooth deterministic f (r), but also for the models with hopping terms of the form tij = (1 + hij)f (rij) with hij being a random variable with zero mean and relatively narrow distribution function, f 2 (rij) h 2 ij r −2d (similar to [37]). of states N i ε (r) separated by a distance r, R i < r < R, from a certain state with energy ε and resonant to it can be written as…”

“…After determining the validity range of RG, we check numerically the fact about the density of states stated in the Appendix A. According to the RG approach, the function ν R (ε) barely depends on the cutoff radius R, a < d: (37) and it converges to a non-singular function. Both these statements are clearly seen from Fig.…”

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

“…Multifractal statistics appears at the Anderson localization transition for single-particle lattice systems [17,[65][66][67][68][69][70][71]. In addition, recent examples have reported (multi)fractal phases extend-ing over a whole range of parameters [72][73][74][75][76][77][78][79][80][81][82][83][84][85][86]. Multifractal wavefunctions have been found for some quantum maps [68,70,87,88].…”

Dynamical phases in a ``multifractal'' Rosenzweig-Porter model

“…In order to find the scaling exponent τ * one should use the knowledge about the scaling of the total bandwidth E BW ∼ N 1−∆ . It is well-known [25,47,53] that in the non-ergodic (e.g. localized and multifractal) phase E BW = W /2 ∼ N 0 , so that ∆ = 1.…”