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

## TL;DR

This paper investigates how soft constraints affect localization-delocalization transitions in long-range random matrix models, revealing unexpected robustness of delocalized phases through numerical and analytical methods.

## Contribution

It introduces a new analytical approach to understand the robustness of delocalized phases under soft constraints in long-range random models.

## Key findings

- Delocalized phases are robust to correlations in long-range hopping.
- Analytical methods confirm numerical observations of robustness.
- Eigenstate calculation method is developed for correlated random matrices.

## 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.