Eigenvectors under a generic perturbation: non-perturbative results from the random matrix approach

TL;DR

This paper derives non-perturbative analytical results for the statistics of eigenvectors of a Hamiltonian perturbed by a GUE random matrix, applicable to arbitrary deterministic or random initial Hamiltonians, validated by simulations.

Contribution

It introduces a supersymmetry-based method to analyze eigenvector statistics under generic perturbations, extending to random initial Hamiltonians like the Rosenzweig-Porter model.

Findings

Analytical formulas for eigenvector statistics under GUE perturbations

Validation of predictions through numerical simulations

Extension to random initial Hamiltonians like the Rosenzweig-Porter model

Abstract

We consider eigenvectors of the Hamiltonian $H_0$ perturbed by a generic perturbation $V$ modelled by a random matrix from the Gaussian Unitary Ensemble (GUE). Using the supersymmetry approach we derive analytical results for the statistics of the eigenvectors, which are non-perturbative in $V$ and valid for an arbitrary deterministic $H_0$. Further we generalise them to the case of a random $H_0$, focusing, in particular, on the Rosenzweig-Porter model. Our analytical predictions are confirmed by numerical simulations.