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

Abstract

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