Sensitivity of (multi)fractal eigenstates to a perturbation of the Hamiltonian

Abstract

We study the response of an isolated quantum system governed by the Hamiltonian drawn from the Gaussian Rosenzweig-Porter random matrix ensemble to a perturbation controlled by a small parameter. We focus on the density of states, local density of states and the eigenfunction amplitude overlap correlation functions which are calculated exactly using the mapping to the supersymmetric nonlinear sigma model. We show that the susceptibility of eigenfunction fidelity to the parameter of perturbation can be expressed in terms of these correlation functions and is strongly peaked at the localization transition: It is independent of the effective disorder strength in the ergodic phase, grows exponentially with increasing disorder in the fractal phase and decreases exponentially in the localized phase. As a function of the matrix size, the fidelity susceptibility remains constant in the ergodic phase and increases in the fractal and in the localized phases at modestly strong disorder. We show that there is a critical disorder strength inside the insulating phase such that for disorder stronger than the critical the fidelity susceptibility decreases with increasing the system size. The overall behavior is very similar to the one observed numerically in a recent work by Sels and Polkovnikov [Phys. Rev. E 104, 054105 (2021)] for the normalized fidelity susceptibility in a disordered XXZ spin chain.

arXiv comment16 pages, 10 figures

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• Experimental test of the Rosenzweig-Porter model for the transition from Poisson to Gaussian unitary ensemble statistics
• The Rosenzweig Porter model revisited for the three Wigner Dyson symmetry classes
• Cat states carrying long-range correlations in the many-body localized phase
• The Wishart--Rosenzweig--Porter random matrix ensemble
• Density of states correlations in Lévy Rosenzweig-Porter model via supersymmetry approach
• Replica approach to the generalized Rosenzweig-Porter model