Random matrix theory of charge distribution in disordered quantum impurity models

We introduce a bare-bone random matrix quantum impurity model, by hybridizing a localized spinless electronic level with a bath of random fermions in the Gaussian Orthogonal Ensemble (GOE). While stripped out of correlations effects, this model reproduces some salient features of the impurity charge distribution obtained in previous works on interacting disordered impurity models. Computing by numerical sampling the impurity charge distribution in our model, we find a crossover from a Gaussian distribution (centered on half a charge unit) at large hybridization, to a bimodal distribution (centered both on zero and full occupations of the charge) at small hybridization. In the bimodal regime, a universal $(-3/2)$ power-law is also observed. All these findings are very well accounted for by an analytic surmise computed with a single random electron level in the bath. We also derive an exact functional integral for the general probability distribution function of eigenvalues and eigenstates, that formally captures the statistical behavior of our model for any number $N$ of fermionic orbitals in the bath. In the Gaussian regime and in the limit $N\to\infty$, we are able to solve exactly the random matrix theory (RMT) for the charge distribution, obtaining perfect agreement with the numerics. Our results could be tested experimentally in mesoscopic devices, for instance by coupling a small quantum dot to a chaotic electronic reservoir, and using a quantum point contact as local charge sensor for the quantum dot occupation.