Ground-state energy of noninteracting fermions with a random energy spectrum

¹ Institut für Physik, Universität Oldenburg - D-26111 Oldenburg, Germany
² Univ. Paris-Sud, CNRS, LPTMS, UMR 8626 - Orsay F-91405, France

Received: 29 August 2018
Accepted: 9 November 2018

Abstract

We derive analytically the full distribution of the ground-state energy of K non-interacting fermions in a disordered environment, modelled by a Hamiltonian whose spectrum consists of N i.i.d. random energy levels with distribution (with ε ≥ 0), in the same spirit as the “Random Energy Model”. We show that for each fixed K, the distribution P_{K, N}(E₀) of the ground-state energy E₀ has a universal scaling form in the limit of large N. We compute this universal scaling function and show that it depends only on K and the exponent α characterizing the small ε behaviour of . We compared the analytical predictions with results from numerical simulations. For this purpose we employed a sophisticated importance-sampling algorithm that allowed us to obtain the distributions over a large range of the support down to probabilities as small as . We found asymptotically a very good agreement between analytical predictions and numerical results.