Ground state energy of noninteracting fermions with a random energy spectrum

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 $p(\epsilon )$ (with $\epsilon \ge 0$), in the same spirit as the `Random Energy Model’. We show that for each fixed $K$, the distribution $P_{K,N}(E_0)$ of the ground-state energy $E_0$ 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 $\alpha$ characterizing the small $\epsilon$ behaviour of $p(\epsilon )\sim {\epsilon }^{\alpha }$. 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 ${10}^{-160}$. We found asymptotically a very good agreement between analytical predictions and numerical results.