Universal correlations for deterministic plus random Hamiltonians

TL;DR

This paper derives an explicit formula for the average energy level correlation in disordered systems with deterministic and random Hamiltonians, revealing a universal short-distance behavior independent of system specifics.

Contribution

It provides a universal correlation function for large matrices with combined deterministic and random Hamiltonians, valid for any unperturbed Hamiltonian and random potential distribution.

Findings

Correlation function explicitly determined for large matrices.

Short-distance behavior is universal, independent of system details.

Compact representation of the correlation function obtained.

Abstract

We consider the (smoothed) average correlation between the density of energy levels of a disordered system, in which the Hamiltonian is equal to the sum of a deterministic H0 and of a random potential $\varphi$. Remarkably, this correlation function may be explicitly determined in the limit of large matrices, for any unperturbed H0 and for a class of probability distribution P$(\varphi)$ of the random potential. We find a compact representation of the correlation function. From this representation one obtains readily the short distance behavior, which has been conjectured in various contexts to be universal. Indeed we find that it is totally independent of both H0 and P($\varphi$).