Particle Diagrams and Statistics of Many-Body Random Potentials

TL;DR

This paper introduces a diagrammatic method to compute statistical properties of many-body random potentials, providing an efficient alternative to traditional techniques and revealing the transition of level density moments from semi-circular to Gaussian as system parameters vary.

Contribution

The paper develops a Feynman-like diagram approach for calculating moments of level densities in many-body random ensembles, applicable to bosons and fermions, and elucidates the transition between different spectral density regimes.

Findings

Method efficiently identifies relevant terms in the large system limit.

Results show a transition from semi-circular to Gaussian level density moments.

Diagram relevance increases with system size, starting at specific interaction points.

Abstract

We present a method using Feynman-like diagrams to calculate the statistical properties of random many-body potentials. This method provides a promising alternative to existing techniques typically applied to this class of problems, such as the method of supersymmetry and the eigenvector expansion technique pioneered in [1]. We use it here to calculate the fourth, sixth and eighth moments of the average level density for systems with $m$ bosons or fermions that interact through a random $k$-body Hermitian potential ($k \le m$); the ensemble of such potentials with a Gaussian weight is known as the embedded Gaussian Unitary Ensemble (eGUE) [2]. Our results apply in the limit where the number $l$ of available single-particle states is taken to infinity. A key advantage of the method is that it provides an efficient way to identify only those expressions which will stay relevant in this limit. It also provides a general argument for why these terms have to be the same for bosons and fermions. The moments are obtained as sums over ratios of binomial expressions, with a transition from moments associated to a semi-circular level density for $m < 2k$ to Gaussian moments in the dilute limit $k \ll m \ll l$. Regarding the form of this transition, we see that as $m$ is increased, more and more diagrams become relevant, with new contributions starting from each of the points $m = 2k, 3k, \ldots, nk$ for the $2n$-th moment.