Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions

TL;DR

This paper connects the density matrix of the impenetrable Bose gas with random matrix averages, deriving large $n$ asymptotics to analyze low-lying states and occupations in boundary conditions.

Contribution

It introduces a novel approach using random matrix theory and Selberg integrals to analyze the Bose gas density matrix in boundary conditions.

Findings

Asymptotic form of the density matrix for large $n$

Effective single particle states scale as $ ootN$

Method applies to Dirichlet and Neumann boundary conditions

Abstract

The density matrix for the impenetrable Bose gas in Dirichlet and Neumann boundary conditions can be written in terms of $<\prod_{l=1}^n| \cos\phi_1-\cos\theta_l| |\cos\phi_2-\cos\theta_l|>$, where the average is with respect to the eigenvalue probability density function for random unitary matrices from the classical groups $Sp(n)$ and $O^+(2n)$ respectively. In the large $n$ limit log-gas considerations imply that the average factorizes into the product of averages of the form $<\prod_{l=1}^n|\cos\phi-\cos\theta_l>$. By changing variables this average in turn is a special case of the function of $t$ obtained by averaging $\prod_{l=1}^n| t-x_l|^{2q}$ over the Jacobi unitary ensemble from random matrix theory. The latter task is accomplished by a duality formula from the theory of Selberg correlation integrals, and the large $n$ asymptotic form is obtained. The corresponding large $n$ asymptotic form of the density matrix is used, via the exact solution of a particular integral equation, to compute the asymptotic form of the low lying effective single particle states and their occupations, which are proportional to $\sqrt{N}$.