Painlev\'e transcendent evaluations of finite system density matrices for 1d impenetrable Bosons

TL;DR

This paper derives exact formulas for the density matrix of a 1D impenetrable Bose gas using Painlevé transcendents, aiding the understanding of Bose-Einstein condensation in finite systems.

Contribution

It provides new Painlevé transcendent representations and recurrence relations for the density matrix of impenetrable Bose gases in finite systems with various boundary conditions.

Findings

Density matrix expressed via Painlevé VI transcendent on a circle

Recurrence relation for density matrix in particle number

Determinant and random matrix average forms for trapped systems

Abstract

The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlev\'e VI transcendent in $\sigma$-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlev\'e V and VI. We discuss how our results can be used to compute the ground state occupations.