Levinson's theorem and scattering phase shift contributions to the partition function of interacting gases in two dimensions

TL;DR

This paper investigates the role of scattering states in the thermodynamics of a 2D plasma, proving Levinson's theorem, and analyzing how scattering and bound states influence the partition function and virial coefficients.

Contribution

It provides a statistical mechanical proof of Levinson's theorem in two dimensions and clarifies the impact of scattering states on the thermodynamics of 2D gases with attractive interactions.

Findings

Scattering states contribute significantly to the 2D plasma partition function.

Proper accounting of scattering removes thermodynamic singularities caused by bound states.

The bound-state contribution vanishes logarithmically as binding energy approaches zero.

Abstract

We consider scattering state contributions to the partition function of a two-dimensional (2D) plasma in addition to the bound-state sum. A partition function continuity requirement is used to provide a statistical mechanical heuristic proof of Levinson's theorem in two dimensions. We show that a proper account of scattering eliminates singularities in thermodynamic properties of the nonideal 2D gas caused by the emergence of additional bound states as the strength of an attractive potential is increased. The bound-state contribution to the partition function of the 2D gas, with a weak short-range attraction between its particles, is found to vanish logarithmically as the binding energy decreases. A consistent treatment of bound and scattering states in a screened Coulomb potential allowed us to calculate the quantum-mechanical second virial coefficient of the dilute 2D electron-hole plasma and to establish the difference between the nearly ideal electron-hole gas in GaAs and the strongly correlated exciton/free-carrier plasma in wide-gap semiconductors such as ZnSe or GaN.