Thermodynamic Properties of the Two-Dimensional Coulomb Gas in the Low-Density Limit

Abstract

The model under consideration is the two-dimensional Coulomb gas of $\pm$ charged hard disks with diameter $\sigma$. For the case of pointlike charges $(\sigma =0)$, the system is stable against collapse of positive-negative pairs of charges in the range of inverse temperatures $0\le \beta <2$, where its full thermodynamics was obtained exactly [L. {Š}amaj and I. Trav{ě}nec, {\it J. Stat. Phys.} {\bf 101}:713 (2000)]. In the present work, we derive the leading correction to the exact thermodynamics of pointlike charges due to the presence of the hard core $\sigma$ (appearing in the dimensionless combination $n{\sigma }^2$, $n$ is the particle density). This permits us to extend the treatment to the interval $2\le \beta <3$ (the Kosterlitz-Thouless phase transition takes place at $\beta =4$). The results, which are exact in the low-density limit $n{\sigma }^2\to 0$, reproduce correctly the singularities of thermodynamic quantities at the collapse point $\beta =2$ and agree very well with Monte-Carlo simulations.

18 pages, 3 figures