Sine-Gordon mean field theory of a Coulomb gas

Sine-Gordon field theory is used to investigate the phase diagram of a neutral Coulomb gas. A variational mean-field free energy is constructed and the corresponding phase diagrams in two and three dimensions are obtained. When analyzed in terms of chemical potential, the sine-Gordon theory predicts the phase diagram topologically identical to the Monte Carlo simulations and a recently developed Debye-H$\mathrm{u}¨$ckel-Bjerrum theory. In two dimensions we find that the infinite-order Kosterlitz-Thouless line terminates in a tricritical point, after which the metal-insulator transition becomes first order. However, when the transformation from the chemical potential to the density is made the whole insulating phase is mapped onto zero density.