Analysis of a new set of renormalization equations for the Kosterlitz-Thouless transition

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Abstract
A new and more direct derivation is given of an extended set of renormalization equations for the two-dimensional Coulomb gas. Our equations are equivalent to those proposed by Minnhagen, but are cast in the form of ordinary renormalization equations. It turns out that the difference from the usual renormalization equations is due to the fact that the boundary conditions to be used are of a mixed type: two of them appear as initial conditions; the remaining ones as conditions at the end of a trajectory. The striking new features found by Minnhagen, such as the nonuniversal jump in the inverse dielectric constant and the presence of a first-order transition at higher densities, are a direct result of the possibility (due to the mixed boundary conditions) of having crossing trajectories. These crossing trajectories were also found by Minnhagen et al., but only in the high-temperature region as multiple solutions of their integral equations. Finally, we show that the rather ad hoc criterion adopted by Minnhagen and co-workers for selecting the equilibrium solution indeed corresponds to the one with the smaller free energy, at least in the cases considered by these authors.