Phase transitions in gases with generalized charges interacting through a logarithmic law I. $d=2$, Isotropic case

The renormalization group method is used to consider the dependence of the critical properties of two-dimensional Coulomb gases with generalized charges on the internal symmetry. It is shown that only gases with definite discrete symmetries are renormalizable. A new method is proposed for obtaining the terms of higher order in the renormalization group equations. The nature of the singularities, the critical exponents, and the long-wavelength behavior of the correlation functions in the low-temperature phase for various renormalizable gases are found.