The Kosterlitz-Thouless transitions by the mayer expansion finiteness of the perturbations

Abstract

The Kosterlitz-Thouless transitions in 2D XY and Coulomb gas models are discussed by the Mayer expansion. After transforming the 2D XY model into a (generalzed) 2D Coulomb gas model by the duality transformation, it is shown that the free energy (pressure) and the two point correlation function 〈cos( θ 0− θ ζ )〉 are expressed as Σa N ( β) and exp[( 2 β ) C 0(ζ) + Σb N(β:ζ)] , respectively, for large inverse temperature β > β c , where { a N ( β)} are the usual virial coefficients and { a N , b N } are the contributions from N-electron system. Moreover C 0( ζ) ≅ − (2 π) −1 log(| ζ| + 1), | a N ( β)| ≤ C 1( N) exp[− βK 3 N] and | b N ( β: N) ≤ C 2( N) exp[− βK 4 N] | C 0( ζ)| ( K i > 0). By comparing this system with the dipole gas system in which these series converge absolutely, it is conjectured that these series converge absolutely for large β.