ar X iv : c on d - m at / 9 50 50 15 v 1 3 M ay 1 99 5 CORRELATIONS IN TWO - COMPONENT LOG - GAS SYSTEMS

A systematic study of the properties of particle and charge correlation functions in the two-dimensional Coulomb gas confined to a one-dimensional domain is under-taken. Two versions of this system are considered: one in which the positive and negative charges are constrained to alternate in sign along the line, and the other where there is no charge ordering constraint. Both systems undergo a zero-density Kosterlitz-Thouless type transition as the dimensionless coupling Γ := q 2 /kT is varied through Γ = 2. In the charge ordered system we use a perturbation technique to establish an O(1/r 4) decay of the two-body correlations in the high temperature limit. For Γ → 2 + , the low-fugacity expansion of the asymptotic charge-charge correlation can be resummed to all orders in the fugacity. The resummation leads to the Kosterlitz renormalization equations. In the system without charge ordering the two-body correlations exhibit an O(1/r 2) decay in the high temperature limit, with a universal amplitude for the charge-charge correlation which is associated with the state being conductive. Low fugacity expansions establish an O(1/r Γ) decay of the two body correlations for 2 < Γ < 4, and an O(1/r 4) decay for Γ > 4. For both systems we derive sum rules which relate the long wavelength behaviour of the Fourier transform of the charge correlations to the dipole carried by the screening cloud surrounding two opposite internal charges. These sum rules are checked for specific solvable models. Our predictions for the Kosterlitz-Thouless transition and the large-distance behaviours of the correlations should be valid at low densities. At higher densities, both systems might undergo a first-order liquid-gas transition analogous to the two-dimensional case.