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 undertaken. 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 anO(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 anO(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 anO(1/r Γ) decay of the two-body correlations for 2<Γ<4 and anO(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 behavior 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.
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Alastuey, A., Forrester, P.J. Correlations in two-component log-gas systems. J Stat Phys 81, 579–627 (1995). https://doi.org/10.1007/BF02179249
- Kosterlitz-Thouless transition
- log-gas systems
- fugacity expansions
- sum rules