A Trickiness of the High-Temperature Limit for Number Density Correlation Functions in Classical Coulomb Fluids

The Debye-H\"uckel theory describes rigorously the thermal equilibrium of classical Coulomb fluids in the high-temperature $\beta\to 0$ regime ($\beta$ denotes the inverse temperature). It is generally believed that the Debye-H\"uckel theory and the systematic high-temperature expansion provide an adequate description also in the region of small {\em strictly positive} values of $\beta>0$. This hypothesis is tested in the present paper on a two-dimensional Coulomb gas of pointlike $+/-$ unit charges interacting via a logarithmic potential which is equivalent to an integrable sine-Gordon field model. In particular, we apply a form factor method to obtain the exact asymptotic large-distance behavior of particle correlation functions, considered in the charge and number density combinations. We first determine the general forms of the leading and subleading asymptotic terms at strictly positive $\beta>0$ and then evaluate their high-temperature $\beta\to 0$ forms. In the case of the {\em charge} correlation function, the leading asymptotic term at a strictly positive $\beta>0$ is also the leading one in the high-temperature $\beta\to 0$ regime. On the contrary, the $\beta\to 0$ behavior of the {\em number density} correlation function is accompanied by an interference between the first two asymptotic terms. Consequently, the large-distance behavior of this function exhibits a discontinuity when going from strictly positive values of $\beta>0$ to the Debye-H\"uckel limit $\beta\to 0$. This is the crucial conclusion of the paper: the large-distance asymptotics and the high-temperature limit do not commute for the density correlation function of the two-dimensional Coulomb gas.