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

The Debye-Hückel theory describes rigorously the thermal equilibrium of classical Coulomb fluids in the high-temperature β → 0 regime (β denotes the inverse temperature). It is generally believed that the Debye-Hückel theory and the systematic high-temperature expansion provide an adequate description also in the region of small strictly positive values of β > 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 β > 0 and then evaluate their high-temperature β → 0 forms. In the case of the charge correlation function, the leading asymptotic term at a strictly positive β > 0 is also the leading one in the hightemperature β → 0 regime. On the contrary, the β → 0 behavior of the 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 β > 0 to the Debye-Hückel limit β → 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.