Finite-Size Corrections for Coulomb Systems in the Debye-Huckel Regime

TL;DR

This paper develops a method using sine-Gordon field theory to compute universal finite-size corrections in the free energy of Coulomb systems within the Debye-Hückel regime, exemplified by systems in disk and annulus geometries.

Contribution

It introduces a novel approach to derive finite-size corrections for Coulomb systems at the Debye-Hückel level using spectral analysis of the Laplace operator.

Findings

Derived explicit grand potential expansions for disk and annulus geometries.

Confirmed the universality of the logarithmic correction coefficient.

Provided a practical method applicable to various geometries in Coulomb systems.

Abstract

It has been argued that for a finite two-dimensional classical Coulomb system of characteristic size $R$, in its conducting phase, as $R\to \infty $ the total free energy (times the inverse temperature $\beta$) admits an expansion of the form: $\beta F=AR^{2}+BR+{1/6}\chi \ln R,$ where $\chi $ is the Euler characteristic of the manifold where the system lives. The first two terms represent the bulk free energy and the surface free energy respectively. The coefficients $A$ and $B$ are non-universal but the coefficient of $\ln R$ is universal: it does not depend on the detail of the microscopic constitution of the system (particle densities, temperature, etc...). By doing the usual Legendre transform this universal finite-size correction is also present in the grand potential. The explicit form of the expansion has been checked for some exactly solvable models for a special value of the coulombic coupling. In this paper we present a method to obtain these finite-size corrections at the Debye--H\"uckel regime. It is based on the sine-Gordon field theory to find an expression for the grand canonical partition function in terms of the spectrum of the Laplace operator. As examples we find explicitly the grand potential expansion for a Coulomb system confined in a disk and in an annulus with ideal conductor walls.