Relations Among Two Methods for Computing the Partition Function of the Two-Dimensional One-Component Plasma

TL;DR

This paper explores the relationship between two methods for computing the partition function of the 2D one-component plasma, revealing connections and properties of expansion coefficients for the Vandermonde determinant power.

Contribution

It establishes a link between the Jack polynomial expansion and the fermionic chain mapping for the 2dOCP partition function, enhancing understanding of their equivalence.

Findings

Derived the connection between the two computational methods.

Explored properties of the expansion coefficients for the Vandermonde power.

Provided insights into the structure of the partition function for even coupling constants.

Abstract

The two-dimensional one-component plasma ---2dOCP--- is a system composed by $n$ mobile particles with charge $q$ over a neutralizing background in a two-dimensional surface. The Boltzmann factor of this system, at temperature $T$, takes the form of a Vandermonde determinant to the power $\Gamma = q^2/(2\pi\varepsilon k_BT)$, where $\Gamma$ is the coupling constant of this Coulomb system. The partition function of the model has been computed exactly for the even values of the coupling constant $\Gamma$, and a finite number of particles $n$, by two means: 1) by recognizing that the Boltzmann factor is the square of a Jack polynomial and expanding it in an appropriate monomial base, and 2) by mapping the system onto a 1-dimensional chain of interacting fermions. In this work the connection among the two methods is derived, and some properties of the expansion coefficients for the power of the Vandermonde determinant are explored.