Coulomb gas

The setup starts with considering ${\displaystyle N}$ charged particles in ${\displaystyle \mathbb {R} ^{d}}$ with positions ${\displaystyle \mathbf {r} _{i}}$ and charges ${\displaystyle q_{i}}$. From electrostatics, the pairwise potential energy between particles labelled by indices ${\displaystyle i,j}$ is (up to scale factor)

$${\displaystyle V_{ij}=q_{i}q_{j}g(|\mathbf {r} _{i}-\mathbf {r} _{j}|),}$$

where ${\displaystyle g(x)}$ is the Coulomb kernel or Green's function of the Laplace equation in ${\displaystyle d}$ dimensions,[2] so

$${\displaystyle {\begin{aligned}g(x)={\begin{cases}-\log |x|&{\text{ if }}d=2,\\{\frac {1}{(d-2)|x|^{d-2}}}&{\text{ if }}d>2.\end{cases}}\end{aligned}}}$$

The free energy due to these interactions is then (proportional to) ${\displaystyle F=\sum _{i\neq j}V_{ij}}$, and the partition function is given by integrating over different configurations, that is, the positions of the charged particles.

The two-dimensional Coulomb gas can be used as a framework for describing fields in minimal models. This comes from the similarity of the two-point correlation function of the free boson ${\displaystyle \varphi }$,

$${\displaystyle \langle \varphi (z,{\bar {z}})\varphi (w,{\bar {w}})\rangle =-\log |z-w|^{2}}$$

to the electric potential energy between two unit charges in two dimensions.[3]

• Sine-Gordon equation
• XY model