Vortex Coulomb gas ($Z_{\mathrm{v}}={\sum }_{\{q\}}{e}^{-2{\pi }^2\beta {\sum }_{x,y}{q}_{x}G(x-y)q_y}$)

In two dimensions, the integer defects $q_x$ on dual faces are vortices of winding $2\pi q_x$. Here $G$ is the inverse positive lattice Laplacian with its boundary or zero-mode convention specified. At separations large compared with the lattice spacing, $G(r)=-(2\pi {)}^{-1}\log (r/a)+\text{constant}$, yielding the pair action $-2\pi \beta {\sum }_{i<j}{q}_i{q}_j\log (r_{ij}/a)$ and core self-energies. Like charges repel and opposite charges attract. With periodic boundaries the total charge is zero and the inverse is taken on the mean-zero subspace; harmonic winding sectors require their own factors.

On a rectangular torus, an integer height for the integer-current representation of the Villain model may have twists $w_x,w_y\in \mathbb{Z}$. Subtract $w_{x}x/L_x+w_{y}y/L_y$ to get a periodic real height. Its Gaussian energy splits into the ordinary gradient energy and $(L_y{w}_x^2/L_x+L_x{w}_y^2/L_y)/(2\beta )$. Height Poisson summation supplies a phase with dipole moments $P_x=\sum q_{x}x$ and $P_y=\sum q_{x}y$. A further Poisson summation of the twists gives, up to a charge-independent factor,Here $G$ is the inverse lattice Laplacian on the mean-zero subspace. This keeps the smooth spin wave determinant separate, while correctly retaining the coupling between vortex dipole moments and global winding sectors at finite volume. Dropping those sectors without stating boundary conditions is not an exact torus duality.

For the Villain model on a periodic two-dimensional lattice, the vortex charge is the oriented curl of its integer bond variable. Summing over all faces cancels every bond twice, proving neutrality. In an infinite plane, a configuration with net winding $Q$ has far-field spin wave action $\pi \beta Q^2\log (L/a)$, so finite-energy localized configurations must have $Q=0$. At an open boundary the total winding instead equals the boundary circulation; neutrality cannot be imposed without a boundary condition.