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The physical angle takes values on a circle, so it is defined modulo $2\pi$. Around a closed curve enclosing a phase vortex,The integer winding number cannot change through a smooth single-valued deformation unless a defect crosses the curve. Thus a globally smooth Gaussian spin wave field omits whole topological sectors of the XY model.

Outside a vortex core of radius $a$, an isolated charge-$n$ phase vortex has $\theta =n\phi$ and $|\nabla \theta |=|n|/r$. Its dimensionless energy in a sample of size $L$ isThere are of order $(L/a{)}^2$ positions for its core, so the positional entropy contributes $2\log (L/a)$. The competition for unit charge is therefore $(\pi \bar{K}-2)\log (L/a)$, apart from core terms. Opposite charges have a logarithmic attraction and form a vortex Coulomb gas.

At low temperature, vortex-antivortex binding leaves no free charges on large scales. Spin waves still destroy true long-range order, but the correlation function decays algebraically with the renormalized phase stiffness. At the Berezinskii–Kosterlitz–Thouless transition, pairs unbind; free vortices screen the interaction and produce a finite correlation length and exponential decay above the transition. The energy-entropy estimate identifies the threshold stiffness $K_R=2/\pi$, where $K_R$ is the long-distance dimensionless stiffness, not necessarily the bare $\bar{K}$. It agrees with the universal stiffness jump and the limiting algebraic exponent $1/(2\pi K_R)=1/4$. The two-dimensional transition is vortex unbinding between quasi-long-range order and disorder, rather than the onset of nonzero spontaneous magnetization.

Angle integration gives a dual representation in any lattice dimension. Discrete summation by parts rewrites the phase asEach normalized angle integral is a Kronecker delta. This proves the integer-current representation of the Villain modelThe PDF does not specify dimension or boundaries. The point-vortex vortex Coulomb gas interpretation below is two-dimensional. In three dimensions the defects are vortex lines, and the dual constrained-current representation still holds but is not a gas of point charges with logarithmic interactions.

First take a connected simply connected planar lattice, with free outer angles, so no torus winding sectors are silently discarded. Assign an integer height $\ell$ to each dual face, fixing the exterior face height to zero. A divergence-free integer current is exactly the oriented height difference on the two faces adjoining its bond. To see existence, integrate the rotated current along a dual path: the change around a closed dual path is the total divergence inside, hence zero. Integer currents give integer heights, and fixing the exterior removes their constant-shift ambiguity. Thereforewhere $L$ is the positive dual Graph Laplacian with the exterior height fixed. This gives the dual integer-height representation of the planar Villain model.

Apply the supplied Poisson summation formula separately to every interior height. For $N_f$ interior faces,Complete the Gaussian square, or use its characteristic function, to evaluate the integral asThus the smooth fluctuations and integer defects separate:The Gaussian factor is $Z_{\mathrm{sw}}=(2\pi \beta {)}^{-(N_b-N_f)/2}/\sqrt{\det L}$. It really is the spin wave factor of the original lattice: the planar Euler identity gives $N_b-N_f=N_x-1$, and the matrix-tree theorem identifies $\det L$ with the spanning-tree count of the original graph. Its reduced primal Laplacian has the same determinant, so the noncompact Gaussian angle integral with one angle fixed has exactly this prefactor. The compact common-angle mode has been normalized to one. The remaining $q$ variables are integer vortex windings on the faces, giving the vortex Coulomb gas.

For periodic boundaries, $L$ has a zero mode, and there are also two noncontractible current sectors. Neutral charges use the inverse $G$ on the mean-zero subspace. An integer dual height may have twists $w_x,w_y\in \mathbb{Z}$ across periods $L_x,L_y$. Subtracting $w_{x}x/L_x+w_{y}y/L_y$ makes the remaining height periodic; the gradient energy gains $(L_y{w}_x^2/L_x+L_x{w}_y^2/L_y)/(2\beta )$. Its height Poisson summation supplies the phase $\exp [2\pi i(w_x{P}_x/L_x+w_y{P}_y/L_y)]$, where $P_x=\sum q_{x}x$ and $P_y=\sum q_{x}y$. A second Poisson summation in $w_x,w_y$ yields the positive harmonic-sector factor in periodic Villain dualityUp to charge-independent normalization, the exact periodic result is $Z_V=Z_{\mathrm{sw}}{\sum }_{\sum q=0}{e}^{-2{\pi }^2\beta q^{T}Gq}\mathcal{W}(q)$. Smooth spin waves still factor out, but the finite-volume global winding sectors couple to vortex dipole moments. The familiar neutral bulk formula suppresses this boundary information; it must not be asserted as an exact finite-torus formula without specifying those sectors.

At long distances in two dimensions the lattice Green function solves $-{\nabla }^{2}G=\delta$ and has $G(r)=-(2\pi {)}^{-1}\log (r/a)+\text{constant}$. Indeed the flux of $-\nabla G$ through a circle is one, fixing that coefficient. For well-separated vortices at $r_i$ with integer charges $q_i$, the vortex Coulomb gas action becomeswith the self-energy convention, boundary terms and any global sectors retained separately. The factor two in the pair sum comes from the two ordered terms in $q^{T}Gq$. Opposite charges have pair action $+2\pi \beta |q_i{q}_j|\log (r/a)$, so separating them costs energy and they attract. Like charges have a negative logarithmic pair coefficient and repel. The ultraviolet lattice regularizes the core, while multiply charged defects have a quadratic self-energy cost. A separated unit vortex-antivortex pair costs $2{\mu }_{\mathrm{core}}+2\pi \beta \log (r/a)$, the energetic basis for vortex-antivortex binding at low temperature. In three dimensions one instead has interacting closed vortex lines with a three-dimensional Green kernel; the logarithmic point-pair formula is specifically two-dimensional.