Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 50 / 3

Replace each XY model bond weight by its periodic Gaussian approximation. With angles integrated using $d\theta /(2\pi )$, the Villain model iswhere $N_b$ is the number of bonds. Changing the angle-measure normalization multiplies $Z_V$ by an explicit constant only. The Poisson summation formula applied to one bond gives the equivalent winding representationThus the same Villain model can be written $Z_V=\int {\prod }_{x}d{\theta }_x/(2\pi ){\sum }_{\{p_e\}}\exp [-\beta {\sum }_e({\Delta }_e\theta -2\pi p_e{)}^2/2]$. The coefficient $1/\sqrt{2\pi \beta }$ is the PDF's value; the converted TeX incorrectly places $\beta$ outside the square root. The cosine-to-Gaussian replacement is a large-$\beta$ approximation, but the rest of the calculation treats $Z_V$ as the exact model for every positive $\beta$.

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.

For a periodic two-dimensional lattice the vortex charge neutrality constraint is $\boxed{\sum _x{q}_x=0}$. In the winding form of the Villain model, $q$ is the oriented plaquette curl of the integer bond variable $p$, up to an orientation sign. Summing all plaquette curls cancels each bond against its neighboring plaquette. This proves neutrality directly. Equivalently, restoring the common dual-height mode modulo its integer shift gives ${\int }_0^{1}d{\varphi }_0{e}^{2\pi i{\varphi }_0\sum q}$, which vanishes unless the total charge is zero; only then is the inverse lattice Laplacian well-defined on the charge source.

On an infinite plane, the same condition is required for a localized finite-energy configuration with constant phase at infinity. Net charge $Q$ produces $\theta \sim Q\arg x$ far away, so its exterior action iswhich diverges unless $Q=0$. With an open boundary the total charge instead equals the boundary winding, and the finite planar duality in part (b) can sum nonneutral configurations. Thus neutrality follows from periodicity or the stated finite-energy boundary condition, not from arbitrary boundaries. In three dimensions the analogous local constraint is that vortex lines cannot end in the bulk.