XY model

The XY model is the infinite-state limit of the clock model: each spin is a unit vector with a continuous angle and the global symmetry is $O(2)$.

A Villain model replaces the cosine Boltzmann factor of the XY model by a periodic Gaussian weight. The Poisson summation formula makes its bond weight equal to ${\sum }_{p\in \mathbb{Z}}\exp [-\beta (\psi -2\pi p{)}^2/2]$. The periodic Gaussian retains integer winding defects and permits an explicit separation into spin waves and vortices in two dimensions. Its replacement of the cosine model is an approximation; subsequent transformations within the Villain model can be exact.

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.

Integrating every angle in the bond Fourier series of the Villain model imposes the Kronecker constraint ${\sum }_{\mu }[n_{x,\mu }-n_{x-\hat{\mu },\mu }]=0$. Thus its dual variables are divergence-free integer currents. On a planar lattice such currents are rotated gradients of integer heights on faces. Noncontractible current sectors must additionally be retained on a torus.

On a finite simply connected planar graph, fix the height of the exterior face to zero. Every divergence-free integer bond current is uniquely the oriented difference of the two neighboring face heights. The positive Graph Laplacian $L$ on the remaining faces gives its energy. Applying the Poisson summation formula to these heights and completing the square givesThe Gaussian prefactor $Z_{\mathrm{sw}}$ equals the spin wave determinant of the original graph, up to angle zero-mode normalization. One way to verify this is that the matrix-tree theorem equates the spanning-tree counts of a planar graph and its dual. Boundaries can admit nonzero total vortex charge; neutrality is an additional periodic or finite-energy condition.