Classical XY model

The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a model of statistical mechanics. It is the special case of the n-vector model for n = 2.

Definition

Given a D-dimensional lattice Λ, per each lattice site $j\in\Lambda$ there is a two-dimensional, unit-length vector $\mathbf{s}_j=(\cos \theta_j, \sin\theta_j)$. A spin configuration, $\mathbf{s}=(\mathbf{s}_j)_{j\in \Lambda}$ is an assignment of the angle $\theta_j\in (-\pi,\pi]$ per each site j in the lattice.

Given a translation-invariant interaction J_ij = J(i − j) and a point dependent external field $\mathbf{h}_{i}=(h_j,0)$, the configuration energy is

$H(\mathbf{s}) = - \sum_{i\neq j} J_{ij}\; \mathbf{s}_i\cdot\mathbf{s}_j -\sum_{j} \mathbf{h}_j\cdot \mathbf{s}_j =- \sum_{i\neq j} J_{ij}\; \cos(\theta_i-\theta_j) -\sum_{j} h_j\cos\theta_j$

The case in which J_i,j = 0 except for ij nearest neighbor is called 'nearest neighbor case'.

The configuration probability is given by the Boltzmann distribution with inverse temperature $\beta\ge 0$:

$P(\mathbf{s}) ={e^{-\beta H(\mathbf{s})} \over Z} \qquad Z=\int_{[-\pi,\pi]^{\Lambda}}\!\prod_{j\in \Lambda}d\theta_j\;e^{-\beta H(\mathbf{s})} \,.$

where Z is the normalization, or partition function.

General properties

• The existence of the thermodynamic limit for the free energy and spin correlations were proved by Ginibre, extending to this case the Griffiths inequality.

• At high temperature, the spontaneous magnetization vanishes:

$M(\beta):=|\langle \mathbf{s}_i\rangle|=0$

Besides, cluster expansion shows that the spin correlations cluster exponentially fast: for instance

$|\langle \mathbf{s}_i\cdot \mathbf{s}_j\rangle| \le C(\beta)e^{-c(\beta)|i-j|}$

• Using the Griffiths inequality in the formulation of Ginibre, Aizenman and Simon proved that the two point spin correlation of the ferromagnetics XY model in dimension D, coupling J > 0 and inverse temperature β is dominated by (i.e. has upper bound given by) the two point correlation of the ferromagnetic Ising model in dimension D, coupling J > 0and inverse temperature β / 2

$\langle \mathbf{s}_i\cdot \mathbf{s}_j\rangle_{J,2\beta} \le \langle \sigma_i\sigma_j\rangle_{J,\beta}$

Hence the critical β of the XY model cannot be smaller than the double of the critical temperature of the Ising model

$\beta_c^{XY}\ge 2\beta_c^{\rm Is}$

One dimension

As in any 'nearest-neighbor' n-vector model with free boundary conditions, if the external field is zero, there exists a simple exact solution.

In the nearest-neighbor, free boundary conditions case, the Hamiltonian is

$H(\mathbf{s}) = - J [\cos(\theta_1-\theta_2)+\cdots+\cos(\theta_{L-1}-\theta_L)]$

therefore the partition function factorizes under the change of coordinates

$\theta_j=\theta_j'+\theta_{j-1} \qquad j\ge 2$

That gives

$\begin{align} Z&=\int_{-\pi}^{\pi}d\theta_1\cdots d\theta_L\; e^{\beta J \cos(\theta_1-\theta_2)}\cdots e^{\beta J \cos(\theta_{L-1}-\theta_L)} =2\pi \prod_{j=2}^L\int_{-\pi}^{\pi}d\theta'_j\;e^{\beta J \cos\theta'_j}= \\ &=2\pi\left[\int_{-\pi}^{\pi}d\theta'_j\;e^{\beta J \cos\theta'_j}\right]^{L-1} \end{align}$

Finally

$f(\beta, 0)=-\frac{1}{\beta}\ln \int_{-\pi}^{\pi}d\theta'_j\;e^{\beta J \cos\theta'_j}$

The same computation for periodic boundary condition (and still h = 0) requires the transfer matrix formalism.

Two Dimensions

• At low temperature, the spontaneous magnetization remains zero,

$M(\beta):=|\langle \mathbf{s}_i\rangle|=0$

but Fröhlich and Spencer proved that the decay of the correlations is only power law.

$|\langle \mathbf{s}_i\cdot \mathbf{s}_j\rangle| \ge\frac{C(\beta)}{1+|i-j|^{\eta(\beta)}}$

(a power law upper bound was found by McBryan and Spencer).

The continuous version of the XY model is often used to model systems that possess order parameters with the same kinds of symmetry, e.g. superfluid helium, hexatic liquid crystals. This is what makes them peculiar from other phase transitions which are always accompanied with a symmetry breaking. Topological defects in the XY model leads to a vortex-unbinding transition from the low-temperature phase to the high-temperature disordered phase. In two spatial dimensions the XY model exhibits a Kosterlitz-Thouless transition from the disordered high-temperature phase into the quasi-long range ordered low-temperature phase.

Three and Higher Dimensions

• At low temperature, infrared bound shows that the spontaneous magnetization is strictly positive:

$M(\beta):=|\langle \mathbf{s}_i\rangle|>0$

See also

• Goldstone boson
• Ising model
• Potts model
• Kosterlitz-Thouless transition
• Topological defect
• Superfluid film

References

• Evgeny Demidov, Vortices in the XY model (2004)