Classical XY model

Given a translation-invariant interaction Jij = J(i − j) and a point dependent external field

The configuration probability is given by the Boltzmann distribution with inverse temperature β ≥ 0: where Z is the normalization, or partition function.

indicates the expectation of the random variable A(s) in the infinite volume limit, after periodic boundary conditions have been imposed.

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

Using the properties of the modified Bessel functions, the specific heat (per spin) can be expressed as[5]

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

which can be treated as the trace of a matrix, namely a product of matrices (scalars, in this case).

The trace of a matrix is simply the sum of its eigenvalues, and in the thermodynamic limit

This transfer matrix approach is also required when using free boundary conditions, but with an applied field

In the case of a discrete lattice of spins, the two-dimensional XY model can be evaluated using the transfer matrix approach, reducing the model to an eigenvalue problem and utilizing the largest eigenvalue from the transfer matrix.

The 2D XY model has also been studied in great detail using Monte Carlo simulations, for example with the Metropolis algorithm.

These can be used to compute thermodynamic quantities like the system energy, specific heat, magnetization, etc., over a range of temperatures and time-scales.

(often, it can be discretized into finitely-many angles, like in the related Potts model, for ease of computation.

Similarly the mean-squared magnetization characterizes the average of the square of net components of the spins across the lattice.

Indeed, at high temperatures this quantity approaches zero since the components of the spins will tend to be randomized and thus sum to zero.

Furthermore, using statistical mechanics one can relate thermodynamic averages to quantities like specific heat by calculating

There is no feature in the specific heat consistent with critical behavior (like a divergence) at this predicted temperature.

The nature of the critical transitions and vortex formation can be elucidated by considering a continuous version of the XY model.

Expanding the original cosine as a Taylor series, the Hamiltonian can be expressed in the continuum approximation as

Kosterlitz and Thouless provided a simple argument of why this would be the case: this considers the ground state consisting of all spins in the same orientation, with the addition then of a single vortex.

Putting these together, the free energy of a system would change due to the spontaneous formation of a vortex by an amount

This indicates that at low temperatures, any vortices that arise will want to annihilate with antivortices to lower the system energy.

Indeed, this will be the case qualitatively if one watches 'snapshots' of the spin system at low temperatures, where vortices and antivortices gradually come together to annihilate.

Meanwhile at high temperatures, there will be a collection of unbound vortices and antivortices that are free to move about the plane.

To identify vortices (or antivortices) present as a result of the Kosterlitz–Thouless transition, one can determine the signed change in angle by traversing a circle of lattice points counterclockwise.

These vortexes are topologically non-trivial objects that come in vortex-antivortex pairs, which can separate or pair-annihilate.

As the configuration is studied at long time scales and at low temperatures, it is observed that many of these vortex-antivortex pairs get closer together and eventually pair-annihilate.

The three dimensional case is interesting because the critical exponents at the phase transition are nontrivial.

Many three-dimensional physical systems belong to the same universality class as the three dimensional XY model and share the same critical exponents, most notably easy-plane magnets and liquid Helium-4.

The values of these critical exponents are measured by experiments, Monte Carlo simulations, and can also be computed by theoretical methods of quantum field theory, such as the renormalization group and the conformal bootstrap.