The classical XY model
is a model of statistical mechanics

Statistical mechanics

Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

. It is the special case of the n-vector model for .

Definition

Given a -dimensional lattice , per each lattice site there is a two-dimensional, unit-length vector
. A spin configuration, is an assignment of the angle per each site in the lattice.

Given a translation-invariant interaction and a point dependent external field
, the configuration energy is
The case in which except for nearest neighbor is called 'nearest neighbor case'.

The configuration probability is given by the Boltzmann distribution

Boltzmann distribution

In chemistry, physics, and mathematics, the Boltzmann distribution is a certain distribution function or probability measure for the distribution of the states of a system. It underpins the concept of the canonical ensemble, providing its underlying distribution...

 with inverse temperature :
where is the normalization, or partition function

Partition function (statistical mechanics)

Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...

.

General properties

• The existence of the thermodynamic limit for the free energy
  Thermodynamic free energy
  The thermodynamic free energy is the amount of work that a thermodynamic system can perform. The concept is useful in the thermodynamics of chemical or thermal processes in engineering and science. The free energy is the internal energy of a system less the amount of energy that cannot be used to...
  
   and spin correlations were proved by Ginibre
  Jean Ginibre
  Jean Ginibre is a French mathematical physicist, known in particular for his contributions to random matrix theory , statistical mechanics , and partial differential equations. He received the Paul Langevin Prize in 1969.Jean Ginibre is Emeritus Professor at Paris-Sud 11 University....
  
  , extending to this case the Griffiths inequality
  Griffiths inequality
  In statistical mechanics, the Griffiths inequality , named after Robert B. Griffiths, is a correlation inequality for ferromagnetic spin systems...
  
  .

• At high temperature, the spontaneous magnetization vanishes:

Besides, cluster expansion

Cluster expansion

In statistical mechanics, the cluster expansion is a power series expansion of the partition function of a statistical field theory around a model that is a union of non-interacting 0-dimensional field theories. Cluster expansions originated in the work of...

 shows that the spin correlations cluster exponentially fast: for instance

• Using the Griffiths inequality
  Griffiths inequality
  In statistical mechanics, the Griffiths inequality , named after Robert B. Griffiths, is a correlation inequality for ferromagnetic spin systems...
  
   in the formulation of Ginibre, Aizenman and Simon proved that the two point spin correlation of the ferromagnetics XY model in dimension , coupling and inverse temperature is dominated by (i.e. has upper bound given by) the two point correlation of the ferromagnetic Ising model
  Ising model
  The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
  
   in dimension , coupling and inverse temperature

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

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
therefore the partition function factorizes under the change of coordinates
That gives
Finally

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

Two Dimensions

• At low temperature, the spontaneous magnetization remains zero,

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

.

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

Kosterlitz-Thouless transition

In statistical mechanics, a part of mathematical physics, the Kosterlitz–Thouless transition, or Berezinsky–Kosterlitz–Thouless transition, is a kind of phase transition that appears in the XY model in 2 spatial dimensions. The XY model is a 2-dimensional vector spin model that possesses U or...

 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:

See also

• Goldstone boson
  Goldstone boson
  In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...
  
  
• Ising model
  Ising model
  The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
  
  
• Potts model
  Potts model
  In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics...
  
  
• Kosterlitz-Thouless transition
  Kosterlitz-Thouless transition
  In statistical mechanics, a part of mathematical physics, the Kosterlitz–Thouless transition, or Berezinsky–Kosterlitz–Thouless transition, is a kind of phase transition that appears in the XY model in 2 spatial dimensions. The XY model is a 2-dimensional vector spin model that possesses U or...
  
  
• Topological defect
  Topological defect
  In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of...
  
  
• Superfluid film
  Superfluid film
  Superfluidity and superconductivity are macroscopic manifestations of quantum mechanics. There is considerable interest, both theoretical and practical, in these quantum phase transitions. There has been a tremendous amount of work done in the field of phase transitions and critical phenomenon in...
  
  

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