 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 nvector model for n = 2.
Contents
Definition
Given a Ddimensional lattice Λ, per each lattice site there is a twodimensional, unitlength vector . A spin configuration, is an assignment of the angle per each site j in the lattice.
Given a translationinvariant interaction J_{ij} = J(i − j) and a point dependent external field , the configuration energy is
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 :
where Z is the normalization, or partition function.^{[1]}
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.^{[2]}
 At high temperature, the spontaneous magnetization vanishes:

 Besides, cluster expansion shows that the spin correlations cluster exponentially fast: for instance
 Using the Griffiths inequality in the formulation of Ginibre, Aizenman and Simon^{[3]} 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

 Hence the critical β of the XY model cannot be smaller than the double of the critical temperature of the Ising model
One dimension
As in any 'nearestneighbor' nvector model with free boundary conditions, if the external field is zero, there exists a simple exact solution.
In the nearestneighbor, 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 h = 0) requires the transfer matrix formalism.^{[4]}
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.^{[5]}
 (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 vortexunbinding transition from the lowtemperature phase to the hightemperature disordered phase. In two spatial dimensions the XY model exhibits a KosterlitzThouless transition from the disordered hightemperature phase into the quasilong range ordered lowtemperature phase.
Three and Higher Dimensions
 At low temperature, infrared bound shows that the spontaneous magnetization is strictly positive:
See also
 Goldstone boson
 Ising model
 Potts model
 KosterlitzThouless transition
 Topological defect
 Superfluid film
References
 ^ Lubensky, Chaikin (2000). Principles of Condensed Matter Physics. Cambridge University Press. pp. 699. ISBN 0521794501. http://books.google.com/books?id=P9YjNjzr9OIC&printsec=frontcover&cad=0#v=onepage&q&f=false.
 ^ Ginibre, J. (1970). "General formulation of Griffiths' inequalities". Comm. Math. Phys. 16 (4): 310–328. doi:10.1007/BF01646537.
 ^ Aizenman, M.; Simon, B. (1980). "A comparison of plane rotor and Ising models". Phys. Letts. A 76. doi:10.1016/03759601(80)904934. http://www.sciencedirect.com/science/article/pii/0375960180904934.
 ^ Mattis, D.C. (1984). "Transfer matrix in planerotator model". Phys.Letts 104 A. doi:10.1016/03759601(84)908168. http://www.sciencedirect.com/science/article/pii/0375960184908168.
 ^ Fröhlich, J.; Spencer, T. (1981). "The Kosterlitz–Thouless transition in twodimensional abelian spin systems and the Coulomb gas". Comm. Math. Phys. 81 (4): 527–602. http://projecteuclid.org/euclid.cmp/1103920388.
 Evgeny Demidov, Vortices in the XY model (2004)
Categories: Lattice models
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