Given a D-dimensional lattice Λ, per each lattice site j ∈ Λ there is a two-dimensional, unit-length vector sj = (cos θj, sin θj)
The spin configuration, s = (sj)j ∈ Λ is an assignment of the … See more

• The existence of the thermodynamic limit for the free energy and spin correlations were proved by Ginibre, extending to this case the Griffiths inequality.
• Using the Griffiths inequality in the formulation of Ginibre, Aizenman and Simon proved that … See more

• Classical Heisenberg model
• Coulomb gas
• Goldstone boson
• Ising model See more

• real-time XY model WebGL simulation
• Interactive Monte Carlo simulation of the Ising, XY and Heisenberg models with 3D graphics (requires WebGL compatible browser) See more

As mentioned above in one dimension the XY model does not have a phase transition, while in two dimensions it has the Berezinski-Kosterlitz-Thouless transition between the phases with exponentially and powerlaw decaying correlation functions. See more

• H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, (Oxford University Press, Oxford and New York 1971); See more