XYmodel

The XY model is a classical spin model used in statistical mechanics and condensed matter physics to study planar spin systems. Each lattice site i carries a unit spin S_i in the plane, represented by an angle θ_i or a two-component vector (cos θ_i, sin θ_i). For nearest-neighbor interactions the Hamiltonian is H = -J ∑⟨i,j⟩ S_i · S_j = -J ∑⟨i,j⟩ cos(θ_i - θ_j), with J > 0 for ferromagnetic coupling.

Symmetry: The model has a global U(1) rotational symmetry: θ_i → θ_i + α. In two dimensions, the Mermin-Wagner theorem forbids spontaneous breaking of continuous symmetries at finite temperature; thus there is no true long-range order. Nevertheless, the 2D XY model exhibits a Berezinskii-Kosterlitz-Thouless (BKT) transition: at low T the system has quasi-long-range order with correlations decaying as a power law, while at high T correlations decay exponentially. The transition is driven by the unbinding of vortex–antivortex pairs; the transition temperature is T_BKT and features a universal jump in the spin stiffness (Nelson-Kosterlitz relation).

In three dimensions the U(1) symmetry can be spontaneously broken at finite temperature, giving a true long-range order and a phase transition in the 3D XY universality class with well-studied critical exponents. The XY model also appears as the classical limit of quantum rotor models and as a simplifying description of various superconducting and superfluid systems.

Methods and variants: The XY model is studied by Monte Carlo simulations, spin-wave theory at low T, and renormalization-group analyses; mappings to a Coulomb gas provide analytical insight into the BKT transition. The model has numerous applications, including planar magnets, thin-film superfluids, Josephson junction arrays, and cold-atom setups.