The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a
lattice model of
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
. In general, the XY model can be seen as a specialization of Stanley's
''n''-vector model for .
Definition
Given a -dimensional
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
, per each lattice site there is a two-dimensional,
unit-length vector
The ''spin configuration'', is an assignment of the angle for each .
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
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability t ...
with inverse temperature :
:
where is the
normalization, or
partition function.
The notation
indicates the expectation of the random variable in the infinite volume limit, after ''
periodic boundary conditions'' have been imposed.
Rigorous results
* The existence of the
thermodynamic limit
In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
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 the two point spin correlation of the ''ferromagnetics'' XY model in dimension , coupling and
inverse temperature
In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:\beta = \frac (where is the temperature and is Boltzmann constant).J. Meixner (1975) "Coldness and Tempe ...
is ''dominated'' by (i.e. has an
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an eleme ...
given by) the two point correlation of the ''ferromagnetic''
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
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 (non-periodic) boundary conditions, if the external field is zero, there exists a simple exact solution. In the free boundary conditions case, the Hamiltonian is