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. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
. In general, the XY model can be seen as a specialization of Stanley's
''n''-vector model for .
Definition
Given a -dimensional
lattice , 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 tha ...
with inverse temperature :
:
where is the
normalization
Normalization or normalisation refers to a process that makes something more normal or regular. Science
* Normalization process theory, a sociological theory of the implementation of new technologies or innovations
* Normalization model, used in ...
, or
partition function.
The notation
indicates the expectation of the random variable in the infinite volume limit, after ''
periodic boundary conditions
Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a ''unit cell''. PBCs are often used in computer simulations and mathematical mod ...
'' 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 (mathematics), 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 ...
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 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 every element of .
Dually, a lower bound or minorant of is defined to be an element of that is less ...
given by) the two point correlation of the ''ferromagnetic''
Ising model
The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that r ...
in dimension , coupling and inverse temperature
Hence the critical of the XY model cannot be smaller than the double of the critical 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