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XY model : ウィキペディア英語版
Classical XY model
The classical XY model
(sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. It is the special case of the 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 \mathbf_=(h_j,0), the configuration energy is
: H(\mathbf) = - \sum_ J_\; \mathbf_i\cdot\mathbf_j -\sum_ \mathbf_j\cdot \mathbf_j =- \sum_ J_\; \cos(\theta_i-\theta_j) -\sum_ h_j\cos\theta_j
The case in which except for nearest neighbor is called ''nearest neighbor'' case.
The configuration probability is given by the Boltzmann distribution with inverse temperature :
:P(\mathbf)=\frac} \qquad Z=\int_d\theta_j\;e^.
where is the normalization, or partition function. The notation \langle A(\mathbf)\rangle indicates the expectation of the random variable in the infinite volume limit, after ''periodic boundary conditions'' have been imposed.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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