(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

You have a latice of particles that all have spin 1, but they can change the direction of their spin so constraint [itex]\left|S_j\right|=1[/itex]. There is only interaction with the closest neighbours so we have the following hamiltonian:

[itex]H = -J \sum_{\left\langle ij \right\rangle} \vec{S_i} \cdot\vec{S_j} - \vec{h} \cdot \sum^{N}_{j = 1} \vec{\vec{S_j}}[/itex]

Choose a good orderparameter to treat this in the molecular field approximation. Calculate the selfconsistent equation for this order parameter and determine the spontaneous magnetisation for [itex]T<T_c=Jq/3k_b[/itex].

2. Relevant equations

[itex]Z=\int_{\left|S_1\right|=1}\cdots \int_{\left|S_N\right|=1} d^3 S_1 \cdots d^3 S_N \exp{\left(-\beta H\right)}[/itex]

[itex]M = \frac{1}{\beta} \nabla_h ln Z[/itex]

3. The attempt at a solution

As order parameter I pick [itex]\vec{M} = \sum_j \vec{S_j}[/itex] and than I approximate the hamiltonian with q nearest neighbors by

[itex]H = \frac{-Jq}{2N} \left(\sum^N_{j=1} \vec{S_j}\right)^2 - \vec{h} \cdot \sum^{N}_{j = 1} \vec{\vec{S_j}}[/itex]

This gives

[itex]Z=\int_{\left|S_1\right|=1}\cdots \int_{\left|S_N\right|=1} d^3 S_1 \cdots d^3 S_N \exp{\left(\frac{\beta Jq}{2N} \left(\sum^N_{j=1} \vec{S_j}\right)^2 + \beta \vec{h} \cdot \sum^{N}_{j = 1} \vec{\vec{S_j}} \right)}[/itex]

But I can't manage the integral. How do I calculate this integral? The rest I presume is correct?

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# Statistical Mechanics: classical Heisenberg Model

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