The Hamiltonian for the Ising model on a lattice $\Lambda$ is: \begin{align} H(S) &= - J \sum\limits_{i \sim j} S_i S_j - B \sum\limits_i S_i \\ &= - J \sum\limits_i\sum\limits_{j: i\sim j} S_i S_j - B \sum\limits_i S_i \end{align} Where $J \in \mathbb{R}$ is the coupling constant and $B \in \mathbb{R}$ is the magnetic field (with constants). If $n_i = |\{j \in \Lambda: i \sim j\}|$ is the number of corners neighbouring $i$, then for a rectangular grid in $d$ dimensions we have: $n_i = 2d$. If we rewrite the coupling term for a given $i$, we get: \begin{align} -J \sum\limits_{j:i \sim j} S_i S_j = - 2 d J S_i \frac{1}{2 d} \sum\limits_{j:i \sim j} S_j \end{align} We can interpret the term: \begin{equation} \frac{1}{2 d} \sum\limits_{j:i \sim j} S_j \end{equation} as average magnetisation around $i$. In the Curie-Weiß-Ising Modells we make the assumption of replacing this $i$ dependent magnetisation environment by the average magnetisation. \begin{equation} \label{E:mean_field_approximation} \frac{1}{2 d} \sum\limits_{j:i \sim j} S_j \approx \frac{1}{N} \sum\limits_{j \neq i} S_j \end{equation} If we include the $2d$ term into $J$ we end up with a new coupling constant.

So now my question:

Do you know of experimental papers where they measured coupling constants for different dimensions? e.g. 2D and 3D lattices.


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