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The energy function of an Ising model is $\sum_{i\sim j} s_i J_{ij} s_j$. In terms of the graph, $J_{ij}$ can be the edge weight.

What is the physical nature of the interactions $J_{ij}$?

Are they just magnetic field interactions between dipoles? If so, that means it is just dipoles in a Cartesian space with interactions scaled as the inverse fourth power of distance. Why do we bother defining regular lattices for them? Why not just make assume a local field at each site that is accumulated from all the other dipole moments?

Or are the interactions due to something else like electron cloud interactions between neighbors?

I am sorry if this sounds like an elementary question. I am a non-physicist who found Ising models interesting for their applications in other systems.

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    $\begingroup$ The Ising model has a huge number of different interpretations, usually with no link to magnetism at all, and the interpretation of the coupling constant $J_{ij}$ is different in each case. Note also that, even if you restrict yourself to the magnetic interpretation, the Ising model was introduced before the proper microscopic origins of magnetism were understood. $\endgroup$ – Yvan Velenik Mar 17 '17 at 7:37
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    $\begingroup$ You can actually show it isn't about classical dipole-dipole interaction, but the exchange interaction of the electrostatic force! (Hence written as $J$) $\endgroup$ – Ofek Gillon Mar 17 '17 at 7:40

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