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The Ising model is defined as $$H(\sigma)=-\sum_{<i j>}J_{ij}\sigma_i\sigma_j$$

what are the coupling/interaction terms physically?

Also how can you manufacture a set of spins to have definite interaction terms?

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In real systems, the $J_{ij}$ couplings correspond to the so called exchange interactions between quantum spins. In a very basic sense, its physical origin lies in the interplay between electronic Coulomb repulsion and the Pauli exclusion principle for fermions. According to the details of the physical system, these exchange interactions receive different names: direct exchange (like the one responsible of the first Hund's rule in atomic physics), superexchange, double exchange, indirect exchange. A canonical and ubiquitous example is the Heisenberg exchange Hamiltonian, $J \hat{\vec S}_i \cdot \hat{\vec S}_j$, where $\hat{\vec S}$'s are the usual spin angular momentum operators. In some condensed matter systems (for example, certain rare earth compounds), the relativistic spin-orbit interaction makes that spins can only point in some particular directions, giving rise to the Ising interaction.

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