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$J$ can be determined experimentally by knowing, for example, $T_c$. Through a theoretical model including the structure and the species involved you can estimate the value for $J$ through the overlap integrals of the electrons involved. Here you can read more about exchange interaction, which is the basics behind magnetism (which is a quantum phenomena).

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Note that in the sum $$=\sum_{S_1=\pm 1}...\sum_{S_N=\pm 1}\langle S_2| T_{_{NN}}^{\dagger}|S_1\rangle\langle S_1| T_{_{NNN}}|S_3\rangle\langle S_3| T_{_{NN}}^{\dagger}| S_2\rangle\langle S_2| T_{_{NNN}}|S_4\rangle...\langle S_1| T_{_{NN}}^{\dagger}|S_N\rangle\langle S_N| T_{_{NNN}}|S_2\rangle$$ every pair $|S_i\rangle\langle S_i|$ occurs twice with some ...

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1.) The easiest way to count the energy is as that of domain walls. Both of the configurations you have drawn have two domain walls. Each domain wall costs energy $E\Delta/2$, so both configurations have energy $E\Delta$ above the ground state. I don't understand either how one would count the spin of a domain as 1. Perhaps that is the spin/site? 2.) To see ...

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