I am slightly confused about the nature of the Ising model to study ferromagnetism. Consider the Heisenberg Hamiltonian with Zeeman term: \[H=-\frac{1}{2} \sum_{i\ne j}J_{ij} S_i\cdot S_j+g\mu_B {B}\sum_iS_i^z\] This can also be written as: \[H=-\frac{1}{2} \sum_{i\ne j} J_{ij}S_i^z\cdot S_j^z+g\mu_B {B}\sum_iS_i^z-\frac{1}{2} \sum_{i\ne j} J_{ij}S_i^-S_j^+\] Ashcroft and Mermin define the Ising model to be simply the exclusion of the last term in the above expansion. Whilst other sources (e.g. Simon, 2013) say that it is saying the atoms are in spin states $+S$ or $-S$. These do not appear equivalent - not even in mean field approximation where (correct me if I am wrong) the last term vanishes anyway due to rotational symmetry. Thus which is the standard definition and are they linked in any way?
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asked Apr 6, 2017 at 16:30
Quantum spaghettificationQuantum spaghettification
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$\begingroup$ Hint: Without the last term, you can simply specify each $S^z_i$ when specifying an eigenstate. So call that $\pm S$. $\endgroup$– AHusainCommented Apr 6, 2017 at 23:27
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$\begingroup$ @AHusain So what happens in this case to all the states, $-S+1$, $-S+2$,...,$S-2$,$S-1$? $\endgroup$– Quantum spaghettificationCommented Apr 7, 2017 at 4:41
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$\begingroup$ It's +/- 1/2. Just rescaled. $\endgroup$– AHusainCommented Apr 7, 2017 at 5:04
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