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I have read a few discussions about the difference between the Heisenberg model (using quantum spin operators) and Ising model (with spins $\pm 1$), notably this one or this Quora post.

All the responses seem to be of theoretical nature, highlighting, e.g., the difference in symmetry or the Mermin-Wagner theorem stating the Heisenberg model does have a phase transition at $d\leq 2$.

I am interested in practical aspects. Imagine I want to simulate some real material (let us consider a $3d$ lattice only). Even an undergraduate can write and run a Monte Carlo code simulating the Ising model a few hours. On the other hand, the Heisenberg model involves a true effort. Writing it requires immense knowledge, intuition and skill (constructing the Hamiltonian and total spin operators and computing the expectation values), and running it demands powerful computers even for a fairly low number of atoms and approximation schemes (see this review of methods).

Given the amount of effort, what is the justification for using the quantum Heisenberg model? What new vital features observed in experiment does it provide? If any, can they not be reproduced by the Ising model with vectorised but classical 3D spins, $-J \sum_{\langle ij\rangle} \mathbf S_i \cdot \mathbf S_j$?

Edit: To clarify that what I am trying to figure out is different from this question, I am not interested in theoretical features, but in practical/experimental results in three dimensions. That is, to what extent do Heisenberg/Ising models correctly predict:

  • critical exponents of magnetisation, heat capacity and susceptibility,
  • collective excitations (can these be captured by 3d Ising model?),

for a range of ferromagnetic materials, and where do they fail?

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  • $\begingroup$ Superpositions, for one. $\endgroup$ – probably_someone Aug 10 '18 at 15:31
  • $\begingroup$ @probably_someone can you elaborate? $\endgroup$ – petervanya Aug 11 '18 at 0:00
  • $\begingroup$ Your question sounds misleading: From a practical point of view, you would use the model which best captures the real physics of your system, rather than the one which is simplest to simulate. Also, your claim that the Heisenberg model is hard to simulate is unjustified, at least for bipartite lattices where there are very good algorithms which are not too hard to code. (Admittedly, harder than the Ising model, but still pretty simple.) So I don't see how your "amount of effort" argument makes sense, except for, say, a homework assignment. $\endgroup$ – Norbert Schuch Aug 13 '18 at 14:52
  • $\begingroup$ BTW, I'd say that symmetry breaking (->Mermin-Wagner) IS a "vital feature". If it isn't, what would be a "vital feature" in your opinion? Other "vital features" might be the magnetization curve, or critical exponents, or the value of the magnetization. Are these "vital features" in your opinion? If not, what would be? $\endgroup$ – Norbert Schuch Aug 13 '18 at 14:56
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    $\begingroup$ Possible duplicate of What new features does the Heisenberg Model have compared to the Ising Model? (I think it needs to be made more clear what exactly would be a satisfactory answer not covered by this question and its answers.) $\endgroup$ – Norbert Schuch Aug 13 '18 at 14:57

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