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Studying magnetic systems, two frequently used approximations are the Heisenberg and Ising models (a discussion about these approximations can be read here):

\begin{equation} \tag{Heisenberg} \hat{H}_H=-\sum_{\langle i j\rangle}J\hat{S}_i\hat{S}_j \end{equation}

\begin{equation} \tag{Ising} \hat{H}_I=-\sum_{\langle ij\rangle}J\hat{S}_i^z\hat{S}_j^z \end{equation}

Is there a theoretical method/way to determine realistic values for the $J$ parameter?

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  • $\begingroup$ If by J you mean inter-site exchange (the angular brackets under the summation refer to nearest neighbours), then refer to this question that I asked earlier : mattermodeling.stackexchange.com/questions/1359/…. If you are talking in the context of DFT, J is usually taken to be 0-20% of the Hubbard U as a thumb rule. $\endgroup$
    – Xivi76
    Jul 17 '20 at 18:09
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    $\begingroup$ It's quantifiable in the same way as the hubbard 'U'. Even the Hubbard 'U' is dependent on the spin operator. There are plenty of notes that explain this. $\endgroup$
    – Xivi76
    Jul 19 '20 at 22:34
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    $\begingroup$ Is this related? mattermodeling.stackexchange.com/a/104/88 $\endgroup$
    – Thomas
    Jul 26 '20 at 3:27
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    $\begingroup$ @Xivi76 since this is now one of the longest standing unanswered questions, do you think maybe you could write an answer? I'm trying to help clean up the unanswered queue a little bit! $\endgroup$ Mar 23 at 15:02
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    $\begingroup$ @NikeDattani Coincidentally, I learnt the procedure mentioned in the question quite recently. I can write up an answer. $\endgroup$
    – Xivi76
    Mar 23 at 20:08
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The equation in your question, be it Heisenberg or Ising exchange, can be calculated by Energy mapping analysis. This has to be the most popular paper that discusses this technique. Basically, you consider different spin configurations and map the exchange 'J' into the total energy values from DFT.

In the simplest form, if you have two magnetic atoms in your unit cell, there are four possible spin configurations - $\uparrow$$\uparrow$ , $\uparrow$$\downarrow$, $\downarrow$$\uparrow$,$\downarrow$$\downarrow$, denoted by $E_1, E_2, E_3, E_4$ respectively. The exchange 'J' can be estimated as $1/4* (E_1 - E_2 - E_3 + E_4$). Now, the energy values for the four different configurations are just the total energy values from your DFT calculation. Typically, you would need to constrain your magnetization on the magnetic atoms in your DFT package. If you want to include anisotropic exchange, you need to consider spin-orbit coupling as well.

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