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The Heisenberg Hamiltonian is $H_{Heis}=-\sum_{ij} J_{ij} \langle \vec{S}_i \cdot \vec{S}_j \rangle$ (generally, save different constants depending on convention). This seems to suggest that the exchange coupling parameters $J_{ij}$ would have units of energy/units of spin squared (i.e., $[\frac{\textrm{meV}}{\mu_B^2}]$). But every resource I've found reports $J_{ij}$ in units of energy, ignoring the spin.

I'm aware that, in atomic units, spin is written in units of $\hbar = 1$, but this only transforms the unit into an absorbable constant, meaning the unit is still there, just unreported. But even when venturing outside of atomic units, as many papers do in reporting exchange parameters in [meV], the units of spin are left out. Why is this? I've also seen some Hamiltonians written with unit vectors $\vec{e}_i$ replacing $\vec{S}_i$, as though seeking to preserve only the direction of the atomic magnetic moment without its magnitude. But this strays from the formalism.

What am I missing here?

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  • $\begingroup$ I'm pretty sure that in this context, the spin operators are just defined without the $\hbar$'s, which have been absorbed into $J_{ij}$. (That is, $\vec{S}_x = \frac{1}{2}\sigma_x$ instead of $\vec{S}_x = \frac{\hbar}{2}\sigma_x$.) There's nothing really going on here. $\endgroup$
    – march
    Jan 19, 2023 at 18:22
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    $\begingroup$ Thank you for your comment. I'm under the impression that one can absorb a unit only if the unit is a number (like $\hbar=1$ in atomic units). But since we do not often report $J_{ij}$ in atomic units, preferring rather meV, then we'd need $\hbar$ to take on a non-numerical unit, like eV*s. This would mean $J_{ij}$ actually has units of $\frac{1}{eV*s^2}$, not eV. Right? $\endgroup$
    – LMacEnul
    Jan 19, 2023 at 21:32

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