What exactly is the magnetic field $H$ at Heisenberg model? The quantum hamiltonian of Heisenberg model is usually given in the form:
$$\textit{H}_{heisenberg}=-\sum_{i,j}J_{ij}\vec{S}_{i}\cdot\vec{S}_{j}-g\mu_{B}\vec{H}\cdot\sum_{i}\vec{S}_{i}$$
such that first term corresponds to exchange interaction between spin, and second term corresponds to interaction of spin with external magnetic field.
I am confused about the second term, because most electrodynamics course say the interaction of magnetic dipole and field is given by $-\vec{m}\cdot\vec{B}$, not $-\vec{m}\cdot\vec{H}$. On the other hand, many magnetism textbook describe interaction as $-\vec{m}\cdot\vec{H}$, usually based on Coulombian view of magnetic monopole. Since this model is used in description of ferromagnetic material, the difference between $\vec{B}$ and $\vec{H}$ can't be just omitted, isn't it?
Another confusing point is that the term $-g\mu_{B}\vec{H}\cdot\vec{S}_i$ is describing interaction of individual atom with magnetic field, so it seems that the field must be local field $\vec{H}_{loc}=\vec{\mathfrak{H}}+\vec{H}_{d}$ (or $\vec{B}_{loc}=\vec{\mathfrak{B}}+\vec{B}_{d}$), rather than external field $\vec{\mathfrak{H}}$ (or $\vec{\mathfrak{B}}$). I failed to find clear manifestation about this distinction, and this makes me even hesitating. Since the difference between Coulombian view of monopole with field H and Amperian view of current loop with field B causes different internal structure of local field, it will directly effect local field we are considering.
In summary, my question is: what is the field H in interaction term $-g\mu_{B}\vec{H}\cdot\vec{S}_i$? Is it magnetic field $\vec{H}$, or magnetic induction $\vec{B}$? Is it external field given in experimental setup, or internal local field directly acting in individual atom?
P.S. I'd consulted the book "Electrodynamics of Materials: Forces, Stresses, and Energies in Solids and Fluids" from Scipione Bobbio, which argues both of Coulombian and Amperian view is identical in thermodynamic sense describing whole field and matter, and the difference in energy term just arises from difference between thermodynamic internal free energy. I'm not sure this view is widely accepted...
 A: You are right; the SI units are just a pain in the $\vec{B}$ (!). And also you are right about the distinction between $\vec{H}$ and $\vec{B}$, for the former couples with ''monopoles'' but the latter couples with ''Amperian currents''. Although such distinction is of no relevance in a microscopic model, as pointed out in the comments, $H$ in these models is almost always ''the field filling the space before introducing the system to it'', the ''external field'' if I may. One can denote it best as $B_0$ or $B_{ext}$. And when it should couple to the magnetic moments $m_i$, the corresponding term is a Zeeman one, $-B_0m_i$, and then you can sum over all moments. Its spatial variation is so slow that it can appear as a mere parameter in the Hamiltonian, not as an operator or a degree of freedom. No ''local field'' concept has run into this definition. In fact, such term will be present in a coarse-grained model through the local and, perhaps, nonlocal interactions given by the $J_{ij}$ couplings.
Lastly, you can of course denote the external field equivalently with $\mu_0H_0 = B_0$.
