Confusion about units of physical magnitudes in the Hamiltonian of the Ising model

Question

I am having trouble with the units used in the Hamiltonian of the Ising model. I have search several notes, I have three examples in the picture below
No one states explicitly what the units of the following terms are: $S_i, J$ and  $ H,B,h$ (these should all represent a magnetic field :)

Do the spin variables $S_i$ have dimensions or not, they just say they can have values $+-1$? I guess they are dimensionless.

If that is true, since most call $J$ an energy interaction factor, I guess $ J$ must have units of energy as the hamiltonian does, so that $-sum_i JS_iS_j$ can have units of energy

What about the units of $H, B$ or $h$? They say they represent magnetic field, but how can that be, if considering $S_i$ is dimensionless, $hS_i$ or $BS_i$ or $HS_i$ all have  units of magnetic field, when they should instead have units of energy. I feel like a magnetic moment $mu$ factor is missing there, unless for some unknown reason it has been secretly set to 1 and no one talks about it.
In fact  before learning about the Ising model, when I studied the canonical ensamble, my professor used $-sum_i S_iBmu$ for the energy of  interaction ot the spins with the magnetic field, that looked ok, but when we did the Ising model he joined the secret club and dropped that $mu$ factor as well.

Supposing $H, B$ or $h$ are magnetic field, and that the  natural and correct way of writting this last term is $-mu B sum_i S_i $. Different people use this letters indifferently without adding any extra factor, but   shouldn't a magnetic permeability factor be missing if I used $H$ or $h$  instead of $B$?. I recall from electromagnetism that $B=tilde mu H$ ( where  I am using $tilde mu$ for the magnetic permeability to distinguish it from the magnetic moment $mu$) and that the energy of a magnetic dipole is $-mu B$. So, if I were to use $H$, I guess I should write  $- mu Bsum_i  S_i  = - mu (tildemu H )sum_i  S_i  $instead of just $-H sum_i S_i $ or $-mu H sum_i S_i $, shouldn't I?

How do I make sense of all this?

Vadim · Accepted Answer

These are dimensionless units. In principle one could start with a dimensional Hamiltonian and define the dimensionless units, e.g., the Zeeman term of a spin coupled to a magnetic field is
$$
H_Z = -frac{gmu_B}{hbar}mathbf{S}cdotmathbf{B},
$$
assuming the quantization axis along the magnetic field, we can write
$$
H_Z = -frac{gmu_B}{hbar}BS_z = -frac{gmu_B}{hbar}Bfrac{hbar}{2}hat{sigma}_z = -hhat{sigma}_z,
$$
where $hat{sigma}_z$ is the Pauli matrix with eigenvalues $pm 1$, whereas the rest of the coefficients are absorbed into constant $h$.
However, as the role of the Ising model is not in making quantitative predictions for the values of magnetization, but in providing insights into critical behavior, one is rarely concerned with actual units and the expressions for the dimensionless quantities used.

Confusion about units of physical magnitudes in the Hamiltonian of the Ising model

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