Is there magnetic dipole-dipole interaction between electrons in the quantum level? Classically, two magnetic moments interact with each other through the magnetic field they create. Consider two electrons, the common Hamiltonian would be
$$
\hat{H} = \frac{\hat{p}_1^2+\hat{p}_2^2}{2m}+\frac{e^2}{4\pi \epsilon_0}\frac{1}{(r_1-r_2)^2},
$$
where $\hat{p}_1$ is the momentum operator of one of the electrons.
However, I am wondering whether there should be a term representing the interaction between the spin magnetic moments of the two electrons.
Thoughts: 


*

*Since spin is essentially a relativistic effect, I would imagine that the interaction between spin magnetic moments, if exists, should come from relativistic quantum mechanics. I have heard of Darwin term and spin orbit coupling, both coming from reducing Dirac equation to Schrodinger equation. However, I have never heard of any interaction term between spin magnetic moments.

*In condensed matter physics, there are models, such as Heisenberg model, that describe the interaction of spins. However, such interaction comes from Coulomb interaction, rather than the interaction from spin magnetic moments.

 A: *

*Yes there is such an interaction. It is sometimes called a spin-spin interaction, though strictly, as you say, it is an interaction between magnetic dipole moments. In helium it makes a contribution to the fine structure similar to that from spin-orbit interaction; in other atoms it also contributes, but less noticeably.

*It is a common misconception that spin is a relativistic effect or a quantum effect or both. Strictly speaking, it is no more a relativistic effect or a quantum effect than everything else. I mean there is a low-velocity limit where spin is still relevant, and there are spin states which behave like a classical limit of spin. In both respects the same can be said of other degrees of freedom such as position and momentum, but we don't normally propose that they have to be relativistic, nor that they are a quantum effect. Having said that, when one constructs a relativistic quantum theory one is more or less forced to include spin, whereas when one constructs classical physics you could leave spin out (from ignorance) and a sensible theory can still be constructed. (But you don't have to leave it out, and you can include it while still doing classical physics by 
letting the spin degree of freedom be described by including it in the angular momentum tensor using numbers not operators, and noting that in such a classical version it can be observed without significantly disturbing it. The reason why spin seems to be so "quantum" in practice is because we almost always encounter it in situations far from such a classical limit).
