Spin polarization due to exchange? Would spin polarized ground state exist if no $e$-$e$ repulsion?

Question

Non-relativistic no-magnetic-field many electron hamiltonian contains no spin operators. How would spin polarization happen in many electron ground state (modeled by LSDA DFT for instance)?

I often heard Spin polarization in solid state system ground state is due to exchange interaction. I think it is mostly due to electron-electron repulsion, no?

For instance, consider a fictitious world where there is no electron-electron repulsion. Then given a Helium nuclei, the two electron will aways pair up and occupy the 1s orbital. $( |1s\rangle |1s \rangle + |1s\rangle |1s\rangle ) (|↑\rangle |↓\rangle - |↓\rangle |↑\rangle )$ The spatial is symmetric, the spin anti-sym.. so overall anti-sym like fermions are supposed to.

Now, consider a different fictitious world where the electron-electron repulsion is huge. Then given a Heilum, the two electrons don't want to occupy the same state (single particle Kohn Sham state) because the electron repulsion is just too costly. In such case the two lowest energy become

$( |1s\rangle |2p\rangle + |2p\rangle |1s\rangle ) (|↑\rangle |↓\rangle - |↓\rangle |↑\rangle )$ this is has zero spin polarization

$( |1s\rangle|2p \rangle - |2p\rangle |1s\rangle ) (|↑\rangle |↑\rangle + |↑\rangle |↑\rangle )$ this is has non-zero polarization

Now we have a chance to have a spin polarized ground state.

Question 1: Is electron-electron repulsion necessary for spin polarization? (so the typical "it's due to exchange interaction" is pretty misleading?)

Question 2: Now given the two states right above, one of which is spin polarized the other is not. Are they energetically degenerate? or the single partile picture with Kohn Sham state is misleading, and somehow their true multiple electron ground state have different energy even though the spatial part and spin part must factor out due to the lack of spin operator in the hamiltonian?

Answer 1

Ferromagnetism is a vast topic. A couple of things worth keeping in mind are:

There exist many mechanisms of interaction between magnetic moments, leading to ferromagnetism
Indeed, the analogy with Helium atom is rather limited - e.g., solid state are complex systems, and the (field theoretical or similar) models applied to a solid state usually approximate only a part of the system responsible for the phenomenon in question. E.g., only the conduction band or only the magnetic moments (embedded in a larger matrix.) E.g., RKKY interaction is the interaction between magnetic moments mediated by free-moving electrons (itinerant electrons - hence itinerant ferromagnetism). The magnetic moments are themselves combinations of electrons confined to orbitals of an atom with a non-zero magnetic moment, which participate in exchange with the conduction electrons and other like magnetic moments. However, in practice this exchange interaction is better expressed by writing Hamiltonian terms like $$ H_{exch}=-J\mathbf{s}\cdot\mathbf{S}, $$ rather than spelling the full ground-state wave-function. The electrons and the magnetic moments are thus treated, as if they were different particles, interacting via what is manifestly a spin-dependent interaction (but which is really exchange between identical electrons).

Exchange is not dipole interaction
Another reason for stressing the exchange nature of the interaction is to contrast it to the dipole-dipole interaction of magnetic moments, which has similar terms, but different nature. Importantly, dipole-dipole interaction behavior with distance is inconsistent with most instances of ferromagnetism, although it was dipole-dipole interaction that motivated the first models of ferromagntism and related phenomena (notably the Ising model.)

The classical introduction to the nature of ferromagnetism (helpfully starting with a Helium atom) is Theory of magnetism by Mattis.