The underlying physics behind quantization axes for the electrons spin has never really been clear to me. I saw the Pauli matrices in my quantum courses, but the definition of different principle axes seems arbitrary.
In the case of a magnetic material, or spins in the presence of a magnetic field, there is a clear axis along which to define the spin. But what about when there is no magnetic field?
For instance, in materials with strong spin-orbit coupling electron wavefunctions are mixed between up and down spin polarization. i.e. spin polarized states are not eigenstates. Furthermore, this paper about Berry curvature in SOC systems says, " it cannot be assumed that their local spin quantization axes are antiparallel, as the interactions with neighboring bands may affect the direction of quantization" (DOI: 10.1103/PhysRevB.72.085110)
But what does up and down mean in this context? Can quantization axes be defined differently depending on the properties of the band structure?
Dear Ryan,
The direction of the spins is arbitrary if a magnetic field is not present, but you always can know that two spins in the same state need to be with opposite direction. The reason is the Pauli's principle of exclusion for fermions and the Berry's phase cannot give you also information about this issue, although its curvature can coupled to one applied electric field to give a velocity. But not the state of the spins.
Thanks Daniel. I see what you are saying, but what does majority and minority electron spin mean in a system without an external field? I have seen many wavefunction written in terms of majority and minority spin. Say for instance, in bulk Pt, where there is no broken space or time symmetry (no magnetic fields).
Ryan, the RKKY or spin-orbit select one given orientation of the spins. For instance in spintronics there are spetial configurations for the giant magnetoresistance, etc There are many different techniques to get a majority of spins in one given direction besides the stable magnetism based mainly in Hund rules.
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