spindegeneracy

Spin degeneracy refers to the degeneracy of electronic energy levels with respect to the spin degree of freedom in quantum systems. In a non-relativistic Hamiltonian that contains no spin-dependent terms and no external magnetic field, the energy of each spatial orbital is independent of spin orientation, yielding a twofold spin degeneracy for spin-1/2 particles: each orbital can accommodate two electrons with opposite spins.

Two key mechanisms can influence or remove spin degeneracy. First, time-reversal symmetry in systems with half-integer spin implies Kramers degeneracy: every energy level is at least doubly degenerate in the absence of magnetic fields. This degeneracy is robust against most perturbations that preserve time-reversal symmetry. Second, spin degeneracy can be lifted when spin becomes coupled to other degrees of freedom or to external fields. Spin-orbit coupling mixes spin and orbital motion and can split spin states, while a magnetic field (Zeeman effect) directly lifts degeneracy by shifting energies of different spin projections.

In solid state physics, spin degeneracy is reflected in band structure: bands are typically twofold degenerate due to spin in the absence of spin-orbit coupling or magnetic order. When spin-orbit coupling is present or magnetic order arises, bands can split into spin-polarized branches, leading to phenomena such as Rashba or Dresselhaus spin splitting.

Implications include changes to the electronic density of states, transport properties, and magnetic responses. Spin degeneracy is a foundational concept for understanding magnetic materials, spintronics, and the behavior of electrons in atoms and solids.