The Zeeman, Spin-Orbit, and Quantum Spin-Hall Interactions in Anisotropic and Low-Dimensional Conductors

When an electron or hole is in a conduction band of a crystal, it can be very different from 2, depending upon the crystalline anisotropy and the direction of the applied magnetic induction {bf B}. In fact, it can even be 0! To demonstrate this quantitatively, the Dirac equation is extended for a relativistic electron or hole in an orthorhombically-anisotropic conduction band with effective masses m_j for j=1,2,3 with geometric mean m_g=\(m_1m2m3\)^1/3. The appropriate Foldy-Wouthuysen transformations are extended to evaluate the non-relativistic Hamiltonian to O\({rm m}c²\)^-4, where {rm m}c² is the particle's Einstein rest energy. For {bf B}||hat{bf e}_μ, the Zeeman g_μ factor is 2{rm m}sqrt{m_μ}/m_g^3/2 + O\({rm m}c²\)^-2. While propagating in a two-dimensional (2D) conduction band with m_3gg m₁,m₂, g_||<<2, consistent with recent measurements of the temperature T dependence of the parallel upper critical induction B_c2,||(T) in superconducting monolayer NbSe₂ and in twisted bilayer graphene. While a particle is in its conduction band of an atomically thin one-dimensional metallic chain along hat{bf e}_μ, g<<2 for all {bf B}={bfnabla}times{bf A} directions and vanishingly small for {bf B}||hat{bf e}_μ. The quantum spin Hall Hamiltonian for 2D metals with m₁=m₂=m_|| is K\[{bf E}times\({bf p}-q{bf A}\)\]_perpσ_perp+O\({rm m}c²\)^-4, where {bf E} and {bf p}-q{bf A} are the planar electric field and gauge-invariant momentum, q=mp|e| is the particle's charge, σ_perp is the Pauli matrix normal to the layer, K=pmμ_B/\(2m_||c²\), and μ_B is the Bohr magneton.