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

The Dirac equation is extended to an orthorhombically anisotropic system with three effective masses. It exhibits the full relativistic and non-relativistic physical properties in low dimension self-consistently. We show the anisotropic Dirac equation satisfies the most general proper Lorentz transformations and improper Lorentz transformations . The appropriate Foldy-Wouthuysen transformations are extended to expand the Hamiltonian near the non-relativistic limit to fourth order in (mc²)^-1, and accurate expressions for Zeeman, spin-orbit and quantum spin Hall interactions for low dimension are obtained. We note that When an electron or hole is traveling in an atomically thin one-dimensional conduction band, its Zeeman and spin-orbit interactions are vanishingly small. In two dimensions, its Zeeman interaction depends strongly on the magnetic field direction, and it nearly vanishes when the field is parallel to the conducting layer. We suppose this might be one of the reasons the upper critical field can greatly exceed the standard Pauli limit in clean monolayer and bilayer superconductors at low temperature.