On the effective models of spin–orbit coupling in a two-dimensional electron gas

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We use the method of invariants to derive one- and two-band effective Hamiltonians of a noncentrosymmetric two-dimensional electron gas, in the presence of magnetic field. A complete classification of the antisymmetric spin–orbit and magnetic coupling terms near the Γ point is developed for all two-dimensional crystal symmetries. The effective Hamiltonian depends on the symmetry of the Bloch bands at the Γ point, which is described by one of the double-valued corepresentations of the two-dimensional magnetic point group. In some bands, the spin–orbit coupling is cubic in the electron momentum and the effective Zeeman interaction is strongly anisotropic. As an example of a two-band effective Hamiltonian, we introduce a simple model of a topological insulator with the intraband and interband spin–orbit coupling and investigate its bulk and boundary properties.