Response of Bloch Electrons to Electric Fields

We consider the response of spinful Bloch electrons to external electric fields; the electrons’ unperturbed Hamiltonian is the one given by Dresselhaus [1], including the spin-orbit interaction. We write matrices for the system observables with respect to the Bloch basis states of the unperturbed Hamiltonian, truncate them to two bands, and use them to obtain their Heisenberg equations of motion. The equation for the velocity contains not only the anomalous velocity term obtained by Karplus and Luttinger [2] [expressed in terms of the now-familiar (position operator) Berry connection [3]], but also additional terms involving “quasi-connections” from the momentum and spin operators. We apply the equations of motion resulting from this procedure to a two-dimensional model of a ferromagnet with the Dresselhaus linear-in-wavevector spin-orbit interaction and an exchange field term. We find for this model that the anomalous velocity is dominated by the term coming from the spin operator rather than the position operator.
[1] G. Dresselhaus, Phys. Rev. 100, 580 (1955)
[2] R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954)
[3] D. Culcer, Y. Yao, Q. Niu, Phys. Rev. 72, 085110 (2005)