Universal semiclassical equations based on the quantum metric for a two-band system

We derive semiclassical equations of motion for an accelerated wave packet in a two-band system. We show that these equations can be formulated in terms of the static band geometry described by the quantum metric. We consider the specific cases of the Rashba Hamiltonian with and without a Zeeman term. The semiclassical trajectories are in full agreement with the ones found by solving the Schrödinger equation. This formalism successfully describes the adiabatic limit and the anomalous Hall effect traditionally attributed to Berry curvature. It also describes the opposite limit of coherent band superposition, giving rise to a spatially oscillating Zitterbewegung motion, and all intermediate cases. At $k=0$, such a wave packet exhibits a circular trajectory in real space, with its radius given by the square root of the quantum metric. This quantity appears as a universal length scale, providing a geometrical origin of the Compton wavelength. The quantum metric semiclassical approach could be extended to an arbitrary number of bands.

Figure 1

(a) Bloch sphere representation showing the pseudospin $\mathbf{S}$, the effective field $\mathrm{\Omega }$, and their polar and azimuthal angles $\theta$ and $\varphi$. The angle between the spin and the field is $\eta$. (b), (c) Dispersion along $k_x$ of the eigenmodes of the (b) Rashba and (c) $\text{Rashba}+\text{Zeeman}$ Hamiltonians. The red arrows show the pseudo-spin orientation of the modes.

Figure 2

Rashba Hamiltonian. Wave packet dynamics from Schrödinger equation (black solid line) and analytical solution of semiclassical Eqs. (5) and (6) (red dashed line).

Figure 3

$\text{Rashba}+\text{Zeeman}$ Hamiltonian. Wave packet dynamics from Schrödinger (black solid lines) and semiclassical (red dashed lines) equations: (a) AHE (single-band initial excitation); (b) ZBW (equal fractions of both bands); (c) both effects together; (d) cyclotronlike orbits at constant $k$ (no potential gradient) with a radius determined by the metric.