Semiclassical Theory of Spin-Orbit Interactions using Spin Coherent States

We formulate a semiclassical theory for systems with spin-orbit interactions. Using spin coherent states, we start from the path integral in an extended phase space, formulate the classical dynamics of the coupled orbital and spin degrees of freedom, and calculate the ingredients of Gutzwiller’s trace formula for the density of states. For a two-dimensional quantum dot with a spin-orbit interaction of Rashba type, we obtain satisfactory agreement with fully quantum-mechanical calculations. The mode-conversion problem, which arose in an earlier semiclassical approach, has hereby been overcome.

Figure 1

Periodic orbits in the two-dimensional harmonic oscillator with Rashba spin-orbit interaction (see text for parameters). Left panels: orbits in the $(x,y)$ plane. Right panels: Spin components $n_x$, $n_y$, and $n_z$ versus time. From top to bottom: Adiabatic orbit $A_x^{+}$ along $x$ axis with polarized spin in $y$ direction, and diabatic orbits $D_{x1}^{+}$ and $D_{x2}^{+}$ oscillating around the $x$ axis with spin rotating near the $(n_x,n_z)$ plane.

Figure 2

Oscillating part of the coarse-grained density of states (Gaussian averaging parameter $\gamma =0.6$) versus energy for the same system as in Fig. 1. Solid line: quantum-mechanical result; dashed line: semiclassical result using the 12 primitive orbits.