Is the Aharonov-Casher phase geometrical or dynamical?

Abstract

We consider two two-dimensional (2D) electronic systems in the presence of a perpendicular homogeneous electric field that generates a Rashba spin-orbit interaction (RSOI): a system of non-interacting electrons in a 2D conductor, modeled using the 2D Schr\"odinger equation (SE), and a single-layer graphene system, modeled using a 2D Dirac equation (DE) for massless fermions. In both cases the RSOI is expressed via an $SU(2)$ Rashba vector potential ${\bf A}_{R}$. We demonstrate that ${\bf A}_{R}$ cannot be eliminated from either the 2D SE or the 2D DE via a gauge transformation. Nevertheless, for a plane wave solution, an $SU(2)$ matrix exists that eliminates ${\bf A}_{R}$ from the resulting 1D SE. This unitary matrix is an Aharonov-Casher (AC) phase factor, and facilitates the calculation of the AC phase in the Schr\"odinger scheme. The plane wave solution for the DE contains two components of ${\bf A}_{R}$: $A_{R, k}$ in the direction of the wave vector ${\bf k}$, and $A_{R, n}$ normal to ${\bf k}$. The latter generates an effective electron mass that cannot be eliminated from the DE. The former generates an AC phase that can be eliminated by a time-dependent unitary transformation. Thus, the Dirac AC phase is time-dependent, i.e., it is a dynamical phase. This is in contradistinction to the Schr\"odinger AC phase which is geometrical.