Is the Aharonov-Casher phase geometrical or dynamical?

• URL: http://arxiv.org/abs/2608.12427v1
• Date: Wed, 12 Aug 2026 11:25:35 GMT
• Title: Is the Aharonov-Casher phase geometrical or dynamical?
• Abstract summary: We consider two two-dimensional (2D) electronic systems in the presence of a homogeneous electric field that generates a Rashba spin-orbit interaction (RSOI)<n>$bf A_R$ cannot be eliminated from either the 2D SE or the 2D DE via a gauge transformation.<n>For a plane wave solution, an $SU(2)$ matrix exists that eliminates $bf A_R$ from the resulting 1D SE.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 Schrödinger 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 Schrödinger 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 Schrödinger AC phase which is geometrical.

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