The Aharonov-Casher phase is the electromagnetic dual of the Aharonov-Bohm phase. It arises when a neutral particle with a magnetic moment encircles, for example, a line charge, or moves on a closed circle around a point charge. The simple plausibility argument for the effect to occur uses the relativistic transformation of the electric field generated by the line charge (or point charge) into the rest frame of the moving particle, in which the neutral particle 'sees' a magnetic field $$ {\mathbf B} = \frac{1}{c^2}{\mathbf v}\times{\mathbf E}. $$ This magnetic field couples to the magnetic moment of the moving particle and leads to the Aharonov-Casher phase. Details are nicely worked out in an answer to the SE-question here. The existence of the phase has been experimentally measured with neutron interferometry in this reference.

A similar plausibility argument is used, when a charged particle with a magnetic moment moves, in atomic physics, through the electric field of the atomic nucleus, or, in solid state physics, through that of the crystal lattice. This is the well-known spin-orbit coupling, which can, in suitable solid-state interferometers, also lead to a phase-difference in interference experiments.

My question is: are there fundamental arguments for or against the notion that spin-orbit interaction effects are essentially a manifestation of the Aharonov-Casher effect? Or are they completely different things?

From my perspective, an essential difference is the absence of the charge of the moving particle in Aharonov-Casher physics, and its presence in spin-orbit physics. I have seen this question and its answer on SE about the Aharonov-Casher effect for charged particles. However, there is no mention of spin-orbit interaction, and the answer was very technical. There are also numerous questions and answers about spin-orbit coupling, which did not clarify the question. The topic may be controversial, but all the different opinions would be highly appreciated.


1 Answer 1


Thanks for the A2A. The spin orbit term, responsible for many very known phenomena such as the Stern-Gerlach deflection or the fine structure in the hydrogen atom spectrum, is exactly the same term responsible for the Aharonov-Casher effect.

The only difference is that in the cases, except the Aharonov-Casher effect, the spin-orbit term in the Hamiltonian results in effects on the motion of the particle; while in the Aharonov-Casher's case, it only adds a phase to the wave function of the particle, i.e., it is a purely quantum mechanical effect.

Thus, all the manifestations of the spin-orbit term are, in addition, accompanied by an Aharonov-Casher phase.

The Aharonov-Casher may be better understood, when the non-relativistic Pauli Hamiltonian is written in the Anandan-Goldhaber form (please see Oh and Ryu (Equation 1): $$H = \frac{1}{2m}\left(\mathbf{p}-\frac{e}{c} \mathbf{A} - \frac{\mu}{c} \mathbf{a}\right)^2 + e A^0 + \mu a^0$$ Where: $\mu = \frac{gq\hbar}{4mc}$ is the magnetic moment $ A^{\mu}=(\mathbf{A}, A^0)$ is the U(1) electromagnetic four-vector potential and: $$a^{\mu} = (\mathbf{a}, a^0) = (-\mathbf{\sigma}\cdot\mathbf{B}, \mathbf{\sigma}\times\mathbf{\frac{E}{2}})$$ Is an $SU(2)$ effective gauge field originating from the degeneracy in the underlying Dirac equation. (Thus, the Pauli equation has a $U(1)\times SU(2)$ gauge symmetry).

The Aharonov-Casher phase in this picture is exactly the Berry phase associated with the effective $SU(2)$ symmetry. $$\phi_{AC} = \mathcal{P} e^{-i\frac{\mu}{\hbar c}\oint a_{\mu} dx^{\mu}}$$ Remarks:

  1. The issue of dependence on the charge. When the electric field is expressed in terms of the scalar potential: $$\mathbf{E} = \frac{1}{q} \mathbb{\nabla}V,$$ the dependence on the electric charge cancels.

  2. When physics is restricted to two space-dimensions (i.e., in 2+1 dimensions), then in the case of point particles with Coulomb interaction, the spin-orbit $SU(2)$ gauge potential $\mathbb{a}$ becomes a pure gauge term, (thus does not affect the equations of motion), and thus its only surviving manifestation is through the Aharonov-Casher phase, which becomes a topological phase. This is why this term is so important to non-Abelian statistics and anyonic quantum computation.

  • 1
    $\begingroup$ Thank you very much for your detailed answer. I have to work through it step by step. Minor remark: I guess in the hamiltonian the expression in round brackets should be squared. $\endgroup$
    – flaudemus
    Commented Feb 20, 2019 at 17:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.