Quaternionic Dirac oscillator

Abstract

We construct an elementary quaternionic slow–fast Hamiltonian dynamical system with one formal control parameter and two slow degrees of freedom as half-integer spin in resonance 1:1:2 with two slow oscillators. Invariant under spin reversal and having a codimension-5 crossing of its fast Kramers-degenerate semi-quantum eigenvalues, our system is the dynamical equivalent of the spin-quadrupole model by Avron et al (1989 Commun. Math. Phys. 124 595–627), exhibiting non-Abelian geometric phases. The equivalence is uncovered through the equality of the spectral flow between quantum superbands and Chern numbers c 2 computed by Avron et al.