Eigen Wavefunctions of a Charged Particle Moving in a Self-Linking Magnetic Field

Abstract

In this paper we solve the one-particle Schr\"{o}dinger equation in a magnetic field whose flux lines exhibit mutual linking. To make this problem analytically tractable, we consider a high-symmetry situation where the particle moves in a three-sphere $(S^3)$. The vector potential is obtained from the Berry connection of the two by two Hamiltonian $H(\v{r})=\hat{h}(\v{r}) \cdot\vec{\sigma}$, where $\v{r}\in S^3$, $\hat{h}\in S^2$ and $\vec{\sigma}$ are the Pauli matrices. In order to produce linking flux lines, the map $\hat{h}:S^3\to S^2$ is made to possess nontrivial homotopy. The problem is exactly solvable for a particular mapping ($\hat{h}$) . The resulting eigenfunctions are SO(4) spherical harmonics, the same as those when the magnetic field is absent. The highly nontrivial magnetic field lifts the degeneracy in the energy spectrum in a way reminiscent of the Zeeman effect.