Classically Bound and Quantum Quasi-Bound States of an Electron on a Plane Adjacent to a Magnetic Monopole

Abstract

In three-dimensional space an electron moving in the field of a magnetic monopole has no bound states. In this paper we explore the physics when the electron is restricted to a two-dimensional plane adjacent to a magnetic monopole. We find bound states in the classical version of the problem and quasi-bound states in the quantum one, in addition to a continuum of scattering states. We calculate the lifetimes of the quasi-bound states using several complementary approximate methods, which agree well in the cases where the lifetimes are relatively short. The threshold monopole magnetic charge required to realise a single quasi-bound state is approximately $18Q_D$, where $Q_D$ is the magnetic charge of a Dirac monopole. We examine the feasibility of achieving this magnetic charge in currently available monopole analogues: spin ice, artificial spin ice, and magnetic needles.

Figures (16)

• Figure 1: An electron, of electric charge $q_e = -\vert q_e \vert$ and mass $m^{*}$, moving in a plane a distance $D$ above a magnetic monopole of charge $Q_{m}$. We use cylindrical coordinates with the axis passing through the monopole and the vertical distance measured from the plane ($z$) or alternatively from the monopole ($z'$). We decompose the magnetic field $\mathbf{B}$ into in-plane ${\bf B}_\parallel$ and out-of-plane ${\bf B}_\perp$ components. The vector potential corresponding to the perpendicular component is denoted by $\mathbf{A}=A_\phi(r)\hat{\hbox{\boldmath $\phi$}}$, where $\hat{\hbox{\boldmath $\phi$}}$ is the unit vector in the direction of increasing azimuthal coordinate.
• Figure 2: The effective classical potential governing the radial motion of the electron, $V_\text{cl}(\rho)/\lambda^2$. The shape of this potential depends only on the ratio $M/\lambda$, where $\hbar M$ is the canonical angular momentum of the electron and $\lambda$ is the strength of the magnetic monopole in units of twice the Dirac monopole charge. (a) $V_\text{cl}(\rho)/\lambda^2$ for $M/\lambda$ decreasing from 0 to $-1$ in steps of 0.1 in the direction of the arrow. (b) $V_\text{cl}(\rho)/\lambda^2$ for $M/\lambda$ increasing from 0 to 0.2 in steps of 0.02 in the direction of the arrow.
• Figure 3: Examples of the effective radial potential $V_{\rm cl}(\rho)$ (black curves) for three different values of the electron's angular momentum: (a) $M/\lambda=-0.01$; (b) $M/\lambda=0$; (c) $M/\lambda=0.01$. The blue line shows an example radial energy of $400 E_0$ for the case $\lambda=100$. The turning points for a particle of that radial energy that starts its radial motion within the potential well are shown as red circles.
• Figure 4: Selected orbits corresponding to the potentials seen in Fig. \ref{['Fig classic potentials for orbit']} (green curves), with one orbit starting within the well and one starting outside the well for the same value of the electron's energy: (a) $M/\lambda=-0.01$; (b) $M/\lambda=0$; (c) $M/\lambda=0.01$. The blue lines labelled 'Potential' indicate the radii at which $V_\text{cl}(\rho)=\epsilon$, i.e. the radial turning points. The inset is a zoomed-in view of the region inside the potential well, where all bound states are located. The particle that starts outside the well moves away from the monopole and is unbound.
• Figure 5: (a) Relationship between angular momentum, $M$, and distance from the plane's origin, $\rho$, for circular orbits. Above the curve all orbits are unbound; below the curve they are non-circular bound orbits (see Fig. \ref{['Fig classical all classical orbits']}). For values of $M/\lambda>0$ there are two circular orbits; otherwise only one circular orbit is present. Some circular orbits are stable while others are unstable. The stable circular orbits are found at the minimum of the effective potential for positive-valued $M/\lambda$; the unstable ones are found at the maximum of the effective potential for all values of $M/\lambda$. (b) The approximate number of bound states, scaled by $\lambda$, as a function of $M/\lambda$. This curve is calculated by modelling the system as a simple harmonic oscillator --- see text for details.