Magnetic flux spectrum of a quantum ring ($E_n=E_L(n-\alpha {)}^2$)

For circumference $L$ and Hamiltonian operator $(p-a{)}^2/(2m)$ with a periodic boundary condition, the plane waves $L^{-1/2}{e}^{2\pi inx/L}$ have energies $E_n=E_L(n-\alpha {)}^2$, where $E_L=2{\pi }^2{\hslash }^2/(m{L}^2)$ and $\alpha =aL/(2\pi \hslash )$. Minimal coupling gives $aL=q{\Phi }_C$, where ${\Phi }_C$ is the magnetic flux oriented consistently with increasing position around the ring. The ordered spectrum is periodic in $\alpha$ with period one, corresponding to magnetic flux quantum $h/|q|$.

On $|\alpha |\le 1/2$, its lowest two energies are $E_L{\alpha }^2$ and $E_L(1-|\alpha |{)}^2$. Their crossings at half-integer $\alpha$ can be opened by a potential that couples the relevant momentum states.

In the plane wave basis of a ring, $V(x)=-\lambda \cos (2\pi x/L)$ has matrix elements $V_{mn}=-(\lambda /2)({\delta }_{m,n+1}+{\delta }_{m,n-1})$. Near $\alpha =1/2$, degenerate perturbation theory on $n=0,1$ givesup to corrections from remote states. Thus the avoided crossing gap is $|\lambda |$ to first order when $|\lambda |\ll E_L$.

For $|\lambda |\gg E_L$, a parabolic approximation about a potential minimum instead gives a local quantum harmonic oscillator spacing $\sqrt{2|\lambda |E_L}$. A local gauge transformation removes the flux from this approximation. Actual ring levels retain exponentially small flux dependence through quantum tunnelling, while nonquadratic corrections to their mean energies are algebraic in $\sqrt{E_L/|\lambda |}$.