Flux-threaded quantum rotor ($E_n={\hslash }^2(n-f{)}^2/(2I),f=e{\Phi }_B/(hc)$)

For the Gaussian-unit flux term $e{\Phi }_B\dot{\varphi }/(2\pi c)$, canonical momentum is shifted by $e{\Phi }_B/(2\pi c)$. Periodic wavefunctions then give the displayed spectrum for integer $n$. The flux period is $hc/|e|$ in these units; the corresponding SI expressions omit $c$. A gauge change transfers the flux to a twisted boundary condition rather than removing it from the physical problem. Half-integer $f$ gives a pair of degenerate nearest-momentum levels.

The thermal trace can be expanded in angular-momentum eigenstates or in lifted paths whose endpoints differ by $2\pi m$. A winding saddle is linear in imaginary time, and its flux weight is the Aharonov-Bohm phase $e^{2\pi ifm}$. The common fixed-endpoint fluctuation factor is $\sqrt{I/(2\pi {\hslash }^2\beta )}$; integrating the initial angle supplies $2\pi$. The Poisson summation formula converts the resulting winding Gaussian sum into the momentum sum with exactly cancelling prefactors. The Gaussian evaluation is exact for this free rotor.