Quantum particle on a ring with magnetic flux through it: Can we gauge away the magnetic field?

Question

Consider a quantum particle on a ring and a non-zero homogeneous magnetic field perpendicular to the disk that the ring defines and is non-zero only in the inside of the perimeter of the ring. Let $\vec{B}=B_0 \hat{z}$ and the flux through the ring be $\Phi$.
For the vector potential, we can choose (in cylindrical coordinates) $\vec{A}=\dfrac{\Phi}{2\pi}\dfrac{1}{\rho}\hat{\phi}$.

If I try to perform a gauge-transformation $\vec{A'}=\vec{A}-\vec{\nabla}f$ to gauge-away the magnetic field by going to a new gauge where $\vec{A'}=\vec{0}$, I find $f=\dfrac{\Phi}{2\pi}\phi$. So, since I have found a function $f$ to do this, it seems as if I have successfully gauged-away the magnetic field, which is physically impossible!

What is happening here?
I suspect that something is not right because $f$ is multi-valued at $x=0$ (which corresponds to $\phi=0, 2\pi, ..)$. If this is the case, how do I fix this and get $\vec{A'}\neq0$? Is there a systematic way to treat cases such as this, i.e. to find a proper $f$ that would deal with this problem and give the correct magnetic field?

Answer 1

Describing the problem as "because $f$ is multi-valued" is one way to describe the problem, but it's not actually the root of the problem. The root of the problem is the singularity at $r=0$. In order to qualify as a gauge transformation, the function needs to obey $$\nabla\times \nabla f(\mathbf{x},t) = 0$$ everywhere.

It looks like this function does if you just take the derivatives, but that's misleading. $-\frac{1}{4\pi}\nabla^2 r^{-1}$ looks like it vanishes by the same critereon when it is, in fact, the Dirac delta function. Same thing happens here. Just like how it is possible to show the existence of the delta function in the divergence of the gradient case using the divergence theorem, an application of Stoke's theorem will show you that $$\oint_{\gamma} \nabla f \cdot \operatorname{d}\vec{s} \neq 0$$ whenever the origin is enclosed by the loop, $\gamma$, proving that the gauge transformation is invalid.