What would it mean for Aharanov-Bohm effect in a gauge in which the vector potential vanishes everywhere outside the solenoid? Is it possible to find a gauge in which the vector potential outside the solenoid (with axis along $z$-axis) is made equal to zero everywhere? If so, wouldn't the phase difference in the Aharanov-Bohm experiment be equal to zero in that gauge when two electron beams interfere anywhere on the screen?

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I did not follow the "holonomy bit" of the answer by Chiral Anomaly because I do not know the topological aspects of the AB effect. Leaving that aside, I get the sense from his answer that it is not possible to find a single gauge that would cover the entire XY plane. Please help me understand that.
Suppose, $\Phi_B$ denotes the flux through an infinitely long solenoid. A valid choice of the vector potential is $${\vec A}=\frac{\Phi_B}{2\pi r^2}(-y\hat{x}+x\hat{y}).$$ Now, choosing a scalar function $$\alpha(\vec r)=-\frac{\Phi_B}{2\pi}\phi,$$ (where $\phi$ is the angle in the XY plane in the plane polar coordinates, $0\leq \phi<2\pi$) and defining a new vector potential ${\vec A}'=A+\nabla\alpha$, we seem to trivially make ${\vec A}'$ vanish everywhere. I can understand that this is wrong because it makes ${\vec A}$ vanish everywhere and thus ${\vec B}$ too. Please point out what's wrong with this gauge.
 A: This answer assumes the question is asking about an infinitely long solenoid, so that the magnetic field outside the solenoid is zero even though it's nonzero inside.
In any contractible region of space where the magnetic field is zero, we can choose the gauge so that the vector potential is zero. But the space outside an infinitely-long solenoid is not contractible. We can choose a gauge so that the vector potential is zero almost everywhere outside the solenoid, but it must remain nonzero somewhere so that the holonomy around the solenoid is nonzero. The holonomy is gauge-invariant, as is interference pattern produced when an electron passes around both sides of the solenoid.
Alternatively, we could cover the space outside the solenoid with two overlapping patches, each of which is contractible by itself. Then we can use different gauge transformations in each patch to make the respective vector potentials zero in each patch. The nonzero holonomy is encoded in the transition functions that relate the vector fields in the two patches where they overlap.
Related: Small confusion about the Aharonov-Bohm effect
