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I noticed that the Aharonov–Bohm effect describes a phase factor given by $e^{\frac{i}{\hbar}\int_{\partial\gamma}q A_\mu dx^\mu}$. I also recognize that electrons in a periodic potential gain a phase factor given by $e^{\frac{i}{\hbar}k_ix^i}=e^{\frac{i}{\hbar}\int k_idx^i}$. I recall that $k_i$ plays a role analogous to momentum in solid state physics. I also recall that the canonical momentum operator is $P_\mu=-i\hbar\partial_\mu-qA_\mu$. Notice that when you operate with the momentum operator on a Bloch electron, $\psi(x)=u(x)e^{\frac{i}{\hbar}k_ix^i}$, you get $e^{\frac{i}{\hbar}k_ix^i}(-i\hbar\partial_i+k_i)u(x)$.

My question is whether a parallel can be drawn between the crystal momentum, $k$, and the vector potential $A$. It seems they play a similar role quantum mechanically, but I have never seen Bloch's theorem described in terms of vector potentials. I suppose one does not even need a nontrivial vector potential for Bloch's theorem to hold. Still, crystal momentum and the vector potential play very similar roles in phase factors and I wonder whether there is any deeper meaning to that.

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Because crystal momentum and the vector potential appear together, introducing the vector potential changes the conserved quantity from just crystal momentum to crystal momentum + electromagnetic momentum.

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While all the statements you made about crystal momentum only apply exactly for Bloch states in which the momentum operator is diagonal, the fact that the phase due to the vector potential is $e^{i \int A}$ is true for all states in the one-charged-particle Hilbert space.
This is of course a manifestation of the fact that momentum and an EM field are physically very different and any analogy you want to draw won't go very far.

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Just as V is the potential energy, $\vec{A}$ is the potential momentum. This is also true in a crystal.

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