# Crystal momentum and the vector potential

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.

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.

There is indeed a similarity between the operator for the kinetic momentum in the presence of a vector potential and the momentum operator-like operator acting on the Bloch functions: the kinetic energy operators constructed from taking the square of each of the two have antisymmetric imaginary parts. The kinetic operators are thus still Hermitian, but they lead to complex wave and Bloch functions, respectively, that generally cannot chosen to be real.

At this point, very important differences have to be noted, though. Take a single particle system with time-reversal symmetry: in this case states with crystal momentum $$+k$$ and $$-k$$ are degenerate. By linear combination of the states $$u_{\pm k}(x)\cdot\exp(i\cdot(\pm k)\cdot x)$$ real wave functions can be constructed (as is always the case for single particle systems with time-reversal symmetry).

The presence of the vector potential, on the other hand, generally means that time-reversal symmetry is broken, and wave functions cannot be chosen to be real.

Laxly speaking, the imaginary terms in the kinetic energy operator in the presence of a vector potential have physical implications (add imaginary terms to time-dependent Schrödinger equation $$\Rightarrow$$ time-reversal symmetry is broken). The imaginary terms in the kinetic operator acting only on the Bloch part of the wave functions are "merely" the result of choosing to decompose the wave functions in this way to take advantage of translational symmetry in solving the Schrödinger equation.

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.

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