Particle in a Magnetic Field: Momentum Representation of the Hamiltonian

We consider a particle in a magnetic field \(\vec{B}\), which derives from the vector potential \(\vec{A}\) by \(\vec{B} = \nabla \times \vec{A}\). The Schrödinger equation in position representation reads

$$
\hat{H}_{\vec{k}} \, u_{n, \vec{k}} = \frac{\hbar^2}{2m} \left(-i \nabla + \vec{k} – \vec{A} \right)^2 u_{n, \vec{k}} = \varepsilon_{n, \vec{k}} \, u_{n, \vec{k}}
$$

for the Bloch wavefunction \(\psi_{n, \vec{k}}(\vec{r}) = e^{i \vec{k} \cdot \vec{r}} u_{n, \vec{k}}(\vec{r})\). Next, we employ rationalized units \(\hbar = c = e = 1\) and expand the Hamiltonian

$$
\hat{H}_{\vec{k}} u_{n, \vec{k}} = \frac{1}{2} \left( – \Delta u_{n, \vec{k}} + 2 i (\vec{A} – \vec{k}) \cdot \nabla u_{n, \vec{k}} + (\vec{A} – \vec{k})^2 u_{n, \vec{k}} \right)
$$

The momentum representation is derived by inserting a “one” composed of position eigenstates as we obtained already position representation:

$$
\langle k | \hat{H}_{\vec{k}} | n, \vec{k} \rangle
= \int d^2{r} \langle k | r \rangle \langle r | \hat{H}_{\vec{k}} | n, \vec{k} \rangle
$$

$$
= \frac{1}{2} \frac{1}{S} \int d^2{r} \, e^{-i \vec{g} \cdot \vec{r}} \left[ – \Delta u_{n, \vec{k}} + 2 i (\vec{A} – \vec{k}) \cdot \nabla u_{n, \vec{k}} + (\vec{A} – \vec{k})^2 u_{n, \vec{k}} \right]
$$

$$
\begin{aligned}
= \frac{1}{2} & [
\vec{g}^2 u_{n, \vec{k}}(\vec{g}) \delta_{\vec{g},\vec{g}^\prime}
+ 2 (-\vec{g}) \vec{A}(\vec{g} -\vec{g}^\prime) u_{n, \vec{k}}(\vec{g})
+ 2 \vec{k} \cdot \vec{g} u_{n, \vec{k}}(\vec{g}) \delta_{\vec{g},\vec{g}^\prime} \\
& + \vec{A}^2(\vec{g} -\vec{g}^\prime) u_{n, \vec{k}}(\vec{g})
– 2 \vec{A}(\vec{g} -\vec{g}^\prime) \cdot \vec{k} u_{n, \vec{k}}(\vec{g})
+ \vec{k}^2 u_{n, \vec{k}}(\vec{g}) \delta_{\vec{g},\vec{g}^\prime}
]
\end{aligned}
$$

$$
\begin{aligned}
= \frac{1}{2} & [
(\vec{k}^2 + \vec{g}^2) u_{n, \vec{k}}(\vec{g}) \delta_{\vec{g},\vec{g}^\prime}
– 2 (\vec{g}) \vec{A}(\vec{g} -\vec{g}^\prime) \cdot (\vec{k} + \vec{g}) u_{n, \vec{k}}(\vec{g}) \\
& + \vec{A}^2(\vec{g} -\vec{g}^\prime) u_{n, \vec{k}}(\vec{g})
]
\end{aligned}
$$

with the convolution

$$
\vec{A}^2(\vec{r}) = \sum \vec{A}(\vec{g} -\vec{g}^\prime) \cdot \vec{A}(\vec{g}^\prime)
$$

So the Hamiltonian in momentum representation is given by

$$
\hat{H}_{\vec{k}}(\vec{g},\vec{g}^\prime) = \frac{1}{2} \left[
(\vec{k}^2 + \vec{g}^2) \delta_{\vec{g},\vec{g}^\prime}
– 2 (\vec{g}) \vec{A}(\vec{g} -\vec{g}^\prime) \cdot (\vec{k} + \vec{g}) + \vec{A}^2(\vec{g} -\vec{g}^\prime)
\right]
$$

in the Coulomb gauge \(\nabla \cdot \vec{A} = 0\) and \(\vec{g} \cdot \vec{A} = 0\), respectively.

This problem is treated by Taillefumier at al. (2018) on arXiv

Taillefumier, Mathieu et al. “Chiral two-dimensional electron gas in a periodic magnetic field: Persistent current and quantized anomalous Hall effect.” Physical Review B 78 (2008): 155330.
https://arxiv.org/abs/0807.0707v1