# Magnetic Bloch Functions and Diophantine Equation

I'm trying to understand the paper by Dana, Avron, and Zak  in which they prove the Diophantine equation for Hall Conductivity for an arbitrary periodic Hamiltonian using just magnetic translation symmetry.

They consider a periodic system with lattice vectors $\mathbf{a},\mathbf{b}$ and a magnetic flux of $\phi = 2\pi p / q$ per plaquette. The commuting magnetic translation operators are given by $T(q\mathbf{a})$ and $T(\mathbf{b})$ (so the magnetic unit cell contains q lattice unit cells in the $\mathbf{a}$ direction). The magnetic translation operators commute with each other and the Hamiltonian and so the eigenstates can be written as $\Psi = \psi_{k_1,k_2}e^{i\mathbf{k}\cdot\mathbf{r}}$ where $\psi_{k_1,k_2}$ are the magnetic Bloch functions.

The part of the paper I am unsure about is that they claim that the phases of the magnetic Bloch functions can be chosen so that they satisfy the following periodicity conditions: $$\psi_{k_1 + 2\pi/qa,k_2} = \psi_{k_1,k_2} \\ \psi_{k_1,k_2+2\pi/b} = \exp(i\sigma k_1 q a) \psi_{k_1,k_2}$$ where $\sigma$ is an integer (and turns out to be the Chern number of the band following TKNN). My question is: how does one show this?

In the paper they say that this can be shown using arguments similar to Weinreich's book (Solids: Elementary Theory for Advanced Students). I think they are referring to section 8.3 in which he shows that Bloch functions (in the absence of a magnetic field) are periodic in quasi-momentum $\mathbf{k}$. He shows this by writing the eigenstate in Bloch form as $\Psi(\mathbf{r}) = \psi_{\mathbf{k}}(\mathbf{\rho})e^{i\mathbf{k}\cdot\mathbf{R}}$ where $\mathbf{r} = \mathbf{\rho} + \mathbf{R}$ and $\mathbf{R}$ is a lattice vector while $\mathbf{\rho}$ is a vector in the proximity/unit cell. Since $\Psi$ is an eigenfunction of the translation operator, it follows that if $\mathbf{\sigma}$ and $\mathbf{\sigma}'$ are vectors at the edges of the unit cell such that they differ by a lattice vector $\mathbf{R}_\sigma$ (i.e. $\mathbf{\sigma}' = \mathbf{\sigma} + \mathbf{R}_\sigma$) then $$\psi_{\mathbf{k}}(\sigma') = \psi_{\mathbf{k}}(\sigma) e^{i\mathbf{k}\cdot \mathbf{R}_\sigma}$$ which is a boundary condition for $\psi_{\mathbf{k}}$. If one considers another wave-vector $\mathbf{k}' = \mathbf{k} + \mathbf{G}$ where $\mathbf{G}$ is a reciprocal lattice vector, then we have that $e^{i\mathbf{k}\cdot \mathbf{R}_\sigma} = e^{i\mathbf{k}' \cdot \mathbf{R}_\sigma}$ and so $\psi_{\mathbf{k}'}$ obeys the same BC as $\psi_{\mathbf{k}}$: $$\psi_{\mathbf{k}'}(\sigma') = \psi_{\mathbf{k}'}(\sigma) e^{i\mathbf{k}\cdot \mathbf{R}_\sigma}$$. Since they obey the same Schroedinger equation and BC, one can always choose the same phase conventions for $\psi_{\mathbf{k}}$ and $\psi_{\mathbf{k}'}$ such that $\psi_{\mathbf{k}}$ is periodic in $\mathbf{k}$.

In order to apply this to the case with magnetic translation symmetry, it seems like I just need to consider the corresponding BC coming from the magnetic translation operators. If I do that however, the above BC would be modified by a position-dependent phase factor and I don't see how I can massage those into the form that appear in the Dana, Avron, Zak paper which have a momentum-dependent phase factor.

Any help would be much appreciated.

 I. Dana, Y. Avron, and J. Zak, Journal of Physics C: Solid State Physics 18, L679 (1985)