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I have a question about a small but meaningful (to me at least) step in the original TKNN paper (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.405). I understand the construction of the magnetic translation operators and the use of the Kubo formula to get to the following formula (Eq. 5):

\begin{equation} \sigma_\mathrm{H} = \frac{ie^2}{2\pi h}\sum \oint dk_j \int d^2r \left(u^*\frac{\partial u}{\partial k_j} - \frac{\partial u^*}{\partial k_j}u\right). \end{equation}

However, the following few lines read: "For nonoverlapping subbands $\psi$ is a single-valued analytic function everywhere in the unit cell, which can only change by an $r$-independent phase factor $\theta$ when $k_1$ is changed by $2\pi/aq$ or $k_2$ by $2\pi/b$. The integrand reduces to $\partial \theta/\partial k_j$."

If you can say $u_k(r) = \left| u(r)\right| e^{i\theta(k)}$ then $u^*\frac{\partial u}{\partial k_j} - \frac{\partial u^*}{\partial k_j}u = 2i\frac{\partial \theta}{\partial k_j}\left|u\right|^2$, and the rest of the results follow.

But why must there exists an $r$-independent phase factor when translating to the boundary of the magnetic Brillouin zone? The values of $k_1$ and $k_2$ are only defined modulo $2\pi/aq$ and $2\pi/b$ to begin with. In the nonmagnetic situation where you have the standard Bloch relations, you find $\psi_{k+Q} = \psi_k$. Clearly the difference in the case of the magnetic unit cell is related to the gauge potential, but I'm having trouble seeing the connection. Specifically, I'm trying to understand how the Berry's phase arrises in this derivation. Could anyone flesh out the origin of $\theta$ as it relates to this derivation in particular?

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The OP's main concern is about:

But why must there exists an r-independent phase factor when translating to the boundary of the magnetic Brillouin zone? The values of $k_1$ and $k_2$ are only defined modulo $2π/a$ and $2π/b$ to begin with.

Answer: for "$k_1$ and $k_2$ are only defined modulo $2π/a$ and $2π/b$", the wave function of $\psi(r;k_1,k_2)$ and $\psi(r;k_1+2\pi/a)$ and $\psi(r;k_1, k_2+2\pi/b)$ must describe the same physical states. Therefore, they must differ at most a global phase factor, that is, a phase factor at most depend only on $k$, not $r$.

Otherwise, suppose $\psi(r;k_1+2\pi/a, k_2)=e^{i\theta(r)}\psi(r; k_1,k_2)$, although the modulo $|\psi|^2$ is unchanged, however some physical quantity, such as the expected momentum, which is defined as $\left<\psi|-i\hbar\nabla|\psi\right>$ will change because the r-dependent phase factor.

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