model is metallic for $J=J'$ and $\Delta=0$ but is insulating for all other values of the parameters. In reciprocal space this Hamiltonian becomes $$\hat{H} = \sum_k \left( 
\begin{array}{cc} \Delta & -\rho_k \\ 
-\rho_k^* & -\Delta, 
\end{array}
\right)$$ where $$\rho_k = J e^{ik} + J' e^{-ik}.$$ At a particular value of $k$ we can write the eigenstate for the lower band as $$\left( 
\begin{array}{c} \alpha_k \\ 
\beta_k
\end{array}
\right) = 
\left( 
\begin{array}{c} \sin\left(\frac{\theta_k}{2}\right) \\ 
e^{-i \phi_k}\cos\left(\frac{\theta_k}{2}\right)
\end{array}
\right),$$ where $$\begin{aligned}
\theta_k &=& \arctan \left( \frac{|\rho_k|}{\Delta}\right) \\
\phi_k &=& \arctan \left( \frac{(J-J')\sin(k)}{(J+J')\cos(k)}\right). \nonumber\end{aligned}$$ The cumulants can now be written in terms of the eigenstates. For example, $$C_1 = \frac{i}{\pi} \int_{-\pi/2}^{\pi/2} dk ( \alpha_k^* \partial_k \alpha_k + \beta_k^*
\partial_k \beta_k ),$$ and the other cumulants can be constructed accordingly (note that the unit cell is $L=2$).

Fully dimerized limit {#sec:FDL}
=====================

Here we show that in the fully dimerized limit the GIMs should all have the same magnitude. In Ref. [@Hetenyi14] we pointed out that the Berry phase can be related to an observable $\hat{O}$ fixed by requiring that $$\label{eqn:cnd}
\partial_K H(K) = i [H(K),\hat{O}].$$ This definition does not uniquely fix the operator $\hat{O}$. For example, for the magnetic field example the matrix 