\pi$ times an integer [@Xiao10; @Vanderbilt93], the others do not change at all due to a gauge transformation. The odd GIMs depend on combinations of the GICs which involve odd combinations of the cumulants, therefore they are not multivalued in general. In both sets of figures (\[fig:CircleR1\] and \[fig:CircleR02\]) the points $A^*$ are not exactly on the $\phi = -\pi/2$ axis, but instead we numerically realize the limit $\phi = \lim_{\delta \Phi
    \rightarrow 0^+} (-\pi/2 + \delta \Phi)$. In the actual calculation we took $\delta \Phi = 2 \pi/1000$. Also, the point $\phi = -\pi/2$ or $\phi =
  3\pi/2$ is excluded from the curves shown in the upper panels of the two figures.

The example with radius unity (Fig. \[fig:CircleR1\]) remains mostly in the fully dimerized limit, as can be seen in the upper panel of the figure. The odd moments and even moments are always equal. Except for a small region near $\phi/\pi = 0.5$ the absolute values of the moments are equal. The lower panel shows the evolution of the probability distribution along the path. Starting from a relatively sharp distribution localized near $x=1$, the maximum moves to the left. Before reaching half the unit cell, the distribution 