(with $\chi^2$ tabulated in Table \[tab:chi2\]), as well as the case $J'$ close to $J$. All of the $J'=J$ distributions are symmetric around the origin. As $\Delta/J \rightarrow 0$ the distribution broadens, and it is clear that a conducting phase is approached [@Resta99]. If $\Delta<0$ then the polarizations are localized near $x = \pm1$ depending on whether $J'$ is smaller or larger than $J$. This is consistent with Fig. \[fig:mom1\].

In Figs. \[fig:CircleR1\] and \[fig:CircleR02\] we show the evolution of the reconstructed probability distributions along two cyclic paths which encircle the topologically non-trivial point of the RM model, one with radius unity, the other with radius $0.2$ in the $\Delta/t$, $\delta/t$ plane. In these calculations the parametrization was different from the previous ones, here $t$ was set to unity, rather than $J$ (see Eq. (\[eqn:tdelta\])). For the points $A^*$, $B$, ... in Figs. \[fig:CircleR1\] and \[fig:CircleR02\] the values of $\Delta/J$ and $J'/J$ are shown in Table \[tab:chi2\_circles\]. The upper panel in both figures show the evolution of the different GIMs(GICs). The even moments are single-valued, the odd ones are not. This follows from gauge invariance properties of the cumulants (Eq. (\[eqn:cmlnts\])). The first cumulant is only gauge invariant modulo $2 