this is emphasized in the inset). The maximum of the probability distribution is always between zero and one. These curves are all cases for which all the moments are equal. As the alternate hoppings ($J'$) are turned on, the shifting occurs in a qualitatively different manner. Initially ($\Delta/J=-2$ in all cases) the curves are centered very near $x=1$. $\Delta/J=-2$ is for most cases well in the region where the moments are equal. As $\Delta$ decreases, the distributions shift, but they do this by becoming asymmetric about their mean, with the density increasing on the side left of the maxima of the distributions. The shape of the distributions changes considerably. This is clearly due to the fact that in these latter cases the moments vary as $\Delta$ is varied, and they are not all equal. The maxima for the cases for which $J'/J \neq 0$ shift much less as $\Delta/J$ is varied. When $\Delta/J$ changes sign (results not shown), the polarization becomes centered around $x=0$ end of the unit cell and the probability distributions are reflections of the ones shown in Fig. \[fig:prob\] across $x = 1/2$.

The probability distributions for the case $J'=J=1$ are also shown separately in Fig. \[fig:prob1\] 