spreads. After passing through the midpoint the system, where the maximum is the smallest, the distribution begins to localize again until $x=0$. From there this tendency is repeated. Indeed, the distribution ends up at $x=-1$ at the end of the process: the Wannier function “walked” to an equivalent position in the next unit cell. For the case of the smaller radius (0.2, Fig. \[fig:CircleR1\]) the initial distribution is broader, and as the cycle is traversed, the maximum of the distribution oscillates with a smaller amplitude, but the “walking” to a new equivalent position still occurs.

In both Figs. \[fig:CircleR1\] and \[fig:CircleR02\] it is clear that the odd moments do not correspond to single-valued functions. The values of the odd moments depend on whether we approach the original point from which the cycle begins ($\delta = 0, \Delta<0$) from the left or the right. At the same time, the probability distributions for some cases with $\delta = 0, \Delta<0$ are shown in Fig. \[fig:prob1\]; they are centered around zero and they spread as $\Delta/J$ is decreased. This suggests the limiting cases from either direction give different results from the result for fixing the Hamiltonian parameters such that $\delta = 0,\Delta<0$.

Conclusion
