lengths, ranging in odd numbers from $3$ to $11$, and shown in Fig. \[fig:spinonehalf\] as increasing from the top. Note that the results for even numbers of spins have been omitted as they showed no qualitative difference. This may be due to finite-size effects of the kind which have been observed in similar studies elsewhere, see for example Ref. [@Kay]. The left-hand column of the figure depicts the Bargmann invariants $\mathcal{C}$ on the unit disk in the complex plane. Each data point of the graph stems from a different value of the coupling parameter $J$. The right-hand column shows the corresponding Bargmann phase $\varphi/\pi$, as a function of the coupling parameter $(J-J_c)/r$, where $r$ refers to the radius of the circle within which the $\mathcal{N}$-polygon circuit is inscribed. Note that in this case the radius was chosen as $10^{-5}$, but the graphs, as they are plotted, are qualitatively independent of the size of this radius.

We would like to point out several features that are apparent from Fig. \[fig:spinonehalf\]. First and foremost, the Bargmann invariant $\mathcal{C}$ is observed to always pass through the origin of the complex plane just when the spin coupling strength attains its critical value of $J_c 