our regular polygon circuit as that of the enveloping cone, the Bargmann phase $\varphi$ asymptotically approaches Berry’s solid angle result as the number of vertices of the polygon is increased.

Of course, in the present work only spin chains of up to $11$ particles were simulated, and the exponential growth in the size of the Hilbert space with increasing particle number precludes one from simulating chains that are much longer than that [@footnote2]. However, from the preceding discussion the trend for Bargmann invariants and phases of longer spin chains should by now be fairly obvious. In particular, it has become clear that, perhaps surprisingly, precious little may be deduced about criticality from the Bargmann phase. Instead, a much clearer indication of the existence of a critical point is the vanishing magnitude of the Bargmann invariant itself.

Signatures of Criticality
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Magnitude of the Bargmann Invariant {#subsection:magsignature}
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It was already apparent from Fig. \[fig:spinonehalf\] that the magnitude of the Bargmann invariant $\mathcal{C}$ is able to assume the role of a diagnostic tool for critical points in the finite-sized interacting spin chain considered. For clarity, the magnitude of the graphs of $\mathcal{C}$ in Fig. \[fig:spinonehalf\] are plotted separately in Fig. \[fig:varyspinnumber\].

