a signature for the critical point. Of note is also the fact that the ‘joined-up’ Bargmann phase extends twice as high for spin-$1$ chains as it does for spin-$1/2$ chains, for the same number of particles. Again, this can be understood from Berry’s solid angle result for non-interacting particles, where the geometric phase is proportional to the spin dimension.

Summary and Discussion {#section:Summary}
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In summary, we have shown how the magnitude and ‘speed’ of the Bargmann invariant can act as a signature of the critical point for the finite one-dimensional spin chain model considered. This stands in stark contrast to the Bargmann phase, which is oblivious to the phase transition as long as the spin chain length remains finite.

It is conjectured that the signatures presented here in the context of a specific spin chain model will uphold their validity also in a wider, more general setting. It would be worthwhile to conduct experimental investigations with a view to testing this conjecture; similarly, any theoretical proof of the proposed conjecture would be of considerable interest.

Finally, it is worth stressing that the observed signatures of criticality are very likely a direct consequence of the finite ‘speed’ with which the 