nature, than of any real, physically measurable consequence.

In the present work, in contrast, we employ a technique based on the geometric phase that is able to detect critical points without the need to contract the circuit’s radius to zero or to extend the number of particles to infinity. We do so by considering discretized circuits that pass directly through the critical point, rather than circumnavigating it. This procedure and the obtained results are outlined in the following.

Procedure and Outline {#section:Outline}
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We start by introducing the concept of a Bargmann invariant [@Bargmann] and its associated phase, which may be regarded as a generalized Berry phase. It will be seen that Bargmann invariants are ideally suited for *numerically* analyzing the gauge-invariant phase induced by evolving states along a *discretized* circuit in parameter space. We then introduce the model of interacting spin-$1/2$ particles with which we will be concerned for the rest of this paper and discuss some of its more salient features. The Bargmann invariant and associated phase is computed numerically for a chosen discretized circuit and plotted as a function of the spin coupling strength $J$. Of note will be the ‘speed’ (to be defined) with which the 