magnitude of the product of overlaps along the circuit in Eq. \[eq:phasedefn2\] to be small. In fact, similar results were observed in a work by Zanardi and Paunkovi$\acute{\text{c}}$ [@Zanardi], who concluded that the ground state overlap function is itself a good characterization of QPTs.

‘Speed’ of the Bargmann Invariant
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Another characteristic that is sensitive to the phase transition is the ‘speed’ with which the Bargmann invariant changes as a function of the coupling parameter, which we define by $v_s :=
|\mathcal{C}_{s+1}-\mathcal{C}_s|/(J_{s+1} - J_s)$. As is evident from Fig. \[fig:speed\], the speed picks up markedly in the vicinity of the critical point. Again, this phenomenon is a direct result of the rapid change of the ground state near a critical point.

\[tc\]\[tc\][$(J-J_c)/r$]{} \[bc\]\[bc\][ $v$]{} ![ Plot of the ‘speed’, $v_s :=
|\mathcal{C}_{s+1}-\mathcal{C}_s|/(J_{s+1} - J_s)$, as a function of the spin coupling strength. The spin chain length is $N=3$ and the circuit is composed of $\mathcal{N} = 100$ vertices. []{data-label="fig:speed"}](3N100speed "fig:"){width=".45\textwidth" height=".3\textwidth"}

Results for the Spin-$1$ Chain {#section:spinone}
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The results presented in Fig. \[fig:spinone\] derive from simulations that are entirely analogous to those that gave rise to Fig. \[fig:spinonehalf\], but they refer to the spin-$1$ chain model, rather than the 