strengths (generating a complex number, the Bargmann invariant, for each value), we start to build up a picture of the general trend. We present our results for varying lengths of the spin chain, contrast the case of spin-$1/2$ particles with that of spin-$1$ particles, and discuss the effect of increasing the number $\mathcal{N}$ of constituent vertices of the circuit. Of particular interest, of course, is what, if anything, may be said about the Bargmann invariant as the coupling parameter crosses its critical point $J_c$. The report concludes with a summary and brief discussion of our work.

Bargmann Invariant and Phase {#section:Bargmann}
============================

In quite general terms one may define the phase difference $\chi$ between two non-orthogonal state vectors, ${\mbox{$| \Psi_1 \rangle$}}$ and ${\mbox{$| \Psi_2 \rangle$}}$, by the relation $$\exp{(i \chi)} \equiv
    {\mbox{$\langle \Psi_1  | \Psi_2 \rangle$}}/|{\mbox{$\langle \Psi_1  | \Psi_2 \rangle$}}|,$$ so that $\chi = \text{arg}{\mbox{$\langle \Psi_1  | \Psi_2 \rangle$}}$. Naturally, individual state vectors are only defined up to arbitrary phase factors; $\chi$ is thus gauge-dependent and its value may not be assigned any direct physical meaning. On the other hand, any *cyclic* combination of state vectors, such as $\text{arg}({\mbox{$\langle \Psi_1  | \Psi_2 \rangle$}}{\mbox{$\langle \Psi_2  | \Psi_3 \rangle$}}{\mbox{$\langle \Psi_3  | \Psi_1 