In order to convince oneself that this is indeed the correct ground state, one can easily show that no individual spin flip is capable of lowering the energy any further.

From the analysis above it follows that the ground state energy $E^{(0)}$ can always be written as a function of $|J-J_c|$, demonstrating non-analyticity at the critical point $J_c$. Moreover, the energy gap vanishes in the region $J \geqslant J_c$.

The Circuit {#section:CriticalPoint}
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In light of the previous discussion, the parameter circuit which suggests itself is the following: a regular polygon residing on the Bloch sphere of which one edge is bisected by the $x$-axis, as illustrated in Fig. \[fig:Bloch2\].

As we move along the vertices of the circuit, the critical $x$-axis is crossed in the direction of positive $z$-axis.

Results for the Spin-$1/2$ Chain {#section:Results}
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We now turn to the results of our simulation of the Bargmann invariant for the spin-$1/2$ chain, which are summarized in Fig. \[fig:spinonehalf\]. The circuit used in the simulation is the regular polygon described in the previous section, but consists of $\mathcal{N}=100$ vertices.

\[tc\]\[tc\] \[bc\]\[bc\] \[tc\]\[tc\] \[bc\]\[bc\]\
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The Bargmann invariant and its phase were computed for different spin chain 