vectors of that family of Hamiltonians to be arranged on the vertices of a regular polygon residing on the surface of the Bloch sphere (i.e. $B = 1$); an example of such a circuit is depicted in Fig. \[fig:Bloch2\]. For a given coupling strength, $J_0$, the Bargmann invariant can now be obtained by numerically computing the ground states at each of the circuit’s vertices and applying Eq. (\[eq:phasedefn2\]). Having chosen our circuit in parameter space, we would then like to investigate the manner in which the Bargmann invariant depends on the choice of spin coupling strength, $J$. Of particular interest are circuits that slice through any critical points of the system. Does the Bargmann invariant undergo a marked shift when the parameters pass through their critical values? Of further interest is the dependence on the number $\mathcal{N}$ of vertices of the regular polygon circuit, as well as the dependence on the number $N$ of spins in the chain.

Leaving the choice of circuit in parameter space open for now, it is evident from the form of Hamiltonian (\[eq:generalham\]) that there are two ‘special’ directions in which the magnetic field vector may point, namely the $x$-direction and any direction in the 