Bargmann invariant changes as a function of $J$ and the behaviour of the magnitude of the Bargmann invariant; we will find that both of these quantities can act as adequate signatures of criticality. At this stage, it may be helpful to depict the procedure schematically:

\[fig:schematic\] \[tc\]\[tc\][$J$]{} \[tc\]\[tc\][$J_c$]{} \[bl\]\[bl\][$\underline{\alpha}_{\mathcal{N}} = \underline{\alpha}_{0}$]{} \[rc\]\[rc\][$\underline{\alpha}_{s+1}$]{} \[rc\]\[rc\][$\underline{\alpha}_{s}$]{}

Let us fix the coupling parameter to some value $J_0$ and choose a particular circuit composed of $\mathcal{N}$ vertices $\underline{\alpha}_s$, each representing the set of parameters needed (together with $J_0$) in order to completely describe the Hamiltonian of the system at that point. At each individual point on the circuit we numerically compute the ground state of the system. The Bargmann invariant associated with the circuit at $J_0$ is then obtained by cyclically multiplying successive overlaps of these ground state wavefunctions. This will be explained in more detail in section \[section:Bargmann\], which is devoted entirely to the subject of Bargmann invariants; for now, we merely note that, in order to be sure to acquire a gauge-invariant quantity in this way, the wavefunctions at the start and end points of the circle need to be taken as *identical*. By repeating this procedure for a range of different coupling 