= 1/2$. In other words, the magnitude $|\mathcal{C}|$ vanishes at the critical point. This is one of the features which we would like to propose as a signature of criticality, and is discussed further in section \[subsection:magsignature\].

Another trend that is immediately apparent from Fig. \[fig:spinonehalf\] is that with increasing spin chain length the Bargamann invariants ‘wind ever more loops’ around the origin. This statement can be made a little more precise by referring to the corresponding Bargmann phases. These graphs are characterized by discontinuities at the critical point: ‘jumps’ by $\pi$ alternating with ‘jumps’ by $2\pi$. However, on joining up those separate pieces in the graphs end-to-end, one notices that the ‘image’ covered by the resulting graphs grows with $N$ as $\pi
(N-1)/2$. This result may be explained by way of referring to the $J=0$ limit of non-interacting spins for which the Berry phase grows linearly with the number of spins. Also note that for spin-$1$ chains, the ‘joined-up’ Bargmann phase covers twice as much ground for any given number of particles (see Fig. \[fig:spinone\]).

In addition, we should note that we obtain Berry’s solid angle result for the case of non-interacting spins, $J=0$. Approximating the solid angle of 