in the system. An important point to note here is that the circuit in parameter space need only pass *near* the critical point for the Berry phase to register it. In other words, the system need not actually undergo the quantum phase transition for the Berry phase to pinpoint its presence and location in parameter space, a consideration which assumes particular importance in the light of the difficulties that may be associated with physically implementing actual QPTs.

Previous Work {#subsection:PreviousWork}
-------------

A relationship between the geometric phase and criticality in spin chains was examined by Carollo and Pachos [@Carollo1; @Carollo2]. Specifically, the authors analyzed the Berry phase of the ground state of a one-dimensional spin-$1/2$ $XY$ model $$H = \sum_{l=-M}^{M} \left( \frac{1 + \gamma}{2}\sigma_l^x
      \sigma_{l+1}^x + \frac{1 - \gamma}{2}\sigma_l^y \sigma_{l+1}^y +
      \lambda \sigma_l^z \right),$$ where $M = (N-1)/2$ for an odd number of spins $N$. The Berry phase, as computed with respect to a rotation of the complete Hamiltonian around the $z$-axis, was shown to exhibit the following property in the thermodynamical limit $N\rightarrow\infty$: $$\label{eq:PT}
  \lim_{\gamma\rightarrow 0}(\varphi/M) =
    \begin{cases}
      0 \,(\text{mod }2\pi), & \text{for} |\lambda| > 1 \nonumber \\
      \text{finite}, & \text{for} |\lambda| < 1. \nonumber
    \end{cases}$$ As the $XY$ 