  We study the geometric phase of the ground state in a one-dimensional
transverse XY spin chain in the vicinity of a quantum multi-critical point. We
approach the multi-critical point along different paths and estimate the
geometric phase by applying a rotation in all spins about z-axis by an angle
$\eta$. Although the geometric phase itself vanishes at the multi-critical
point, the derivative with respect to the anisotropy parameter of the model
shows peaks at different points on the ferromagnetic side close to it where the
energy gap is a local minimum; we call these points `quasi-critical'. The value
of the derivative at any quasi-critical point scales with the system size in a
power-law fashion with the exponent varying continuously with the parameter
$\alpha$ that defines a path, upto a critical value $\alpha = \alpha_{c}=2$.
For $\alpha > \alpha_{c}$, or on the paramagnetic side no such peak is
observed. Numerically obtained results are in perfect agreement with analytical
predictions.
