Path-dependent scaling of geometric phase near a quantum multi-critical point

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 the 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 a that defines a path, up to 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.