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

TL;DR

This paper investigates how the geometric phase behaves near a quantum multi-critical point in a 1D transverse XY spin chain, revealing path-dependent scaling and identifying quasi-critical points through derivative peaks.

Contribution

It introduces a path-dependent analysis of the geometric phase near multi-critical points and characterizes the scaling behavior of its derivative at quasi-critical points.

Findings

Derivative peaks occur at quasi-critical points close to the multi-critical point.

Scaling exponent of the derivative varies with the path parameter α, up to a critical value.

No peaks are observed on the paramagnetic side for α > α_c.

Abstract

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.