Geometric phase in the Kitaev honeycomb model and scaling behavior at critical points

TL;DR

This paper investigates the geometric phase in the Kitaev honeycomb model, revealing its behavior at critical points and demonstrating its universality and effectiveness in characterizing topological quantum phase transitions.

Contribution

It derives the geometric phase for the Kitaev honeycomb model and shows its non-analytic behavior at critical points, highlighting its role in understanding topological phase transitions.

Findings

Geometric phase is non-analytic at critical points.

Finite-size scaling reveals universal behavior.

Second derivative diverges at phase transition.

Abstract

In this paper a geometric phase of the Kitaev honeycomb model is derived and proposed to characterize the topological quantum phase transition. The simultaneous rotation of two spins is crucial to generate the geometric phase for the multi-spin in a unit-cell unlike the one-spin case. It is found that the ground-state geometric phase, which is non-analytic at the critical points, possesses zigzagging behavior in the gapless $B$ phase of non-Abelian anyon excitations, but is a smooth function in the gapped $A$ phase. Furthermore, the finite-size scaling behavior of the non-analytic geometric phase along with its first- and second-order partial derivatives in the vicinity of critical points is shown to exhibit the universality. The divergent second-order derivative of geometric phase in the thermodynamic limit indicates the typical second-order phase transition and thus the topological quantum phase transition can be well described in terms of the geometric-phase.