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

In this paper a geometric phase is proposed to characterise the topological quantum phase transition of the Kitaev honeycomb model. The simultaneous rotation of two spins is crucial for generating 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 behaviour 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 behaviour 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 the geometric phase in the thermodynamic limit indicates the typical second-order phase transition and thus the topological quantum phase transition can be well detected by the geometric phase