Phase transition of Kitaev spin liquid described by quantum geometric tensor

TL;DR

This paper studies the topological phase transition in Kitaev spin liquids under magnetic fields by calculating quantum geometric tensors, revealing robustness of Berry curvature and correlations with phase diagrams at finite temperatures.

Contribution

It introduces a comprehensive analysis of the quantum geometric tensor, Berry curvature, and Fubini-Study metric in the Kitaev model, including finite temperature effects and their relation to phase transitions.

Findings

Berry curvature peaks at phase boundaries and is robust against local perturbations.

Fubini-Study metric correlates with phase diagram and peaks at phase transition points.

Mean Uhlmann curvature shows extrema when transitioning between phases.

Abstract

Weinvestigate the topological phase transition of Kitaev spin liquid in an external magnetic field by calculating the Berry curvature and the Fubini-Study metric. Employing Jordan-Wigner transformation and effective perturbative theory to transform the Hamiltonian into fermionic quadratic form, the Berry curvature is calculated by choosing the effective magnetic field as the parameter, and we find that the xy-component of the Berry curvature has the same behavior around the critical lines with the phase diagram and the behavior of Berry curvature around the critical line will not be influenced by local perturbation, i.e. it has the robustness against the local perturbation. Especially, we relate the Berry curvature with the derivative of effective magnetic susceptibility which can be regarded as the signature of topological phase transition besides, we related the second nonlinear susceptibility with the non-Abelian Berry connection. Then we analytically calculate the generalized Berry curvature in the mixed state called mean Uhlmann curvature which can be related with the spectral function, the curves that mean Uhlmanncurvaturechangingtemperaturewithdifferentcouplingconstant reveal that it will have an extrema when adjust $J_x$ from $A$ phase to $B$ phase. At last we analytically calculate the quantum geometric tensor in the effective magnetic field space whose imaginary part is the Berry curvature and real part is the Fubini-Study metric, and we find that the zz-component of Fubini-Study metric and the phase diagram are highly correlated which will peak at the cross point of three phases, then the Fubini-Study metric is extended to the finite temperature with arising an additional term called Fisher-Rao metric caused by mixed state.