Quantum fidelity for one-dimensional Dirac fermions and two-dimensional Kitaev model in the thermodynamic limit

TL;DR

This paper investigates the scaling behavior of quantum fidelity in one-dimensional Dirac fermions and the two-dimensional Kitaev model, revealing power-law scalings linked to critical exponents and analyzing geometric phases.

Contribution

It derives and verifies generic scaling forms of fidelity at quantum critical points for both thermodynamic and non-thermodynamic limits, including geometric phase analysis.

Findings

Fidelity follows power-law scaling in gapless and gapped phases.

Scaling forms are derived and verified for the Kitaev model.

Geometric phase vanishes under spin rotation in the Kitaev model.

Abstract

We study the scaling behavior of the fidelity ($F$) in the thermodynamic limit using the examples of a system of Dirac fermions in one dimension and the Kitaev model on a honeycomb lattice. We show that the thermodynamic fidelity inside the gapless as well as gapped phases follow power-law scalings, with the power given by some of the critical exponents of the system. The generic scaling forms of $F$ for an anisotropic quantum critical point for both thermodynamic and non-thermodynamic limits have been derived and verified for the Kitaev model. The interesting scaling behavior of $F$ inside the gapless phase of the Kitaev model is also discussed. Finally, we consider a rotation of each spin in the Kitaev model around the z axis and calculate $F$ through the overlap between the ground states for angle of rotation $\eta$ and $\eta+d\eta$, respectively. We thereby show that the associated geometric phase vanishes. We have supplemented our analytical calculations with numerical simulations wherever necessary.