  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.
