  We compute the quantum Fisher information (QFI) associated with two different
local operators in the Kitaev honeycomb model, and find divergent behaviour in
the second derivatives of these quantities with respect to the driving
parameter at the quantum phase transition between the gapped and gapless phases
for both fully anti-ferromagnetic and fully ferromagnetic exchange couplings.
The QFI associated with a local magnetization operator behaves differently from
that associated with a local bond operator depending on whether the critical
point is approached from the gapped or gapless side. We show how the behaviour
of the second derivative of the QFI at the critical point can be understood in
terms of diverging length scales in the correlators of the local generators.
