  We study the response of the Kitaev spin liquid (KSL) to a local magnetic
field perpendicular to the Kitaev honeycomb lattice. The local magnetic field
induces a dynamical excitation of a flux pair in the spin liquid and the system
can be described by a generally particle-hole asymmetric interacting resonant
level model. The dynamical excitation of the flux pair closes the flux gap in
the spectrum of the spin correlation function locally for the gapless KSL even
from the perturbative response to a weak magnetic field. Beyond the
perturbative regime, the p-h asymmetry competes with the magnetic field and
results in a rich phase diagram. Moreover, the magnetic field breaks the gauge
equivalence of the ferromagnetic and anti-ferromagnetic Kitaev couplings of the
ground state and leads to very different behaviors for the two cases. The
anti-ferromagnetic case experiences a first order phase transition to the
polarized state during magnetization whereas the ferromagnetic case does not.
This study can be generalized to the Kitaev model in a uniform magnetic field
and may help understand issues in recent experiments on KSL candidates.
