  The continuous quantum phase transition and the nature of quantum critical
point (QCP) in a modified Kondo lattice model with Ising anisotropic exchange
interactions is studied within the density-matrix renormalization group
algorithm. We investigate the effect of quantum fluctuations on critical Kondo
destruction QCP, by probing static and dynamic properties of the magnetic order
and the Kondo effect. In particular, we identify that local Kondo physics
itself becomes critical at the magnetic phase transition point, providing
unbiased evidences for local quantum criticality between two insulators without
resorting to the change of Fermi surface.
