  We show that spatially local, yet low-energy, fluctuations can play an
essential role in the physics of strongly correlated electron systems tuned to
a quantum critical point. A detailed microscopic analysis of the Kondo lattice
model is carried out within an extended dynamical mean-field approach. The
correlation functions for the lattice model are calculated through a
self-consistent Bose-Fermi Kondo problem, in which a local moment is coupled
both to a fermionic bath and to a bosonic bath (a fluctuating magnetic field).
A renormalization-group treatment of this impurity problem--perturbative in
$\epsilon=1-\gamma$, where $\gamma$ is an exponent characterizing the spectrum
of the bosonic bath--shows that competition between the two couplings can drive
the local-moment fluctuations critical. As a result, two distinct types of
quantum critical point emerge in the Kondo lattice, one being of the usual
spin-density-wave type, the other ``locally critical.'' Near the locally
critical point, the dynamical spin susceptibility exhibits $\omega/T$ scaling
with a fractional exponent. While the spin-density-wave critical point is
Gaussian, the locally critical point is an interacting fixed point at which
long-wavelength and spatially local critical modes coexist. A Ginzburg-Landau
description for the locally critical point is discussed. It is argued that
these results are robust, that local criticality provides a natural description
of the quantum critical behavior seen in a number of heavy-fermion metals, and
that this picture may also be relevant to other strongly correlated metals.
