  Quantum criticality in certain heavy-fermion metals is believed to go beyond
the Landau framework of order-parameter fluctuations. In particular, there is
considerable evidence for Kondo destruction: a disappearance of the static
Kondo singlet amplitude that results in a sudden reconstruction of Fermi
surface across the quantum critical point and an extra critical energy scale.
This effect can be analyzed in terms of a dynamical interplay between the Kondo
and RKKY interactions. In the Kondo-destroyed phase, a well-defined Kondo
resonance is lost, but Kondo singlet correlations remain at nonzero
frequencies. This dynamical effect allows for mass enhancement in the
Kondo-destroyed phase. Here, we elucidate the dynamical Kondo effect in
Bose-Fermi Kondo/Anderson models, which unambiguously exhibit Kondo-destruction
quantum critical points. We show that a simple physical quantity---the
expectation value $\langle {\bf S}_{f} \cdot {\bf s}_{c} \rangle$ for the dot
product of the local ($f$) and conduction-electron ($c$) spins---varies
continuously across such quantum critical points. A nonzero $\langle {\bf
S}_{f} \cdot {\bf s}_{c} \rangle$ manifests the dynamical Kondo effect that
operates in the Kondo-destroyed phase. Implications are discussed for the
stability of Kondo-destruction quantum criticality as well as the understanding
of experimental results in quantum critical heavy-fermion metals.
