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.
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