  In the vicinity of the insulator-to-superfluid quantum phase transition in
its core, a dislocation in a He-4 crystal supports particle-hole excitations
with arbitrary small gaps. These exotic analogs of Frenkel interstitial-vacancy
pairs should manifest themselves in various threshold and thermoactivation
effects. In Worm Algorithm simulations, we reveal the presence of corresponding
small gaps via anomalous thermoactivation behavior of particle number
fluctuations, which we unambiguously associate with dislocations by
"visualization" techniques. Experimentally, the related threshold and
thermoactivation dependencies could be observed in the ultrasound absorption.
