  We show that a quantum phase transition can occur in a phonon system in the
presence of dislocations. Due to the competing nature between the topological
protection of the dislocation and anharmonicity, phonons can reach a quantum
critical point at a frequency determined by dislocation density and the
anharmonic constant, at zero temperature. In the symmetry-broken phase, a novel
phonon state is developed with a dynamically-induced dipole field. We carry out
a renormalization group analysis and show that the phonon critical behavior
differs wildly from any electronic system. In particular, at the critical
point, a single phonon mode dominates the density of states and develops an
exotic logarithmic divergence in thermal conductivity. This phonon quantum
criticality provides a completely new avenue to tailor phonon transport at the
single-mode level without using phononic crystals.
