  We discuss elastic instabilities of the atomic crystal lattice at zero
temperature. Due to long-range shear forces of the solid, at such transitions
the phonon velocities vanish, if at all, only along certain crystallographic
directions, and, consequently, the critical phonon fluctuations are suppressed
to a lower dimensional manifold and governed by a Gaussian fixed-point. In case
of symmetry-breaking elastic transitions, a characteristic critical phonon
thermodynamics arises that is found, e.g., to violate Debye's $T^3$-law for the
specific heat. We point out that quantum critical elasticity is triggered
whenever a critical soft mode couples linearly to the strain tensor. In
particular, this is relevant for the electronic Ising-nematic quantum phase
transition in a tetragonal crystal as discussed in the context of certain
cuprates, ruthenates and iron-based superconductors.
