  Deconfined quantum critical points (DQCPs) are putative phase transitions
beyond the Landau paradigm with emergent fractionalized degrees of freedom. The
original example of a DQCP is the spin-1/2 quantum antiferromagnet on the
square lattice which features a second order transition between valence bond
solid (VBS) and N\'eel order. The VBS order breaks a lattice symmetry, and the
corresponding VBS order parameter may couple to lattice distortion modes
(phonons) at appropriate momenta. We investigate a field-theoretic description
of the DQCP in the presence of such a spin-lattice coupling. We show that
treating phonons as classical lattice distortions leads to a relevant
monopole-phonon interaction inducing an instability towards a distorted lattice
by an analogous mechanism to the spin-Peierls instability in one dimension.
Consequently, there is a breakdown of the DQCP which generally becomes a strong
first-order transition. Taking into account the full quantum nature of the
phonons, we argue that the continuous DQCP persists above a critical phonon
frequency. Lastly, we comment on the connection to general gapless, deconfined
gauge theories.
