  A complicating factor in the realization and observation of quantum spin
liquids in materials is the ubiquitous presence of other degrees of freedom, in
particular lattice distortion modes (phonons). These provide additional routes
for relieving magnetic frustration, thereby possibly destabilizing spin-liquid
ground states. In this work, we focus on triangular-lattice Heisenberg
antiferromagnets, where recent numerical evidence suggests the presence of an
extended U(1) Dirac spin liquid phase which is described by compact quantum
electrodynamics in 2+1 dimensions (QED$_3$), featuring gapless spinons and
monopoles as gauge excitations, and believed to flow to a strongly-coupled
fixed point with conformal symmetry. Using complementary perturbation theory
and scaling arguments, we show that a symmetry-allowed coupling between
(classical) finite-wavevector lattice distortions and monopole operators of the
U(1) Dirac spin liquid generally induces a spin-Peierls instability towards a
(confining) 12-site valence-bond solid state. We support our theoretical
analysis with state-of-the-art density matrix renormalization group
simulations. Away from the limit of static distortions, we demonstrate that the
phonon energy gap establishes a parameter regime where the spin liquid is
expected to be stable, and show that the monopole-lattice coupling leads to
softening of the phonon in analogy to the Kohn anomaly. We discuss the
applicability of our results to similar systems, in particular the Dirac spin
liquid on the Kagome lattice.
