Abstract
Quantum fluctuations can inhibit long-range ordering in frustrated magnets and potentially lead to quantum spin liquid (QSL) phases. A prime example are gapless QSLs with emergent U(1) gauge fields, which have been understood to be described in terms of quantum electrodynamics in 2+1 dimension (QED(3)). Despite several promising candidate materials, however, a complicating factor for their realisation is the presence of other degrees of freedom. In particular lattice distortions can act to relieve magnetic frustration, precipitating conventionally ordered states. In this work, we use field-theoretic arguments as well as extensive numerical simulations to show that the U(1) Dirac QSL on the triangular and kagome lattices exhibits a weak-coupling instability due to the coupling of monopoles of the emergent gauge field to lattice distortions, leading to valence-bond solid ordering. This generalises the spin-Peierls instability of one-dimensional quantum critical spin chains to two-dimensional algebraic QSLs. We study static distortions as well as quantum-mechanical phonons. Even in regimes where the QSL is stable, the singular spin-lattice coupling leads to marked temperature-dependent corrections to the phonon spectrum, which provide salient experimental signatures of spin fractionalisation. We discuss the coupling of QSLs to the lattice as a general tool for their discovery and characterisation.