  The existence and stability of spin-liquid phases represent a central topic
in the field of frustrated magnetism. While a few examples of spin-liquid
ground states are well established in specific models (e.g. the Kitaev model on
the honeycomb lattice), recent investigations have suggested the possibility of
their appearance in several Heisenberg-like models on frustrated lattices. An
important related question concerns the stability of spin liquids in the
presence of small perturbations in the Hamiltonian. In this respect, the
magnetoelastic interaction between spins and phonons represents a relevant and
physically motivated perturbation, which has been scarcely investigated so far.
In this work, we study the effect of the spin-phonon coupling on prototypical
models of frustrated magnetism. We adopt a variational framework based upon
Gutzwiller-projected wave functions implemented with a spin-phonon Jastrow
factor, providing a full quantum treatment of both spin and phonon degrees of
freedom. The results on the frustrated $J_1-J_2$ Heisenberg model on one- and
two-dimensional (square) lattices show that, while a valence-bond crystal is
prone to lattice distortions, a gapless spin liquid is stable for small
spin-phonon couplings. In view of the ubiquitous presence of lattice
vibrations, our results are particularly important to demonstrate the
possibility that gapless spin liquids may be realized in real materials.
