Quantum spin liquids (QSLs) are magnetic phases that evade long-range order down to the lowest temperatures due to strong quantum fluctuations. Their lack of conventional order parameters, however, makes experimental identification challenging. In this work, we investigate how distinct QSL phases affect phonon dynamics through spinon-phonon coupling. By computing the phonon self-energy, we show that the phonon spectrum remains unrenormalized by spinon interactions, while sound attenuation exhibit distinct signatures of the underlying QSL phase. Furthermore, the phonon thermal conductivity generally exhibits a linear-in-temperature dependence at low temperatures, in stark contrast to the $T^2$ behavior expected when the scattering is due to ordinary electrons. These response functions therefore provide experimentally accessible fingerprints for distinguishing different QSL backgrounds. Remarkably, we find that a chiral QSL coupled to phonons does not generate a phonon thermal Hall effect despite explicitly breaking time-reversal symmetry. Our results establish phonon transport as a potential probe for identifying and characterizing quantum spin liquids.

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• Exotic phases of matter
• Phonons
• Quantum spin liquid
• Spinon