Spectroscopic fingerprints of gapped quantum spin liquids, both conventional and fractonic

We explain how gapped quantum spin liquids, both conventional and “fractonic,” may be unambiguously diagnosed experimentally using the technique of multidimensional coherent spectroscopy. “Conventional” nonchiral gapped quantum spin liquids (e.g., Z_{2} spin liquid) do not have sharp signatures in linear response, but they do have clear fingerprints in nonlinear response, accessible through the already existing experimental technique of two-dimensional coherent spectroscopy. Type I fracton phases (e.g., X-cube) are (surprisingly) even easier to distinguish, with strongly suggestive features even in linear response, and unambiguous signatures in nonlinear response. Type II fracton systems, like Haah's code, are most subtle, and they may require consideration of high-order nonlinear response for unambiguous diagnosis.