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Motivated by spin-wave continuum (SWC) observed in recent neutron scatteringexperiments in Herbertsmithite, we use Gutzwiller-projected wave functions tostudy dynamic spin structure factor $S(\mathbf{q},\omega)$ of spin liquidstates on the kagome lattice. Spin-1 excited states in spin liquids arerepresented by Gutzwiller-projected two-spinon excited wave functions. Weinvestigate three different spin liquid candidates, spinon Fermi-surface spinliquid (FSL), Dirac spin liquid (DSL) and random-flux spin liquid (RSL). FSLand RSL have low energy peaks in $S(\mathbf{q},\omega)$ at $K$ points in theextended magnetic Brillouin zone, in contrast to experiments where low energypeaks are found at $M$ points. There is no obviuos contradiction between DSLand neutron scattering measurements. Besides a fractionalized spin (i.e.spin-1/2), spinons in DSL carry a fractionalized crystal momentum which ispotentially detectable in SWC in the neutron scattering measurements.

Fractionalized spin-wave continuum in spin liquid states on the kagome lattice