  The charge dynamics of spin-liquid states described by U(1) gauge theory
coupling to fermionic spinons is discussed in this paper. We find that the
gapless spinons give rise to a power law optical conductivity inside the charge
gap. The theory is applied to explain the unusual optical conductivity observed
recently in the organic compound $\kappa-(BEDT-TTF)_2Cu_2(CN)_3$. We also
propose an optical experiment to search for the in-gap excitations in the
Kagome spin liquid insulator.
