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

Power-law Conductivity inside the Mott gap: application to $\kappa-(BEDT-TTF)_2Cu_2(CN)_3$