  Recent terahertz conductivity measurements observed low-power-law frequency
dependence of optical conduction within the Mott gap of the Kagome lattice
spin-liquid candidate Herbertsmithite. We investigate mechanisms for this
observed sub-gap conductivity for two possible scenarios in which the
ground-state is described by: 1) a U(1) Dirac spin-liquid with emergent
fermionic spinons or 2) a nearly critical Z2 spin-liquid in the vicinity of a
continuous quantum phase transition to magnetic order. We identify new
mechanisms for optical-absorption via magneto-elastic effects and spin- orbit
coupling. In addition, for the Dirac spin-liquid scenario, we establish an
explicit microscopic origin for previously proposed absorption mechanisms based
on slave-particle effective field theory descriptions.
