Signature of a Spin Liquid State in the Low-Frequency Optical Conductivity of the S $=$ 1/2 Kagome Antiferromagnet Herbertsmithite

Herbertsmithite (ZnCu$_{\mathrm{3}}$(OH)$_{\mathrm{6}}$Cl$_{\mathrm{2}})$ is an antiferromagnetic Mott insulator composed of a planar kagome arrangement of S $=$ 1/2 copper atoms separated by nonmagnetic zinc atoms. It has recently emerged as one of the best candidates for exhibiting a quantum spin liquid state, showing no magnetic order down to 50 mK despite an exchange energy of 200 K. Here we report a signature of a spin liquid state in the terahertz optical conductivity of Herbertsmithite, measured via Terahertz Time-Domain Spectroscopy. A power-law dependence on frequency with exponent $\sim$ 1.4 is observed in the in-plane conductivity at low frequency, which increases in magnitude as temperature is decreased. This contribution to the conductivity is notably absent in the out-of-plane direction. Theory has predicted that the existence of a Dirac spin liquid with a gauge field serving to couple the spin and charge degrees of freedom would give rise to a power-law conductivity with exponent $\sim$ 2 inside the Mott gap. We discuss this prediction as well as other possible sources of the observed behavior.