  Quantum spin liquids (QSLs) are fluid-like states of quantum spins where its
long-range ordered state is destroyed by quantum fluctuations. The ground state
of QSL and its exotic phenomena, which have been extensively discussed for
decades, have yet to be identified. We employ thermal transport measurements on
newly discovered QSL candidates, $\kappa$-(BEDT-TTF)2Cu2(CN)3 and
EtMe3Sb[Pd(dmit)2]2, and report that the two organic insulators possess
different QSLs characterized by different elementary excitations. In
$\kappa$-(BEDT-TTF)2Cu2(CN)3, heat transport is thermally activated at low
temperatures, suggesting presence of a spin gap in this QSL. In stark contrast,
in EtMe3Sb[Pd(dmit)2]2, a sizable linear temperature dependence of thermal
conductivity is clearly resolved in the zero-temperature limit, showing gapless
excitation with a long mean free path (~1,000 lattice distances). Such a long
mean free path demonstrates a novel feature of QSL as a quantum-condensed state
with long-distance coherence.
