  Quantum spin liquid states, in which spins are quantum-mechanically
delocalized in direction, have been so far studied for charge-localized Mott
insulators arising from strong repulsive interaction. Recently, however, it was
found that the doped Mott insulator with a triangular lattice,
$\kappa$-(ET)$_4$Hg$_{2.89}$Br$_8$, exhibits both spin-liquid-like magnetism
and metallic electrical conduction. Thus, it is now possible to experimentally
explore how the spin liquidity affects the electrical conduction, an issue that
has received a great deal of theoretical attention. Here, with a newly
developed method to combine uniaxial and hydrostatic pressures, we investigate
the electrical conduction in the doped Mott insulator with controlling the
triangular lattice geometry and the repulsion strength which determines the
spin-liquidity and Mottness, respectively. We found that, in a strongly
interacting regime, the electronic state drastically changes from an insulator
to a Fermi liquid via a non-Fermi liquid with varying geometrical frustration,
which suggests that spin liquidity promotes delocalization of charges. This
result indicates that frustration in spin degrees of freedom has a decisive
impact on the transport of charges through the entanglement of spin and charge
in a doped Mott insulator.
