  Organic solids host various electronic phases. Especially, a milestone
compound of organic solid, $\beta'$-$X$[Pd(dmit)$_2$]$_2$ with $X$=EtMe$_3$Sb
shows quantum spin-liquid (QSL) properties suggesting a novel state of matter.
However, nature of the QSL has been largely unknown. Here, we computationally
study five compounds comprehensively with different $X$ using 2D $ab$ $initio$
Hamiltonians and correctly reproduce experimental phase diagram with
antiferromagnetic order for $X$=Me$_4$P, Me$_4$As, Me$_4$Sb, Et$_2$Me$_2$As and
a QSL for $X$=EtMe$_3$Sb without adjustable parameters. We find that the QSL
for $X$=EtMe$_3$Sb exhibits 1D nature characterized by algebraic decay of spin
correlation along one direction, while exponential decay in the other
direction, indicating dimensional reduction from 2D to 1D. The 1D nature indeed
accounts for the experimental specific heat, thermal conductivity and magnetic
susceptibility. The identified QSL, however, preserves 2D nature as well
consistently with spin fractionalization into spinon with Dirac-like gapless
excitations and reveals duality bridging the 1D and 2D QSLs.
