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We have systematically investigated the variation of the exchange parametersand the ground state in the $S = 1/2$ kagome-lattice antiferromagnet(Rb$_{1-x}$Cs$_{x}$)$_2$Cu$_3$SnF$_{12}$, via magnetic measurements usingsingle crystals. One of the parent compounds, Rb$_2$Cu$_3$SnF$_{12}$, which hasa distorted kagome lattice accompanied by four sorts of nearest-neighborexchange interaction, has a disordered ground state described by a pinwheelvalence-bond-solid state. The other parent compound, Cs$_2$Cu$_3$SnF$_{12}$,which has a uniform kagome lattice at room temperature, has an ordered groundstate with the $q$ = 0 spin structure. The analysis of magneticsusceptibilities shows that with increasing cesium concentration $x$, theexchange parameters increase with the tendency to be uniform. It was found thatthe ground state is disordered for $x$ < 0.53 and ordered for $x$ > 0.53. Thepseudogap observed for $x$ < 0.53 and the N\'{e}el temperature for $x$ > 0.53approach zero at $x_{\rm c}$ $\simeq$ 0.53. This is indicative of theoccurrence of a quantum phase transition at $x_{\rm c}$.

Quantum phase transition between disordered and ordered states in the  spin-1/2 kagome lattice antiferromagnet  (Rb$_{1-x}$Cs$_{x}$)$_2$Cu$_3$SnF$_{12}$