Magnetization steps on the spin liquid ground state of the $S $= 1/2 kagome-like antiferromagnet Cu$_{3}$V$_{2}$O$_{7}$(OH)$_{2}$2H$_{2}$O

The ground state of the $S$ = 1/2 kagome antiferromagnet (KAFM) is expected to be a spin liquid with a finite spin gap \textit{$\Delta $ }$\sim \quad J$ / 20. Here, we report the magnetic properties of $S$ = 1/2 KAFM Cu$_{3}$V$_{2}$O$_{7}$(OH)$_{2}$$\cdot$2H$_{2}$O studied by magnetization, specific heat and V NMR measurements. Neither magnetic long- range order nor a spin gap has been detected down to 60 mK, in spite of a large antiferromagnetic interaction $J$ = 86 K, suggesting a gapless spin liquid. Surprisingly, we observed three step-like increases in magnetization at $H_{S1}$ = 4.3, $H_{S2}$ = 25.5, and $H_{S3}$ = 46 T, which implies that there exist at least four kinds of spin liquid or other quantum state under magnetic fields.