  Magnetic properties of the spin-1/2 kagome-like compound volborthite are
studied using a high-quality polycrystalline sample. It is evidenced from
magnetization and specific heat measurements that the spins on the kagome
lattice still fluctuate at low temperature, down to T = 60 mK that corresponds
to 1/1500 of the nearest-neighbor antiferromagnetic interaction, exhibiting
neither a conventional long-range order nor a spin gap. In contrast, 51V NMR
experiments revealed a sharp peak at 1 K in relaxation rate, which indicates
that a certain exotic order occurs. Surprisingly, we have observed three
"steps" in magnetization as a function of magnetic field, suggesting that at
least four liquid-like or other quantum states exist under magnetic fields.
