Geometrical quadrupolar frustration in DyB$_4$

Physical properties of DyB$_4$ have been studied by magnetization, specific heat, and ultrasonic measurements. The magnetic entropy change and the ultrasonic properties in the intermediate phase II indicate that the degeneracy of internal degrees of freedom is not fully lifted in spite of the formation of magnetic order. The ultrasonic attenuation and the huge softening of $C_{44}$ in phase II suggests existence of electric-quadrupolar (orbital) fluctuations of the 4$f$-electron. These unusual properties originate from the geometrical quadrupolar frustration.