Anomalous specific heat behaviour in the quadrupolar Kondo system PrV2Al20

We have measured the specific heat of PrV$_2$Al$_{20}$ at very low temperatures, using high quality single crystals with the residual resistivity ratio ~ 20. The high-quality single crystals exhibit clear double transitions at $T_{\rm Q} =$ 0.75 K and $T^* =$ 0.65 K. These transitions are clearer and shift to higher temperature in higher quality single crystals. Besides, there was no hysteresis in those transitions in warming and cooling process of the heat capacity measurements. In the ordered state below $T^*$, the specific heat shows a $T^4$ dependence, indicating the gapless mode associated with the quadrupole and/or octupole ordering.