  We present the ultra-low-temperature specific heat and thermal conductivity
measurements on the single crystals of YbMgGaO$_4$, which was recently argued
to be a promising candidate for quantum spin liquid (QSL). In the zero magnetic
field, a large magnetic contribution of specific heat is observed, and exhibits
a power-law temperature dependence ($C_m \sim T^{0.74}$). On the contrary, we
do not observed any significant contribution of thermal conductivity from
magnetic excitations. In magnetic fields $H \ge$ 6 T, the exponential
$T$-dependence of $C_m$ and the enhanced thermal conductivity indicate a magnon
gap of the fully-polarized state. The absence of magnetic thermal conductivity
at the zero field in this QSL candidate puts a strong constraint on the
theories of its ground state.
