  The most fascinating feature of certain two-dimensional (2D) gapless quantum
spin liquid (QSL) is that their spinon excitations behave like the fermionic
carriers of a paramagnetic metal. The spinon Fermi surface is then expected to
produce a linear increase of the thermal conductivity with temperature that
should manifest via a residual value ($\kappa_0/T$) in the zero-temperature
limit. However, this linear in T behavior has been reported for very few QSL
candidates. Here, we studied the ultralow-temperature thermal conductivity of
an effective spin-1/2 triangular QSL candidate Na$_2$BaCo(PO$_4$)$_2$, which
has an antiferromagnetic order at very low temperature ($T_N \sim$ 148 mK), and
observed a finite $\kappa_0/T$ extrapolated from the data above $T_N$.
Moreover, while approaching zero temperature, it exhibits series of quantum
spin state transitions with applied field along the $c$ axis. These
observations indicate that Na$_2$BaCo(PO$_4$)$_2$ possibly behaves as a gapless
QSL with itinerant spin excitations above $T_N$ and its strong quantum spin
fluctuations persist below $T_N$.
