  We report an in-plane thermal transport study of the honeycomb Kitaev
material Na$_2$Co$_2$TeO$_6$ at subKelvin temperatures. In zero field, the
$\kappa(T)$ displays a rather weak $T$-dependence but has a non-zero residual
term $\kappa_0/T$, indicating strong phonon scattering by magnetic excitation
and the possibility of itinerant spinon-like excitations coexisting with an
antiferromagnetic order below 27 K. We propose the zero-field ground state is a
novel fractionalized antiferromagnetic (AF*) state with both magnetic order and
fractionalized excitations. With both the heat current and external field along
the $a*$ (Co-Co bond) direction, the $\kappa_{a*}$ exhibits two sharp minima at
7.5 T and 10 T, and its value at 8.5 T is almost the same as the pure phononic
transport for the high-field polarized state. This confirms the phase
boundaries of the reported field-induced intermediate state and suggest its
gapless continuum excitations possibly transport heat. No such intermediate
phase was found in the $\kappa_a$ for the current and field along the $a$
(zigzag chain) direction. Finally, Na$_2$Co$_2$TeO$_6$ displays a strongly
anisotropic magneto-thermal conductivity since the in-plane (out-of-plane)
field strongly enhances (suppresses) the $\kappa_{a*}$ and $\kappa_a$.

Strongly Correlated Electrons