  Whether or not anomalies in the thermal conductivity in insulating cuprates
can be attributed to antiferromagnetic order and magnons in a 2D Mott insulator
remains an intriguing open question. To shed light on this issue, we
investigate the thermal conductivity $\kappa$ and its relationship with the
specific heat $c_v$ in the half-filled 2D single-band Hubbard model, using the
numerically exact determinant quantum Monte Carlo algorithm and maximum entropy
analytic continuation. At low temperatures where the charge degrees of freedom
are gapped-out and $c_v$ exhibits a clear magnon peak, we observe that thermal
conductivity $\kappa$ also tends to form a peak at similar temperatures.
Reducing temperature further produces a sharp upturn in $\kappa$, associated
with an increasing mean-free path. We identify this as the high-temperature
side of the anomalous peak in insulating cuprates, where the mean-free path
eventually is cut-off by other scattering effects, including phonons, disorder,
and physical size. Different scattering effects in our model are identified and
analyzed in the thermal diffusivity.
