  We investigate thermal transport in square ice, a two-dimensional analogue of
spin ice, exploring the role played by emergent magnetic monopoles in
transporting energy. Using kinetic Monte Carlo simulations based on energy
preserving extensions of single-spin-flip dynamics, we explicitly compute the
(longitudinal) thermal conductivity, $\kappa$, over a broad range of
temperatures. We use two methods to determine $\kappa$: a measurement of the
energy current between thermal baths at the boundaries, and the Green-Kubo
formula, yielding quantitatively consistent values for the thermal
conductivity. We interpret these results in terms of transport of energy by
diffusion of magnetic monopoles. We relate the thermal diffusivity, $\kappa/C$
where $C$ is the heat capacity, to the diffusion constant of an isolated
monopole, showing that the subdiffusive monopole implies $\kappa/C$ vanishes at
zero temperature. Finally, we discuss the implications of these results for
thermal transport in three-dimensional spin ice, in spin ice materials such as
Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$, and outline some open questions for
thermal transport in highly frustrated magnets.
