Thermal Conductivity of Square Ice

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.