Thermal conductivity of square ice

We investigate thermal transport in square ice, a two-dimensional analog 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 motion of the 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 Dy2Ti2O7 and Ho2Ti2O7, and outline some open questions for thermal transport in highly frustrated magnets.