Coulomb interaction between monopoles. The static properties of spin ice are accurately described by the monopole model [@KaiserDH]. The dynamic properties can also be described by assuming an effective monopole mobility [@JaubertHoldsworth; @Bovo; @Paulsen_Wien], but there have been few studies of the microscopic origin of the monopole motion [@Tomasello].

![image](figure1.pdf){width="10cm"}

The field and energy scales involved in monopole motion are illustrated in Fig. 1, a-e. When a monopole hops to a neighbouring site a spin is flipped (Fig. 1a). For an isolated monopole (far from any others) this spin flip takes place at nominally zero energy cost (Fig. 1b) because contributions from near-neighbour antiferromagnetic superexchange and ferromagnetic dipole–dipole coupling individually cancel. The cancellation of the field contribution relies on the dipolar self-screening [@Isakov] that maps the long-range interacting system [@Gingras] to the degenerate Pauling manifold of the near neighbour spin ice model [@BramwellHarris98]. This surprising cancellation is a key result of the many–body physics of spin ice. In practice, a monopole hop may also involve a finite energy change arising from longitudinal fields at the spin site: the main source of fields is nearby monopoles [@CMS] (Fig. 1b), while further contributions arise from corrections to the mapping, which give a 