If spin liquids have been famously defined by what they are not, i.e., ordered, the past years have seen the frontier between order and spin liquid starting to fade, with a growing number of materials whose low-temperature physics cannot be explained without co-existence of (partial) magnetic order and spin fluctuations. Here, we study an example of such co-existence in the presence of magnetic dipolar interactions, related to spin ice, where the order is long range and the fluctuations support a Coulomb gauge field. Topological defects are effectively coupled via energetic and entropic Coulomb interactions, the latter one being stronger than for the spin-ice ground state. Depending on whether these defects break the divergence-free condition of the Coulomb gauge field or the long-range order, they are respectively categorized as monopoles — as in spin ice — or monopole holes, in analogy with electron holes in semiconductors. The long-range order plays the role of a fully-occupied valence band, while the Coulomb spin liquid can be seen as an empty conducting band. These results are discussed in the context of other lattices and models which support a similar co-existence of Coulomb gauge field and long-range order. We conclude this work by explaining how dipolar interactions lift the spin-liquid degeneracy at very low energy scale by maximizing the number of flippable plaquettes, in light of the equivalent quantum dimer model.
