We study Ising pyrochlores by means of Monte Carlo simulations. We cover a set of exchange constants ranging from the frustrated ferromagnetic case (spin-ice) to the fully-ordered "all-in-all-out" antiferromagnet in the dipolar model, reinterpreting the results-as in an ionic system-in terms of a temperature vs magnetic charge density phase diagram. In spite of its spin nature and the presence of both double and single nonconserved magnetic charges, the dipolar model gives place to a phase diagram which is quite comparable with those previously obtained for on-lattice systems of electric charges, and on spin ice models with a conserved number of single magnetic charges. The contrast between these systems, to which we add results from the nearest-neighbors model, put forward other features of our phase diagram-notably, a monopole fluid with charge order at high monopole densities that persists up to arbitrarily high temperatures-that can only be explained taking into account construction constraints forced by the underlying spin degrees of freedom.

We demonstrate the appearance of thermal order by disorder in Ising pyrochlores with staggered antiferromagnetic order frustrated by an applied magnetic field. We use a mean-field cluster variational method, a low-temperature expansion, and Monte Carlo simulations to characterize the order-by-disorder transition. By direct evaluation of the density of states, we quantitatively show how a symmetry-broken state is selected by thermal excitations. We discuss the relevance of our results to experiments in 2D and 3D samples and evaluate how anomalous finite-size effects could be exploited to detect this phenomenon experimentally in two-dimensional artificial systems, or in antiferromagnetic all-in-all-out pyrochlores like Nd 2 Hf 2 O 7 or Nd 2 Zr 2 O 7 , for the first time. DOI: 10.1103/PhysRevLett.117.167203 Order by disorder (OBD) is the mechanism whereby a system with a nontrivially degenerate ground state develops long-range order by the effect of classical or quantum fluctuations [1]. From a theoretical point of view, the OBD mechanism is a relatively common occurrence in geometrically frustrated spin models [2], such as the fully frustrated domino model-where it was discussed for the first time [1]-or the Ising antiferromagnet on the three-dimensional fcc lattice [3]. Many other theoretical realizations exist. However, definitive experimental evidence for this mechanism has remained elusive. Strong evidence for quantum OBD in the antiferromagnetic (AFM) XY insulating rare-earth pyrochlore oxide Er 2 Ti 2 O 7 has been reported [4][5][6][7], but a conclusive proof of thermal OBD remains unseen in the laboratory so far. The difficulty lies in establishing whether order is selected through the OBD mechanism (a huge disproportion in the density of low-energy excitations associated with particular ground states) or is due to energetic contributions not taken into account that actually lift the ground-state degeneracy.In this work we study OBD in Ising spin systems where the staggered order is inhibited by a magnetic field. We analyze theoretically and numerically the threedimensional pyrochlore system and its two-dimensional projection (the checkerboard lattice). We demonstrate the existence of singular finite-size effects (FSE) and we show how they can be exploited to detect OBD. Our results suggest that thermal OBD could be finally observed experimentally in natural staggered structures based on the pyrochlores [8][9][10], as well as in artificially designed two-dimensional magnetic [11] or colloidal systems [12].More precisely, we first study an Ising pyrochlore with h111i anisotropy and AFM nearest-neighbor interactions. In the absence of magnetic field (B), the ground state is the all-spins-in-all-spins-out Néel state [13]. A strong field along the crystalline direction [110] can break this order, turning it into a disordered state with three-spins-in-onespin-out and three-out-one-in elementary units. This type of disordered system of magnetic charges (see below) had been studied before in the context of spi...

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