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...

In this article we study the classical nearest-neighbour spin-ice model (nnSI) by means of Monte Carlo simulations, using the Wang-Landau algorithm. The nnSI describes several of the salient features of the spin-ice materials. Despite its simplicity it exhibits a remarkably rich behaviour. The model has been studied using a variety of techniques, thus it serves as an ideal benchmark to test the capabilities of the Wang Landau algorithm in magnetically frustrated systems. We study in detail the residual entropy of the nnSI and, by introducing an applied magnetic field in two different crystallographic directions ([111] and [100],) we explore the physics of the kagome-ice phase, the transition to full polarisation, and the three dimensional Kasteleyn transition. In the latter case, we discuss how additional constraints can be added to the Hamiltonian, by taking into account a selective choice of states in the partition function and, then, show how this choice leads to the realization of the ideal Kasteleyn transition in the system.

In this work we address the classical statistical mechanical problem of calculating the residual entropy of ice models. The numerical work found in the literature is usually based on extrapolating to infinite-size results obtained for finite-size systems with periodic boundary conditions. In this work we investigate how boundary conditions affect the calculation of the residual entropy for square, cubic, and hexagonal lattices using periodic, antiperiodic, and open boundary conditions. We show that periodic boundary conditions lead to noticeable oscillations in the entropy as a function of lattice size, and we calculate in open finite systems the contribution to the entropy from the open boundary. For our calculations we introduce a variation on multicanonical simulation methods that directly calculate the number of states in the ground state without the need of a Hamiltonian.