Field-tuned order by disorder in Ising frustrated magnets with antiferromagnetic interactions

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 characterise 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.