Fig. 2: Ho₂Ir₂O₇ under the application of a [100] magnetic field.

a Measurements and (b) Monte Carlo simulations of the magnetisation at various temperatures. The inset to a shows a measurement of the magnetisation at 1.8 K upon sweeping the field continuously between the positive and negative field limits. Inset to b: a single tetrahedron of the Ho³⁺ sublattice. Magenta arrows indicate the local effective field h_loc due to the ordered Ir moments. Under an externally applied [100] magnetic field (black arrow) the Ho moments order into a 2I2O configuration, oriented either parallel (green arrow) or antiparallel (red arrow) to h_loc. c Magnetoresistance measurements and (d) Monte Carlo simulations of the density of single monopoles (defined as the fraction of 3I1O/1I3O tetrahedra) at various temperatures. Inset to (c): a measurement of the resistance at 0.6 K upon sweeping the field continuously between the positive and negative field limits. The temperature dependence of the resistivity is dominated by the insulating behaviour below the MIT, while the negative magnetoresistance at fixed temperature results from a combination of the paramagnetic response of spin ice at fixed monopole density and variations in the monopole density via the mechanisms described in the text.