Fig. 3: Ho₂Ir₂O₇ under the application of a [111] magnetic field.

a Measurements and b Monte Carlo simulations of magnetisation at 1.8 K. Arrows in (a) indicate the direction of the observed hysteresis loop; the virgin curve lies between the upsweep and downsweep for H > 0. Inset to (a): tetrahedra of the Ho³⁺ sublattice. Magenta arrows indicate the local effective field h_loc for both Ir domain types. An external [111] magnetic field (black arrow) orders the Ho sublattice into a 3I1O/1I3O monopole crystal, with moments either parallel (green arrows) or antiparallel (red arrows) to h_loc. For this direction of applied field, the alignment favours type-A Ir domains. The 100:0 and 0:100 simulations in the inset to (b) represent type-A and type-B single-domain crystals, respectively. Other curves in (b) are weighted averages of these. Due to plastic deformation of the Ir domains, the magnetisation of the Ho moments follows the 50:50 curve for the virgin sweep and 70:30/30:70 curves thereafter, indicated by arrows. c Magnetoresistance measurements. d Monte Carlo simulations of the single monopole density (defined as the fraction of 3I1O/1I3O tetrahedra), simulated using a 50:50 domain ratio for the virgin curve (1.8 K only, labelled ‘I’) and a 70:30/30:70 average thereafter. Single-domain (100:0/0:100) curves are also shown. The 5 K (10 K) curves in (d) are shifted by 70% (90%) to allow for a clearer comparison of the form of the monopole density and experimental magnetoresistance curves (the vertical offset is not relevant to the scattering mechanisms discussed in the text). For higher temperature data see Supplementary Fig. 3.