Fig. 2: Spin textures of the ordered spin-ice states for different applied fields.
Spin-ice magnetic phases displayed as a 2D projection of the HTO unit cell down the z-axis. The orange squares are tetrahedra. The square in the center (gray) is not a tetrahedron; the diagonally opposing spins are in the same lattice plane. The +/− signs indicate the spin directions along the z-axis2,16,17,21. a (2:2)0 state with a net magnetic moment (\(\hat{{{{{{{{\bf{m}}}}}}}}}\)) in the z-direction ((2:2)0 states also form with \(\hat{{{{{{{{\bf{m}}}}}}}}}\) in the x- or y-directions, depending on the field direction); (b) (2:2)X state (with \(\hat{{{{{{{{\bf{m}}}}}}}}}\parallel\)[110]) in which the α-spins (blue) are polarized and the β-spins (red) are antiferromagnetically aligned in chains (highlighted by the open arrows). The (2:2)X state also forms when B is directed along any of the family of \(\left\langle 110\right\rangle\) directions; and (c) (3:1) state with one spin flip per tetrahedron with \(\hat{{{{{{{{\bf{m}}}}}}}}}\parallel\)[111] (the (3:1) state forms when B is directed along any of the family of \(\left\langle 111\right\rangle\) directions with ∣B∣ ≥ 2 T). The spins denoted by the light blue boxes in panels (a) and (c) indicate the spin sublattice that becomes decoupled from the field when the magnetic field is directed exactly along the [112] direction. The direction of the magnetic field is indicated for each spin texture. The total energies per 16-site unit cell, i.e., interaction energies summed over 1st, 2nd and 3rd nearest neighbors and the Zeeman energies, are calculated for each of these ordered state, details regarding this calculation are provided in Supplementary Note 7.
