Hidden order in URu2Si2: Symmetry-induced antitoroidal vortices

We discuss possible approaches to the problem of the URu2Si2 "hidden order" (HO) which remains unsolved after tremendous research efforts. Suppose there is no spatial symmetry breaking at the HO transition temperature and solely the time-reversal symmetry breaking emerges owing to some sort of magnetic order. As a result of its 4/mmm symmetry, each uranium atom is a three-dimensional magnetic vortex; its intra-atomic magnetization M(r) is intrinsically noncollinear, so that its dipole, quadrupole, and toroidal moments vanish, thus making the vortex "hidden." The first nonzero magnetic multipole of the uranium vortex is the toroidal quadrupole. In the unit cell, two uranium vortices can have either the same or opposite signs of M(r); this corresponds to either ferrovortex or antiferrovortex structures with I4/mmm or P(I)4/mmm magnetic space groups, respectively. Our first-principles calculations suggest that the vortex magnetic order of URu2Si2 is rather strong: the total absolute magnetization vertical bar M(r)vertical bar is about 0.9 mu(B) per U atom, detectable by neutron scattering in spite of the unusual form factor. The ferrovortex and antiferrovortex phases have almost the same energy and they are energetically favorable compared to the nonmagnetic phase.