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We discuss possible approaches to the problem of the URu2Si2 "Hidden Order"(HO) which remains unsolved after tremendous efforts of researches. Supposethere is no spatial symmetry breaking at the HO transition temperature andsolely the time-reversal symmetry breaking emerges owing to some sort ofmagnetic order. As a result of its 4/mmm symmetry, each uranium atom is athree-dimensional magnetic vortex; its intra-atomic magnetization M(r) isintrinsically non-collinear, so that its dipole, quadrupole and toroidalmoments vanish, thus making the vortex "hidden". The first non-zero magneticmultipole 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); thiscorresponds to either ferro-vortex or antiferro-vortex structures with I4/mmmor P_I4/mmm magnetic space groups, respectively. Our first-principlescalculations suggest that the vortex magnetic order of URu2Si2 is ratherstrong: the total absolute magnetization |M(r)| is about 0.9 mu_B per U atom,detectable by neutron scattering in spite of the unusual formfactor. Theferro-vortex and antiferro-vortex phases have almost the same energy and theyare energetically favorable compared to the non-magnetic phase.

The hidden order in URu2Si2: Symmetry-induced anti-toroidal vortices