On the hidden order in URu₂Si₂ - Antiferro hexadecapole order and its consequences

An antiferro ordering of an electric hexadecapole moment is discussed as a promising candidate for the long standing mystery of the hidden order phase in URu₂Si₂. Based on localized f-electron picture, we discuss the rationale of the selected multipole and the consequences of the antiferro hexadecapole order of xy(x² - y²) symmetry. The mean-field solutions and the collective excitations from them explain reasonably significant experimental observations: the strong anisotropy in the magnetic susceptibility, characteristic behavior of pressure versus magnetic field or temperature phase diagrams, disappearance of inelastic neutron-scattering intensity out of the hidden order phase, and insensitiveness of the NQR frequency at Ru-sites upon ordering. A consistency with the strong anisotropy in the magnetic responses excludes all the multipoles in two-dimensional representations, such as (O_yz; O_zx). The expected azimuthal angle dependences of the resonant X-ray scattering amplitude are given. The (x² - y²)-type antiferro quadrupole should be induced by an in-plane magnetic field along [110], which is reflected in the thermal expansion and the elastic constant of the transverse (c₁₁ - c₁₂)=2 mode. The (x² - y²)-type [(xy)-type] antiferro quadrupole is also induced by applying the uniaxial stress along [110] direction [[100] direction]. A detection of these induced antiferro quadrupoles under the in-plane magnetic field or the uniaxial stress using the resonant X-ray scattering provides a direct redundant test for the proposed order parameter.