Nature of the tensor order in Cd2Re2O7

The pyrochlore metal Cd2Re2O7 has been recently investigated by second-harmonic generation (SHG) reflectivity. In this paper, we develop a general formalism that allows for the identification of the relevant tensor components of the SHG from azimuthal scans. We demonstrate that the secondary order parameter identified by SHG at the structural phase transition is the x(2) -y(2) component of the axial toroidal quadrupole. This differs from the 3z(2) -r(2) symmetry of the atomic displacements associated with the I4m(2) crystal structure that was previously thought to be its origin. Within the same formalism, we suggest that the primary order parameter detected in the SHG experiment is the 3z(2) -r(2) component of the magnetic quadrupole. We discuss the general mechanism driving the phase transition in our proposed framework, and suggest experiments, particularly resonant x-ray scattering ones, that could clarify this issue.