Nonlinear optical signatures of the tensor order in Cd2Re2O7

Growing evidence indicates that the superconducting pyrochlore Cd2Re2O7 exhibits a structural phase transition at T-c = 200 K with an unusual tensor character(1-3). The structural order parameter for this state is two-dimensional, and spanned by distinct but nearly degenerate crystallographic structures I4(1)22 and I (4) over bar m2 ( ref. 1). Symmetry rules imply that the low-energy excitations of the ordered state are Goldstone phonons, or long wavelength fluctuations between the two crystal structures. These are the structural equivalents of magnons in an XY antiferromagnet, with the two crystal structures analogous to orthogonal spin directions in the xy-plane. Goldstone phonons have been observed in Raman spectroscopy(3), but high-resolution X-ray and neutron scattering experiments have produced conflicting assignments of the static low-temperature structure(4-6). Here, we use optical second-harmonic generation with polarization sensitivity to assign the I (4) over bar m2 structure unambiguously and verify an auxiliary condition on the structure that is implied by the order parameter symmetry. We also show that the temperature dependence of the order parameter is consistent with thermal occupation of the Goldstone mode. The methodology may be applied more widely in characterizing ordered states in matter.