Abstract
Growing evidence indicates that the superconducting pyrochlore Cd2Re2O7 exhibits a structural phase transition at Tc=200 K with an unusual tensor character1,2,3. The structural order parameter for this state is two-dimensional, and spanned by distinct but nearly degenerate crystallographic structures I4122 and 
(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 spectroscopy3, but high-resolution X-ray and neutron scattering experiments have produced conflicting assignments of the static low-temperature structure4,5,6. Here, we use optical second-harmonic generation with polarization sensitivity to assign the 
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.
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Acknowledgements
Research at SFU was supported by the National Science and Engineering Research Council of Canada, the Canadian Institute for Advanced Research Program in Quantum Materials, the Sloan Foundation and the Research Corporation. Research at Oak Ridge/UT is sponsored by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, US Department of Energy, under contract DE-AC05-00OR22725 with Oak Ridge National Laboratory, managed and operated by UT-Battelle, LLC. We thank Z.-G. Ye for use of his polarized light microscope; we also thank him, D. Broun, B. Gaulin, I. Herbut, C. Kendziora, G. Leach and D. Singh for helpful discussions.
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J.C.P. and J.S.D. conceived and designed the experiments; J.C.P. and M.D.C. carried them out, with crystals prepared and characterized by J.H., R.J. and D.M. J.C.P., M.D.C., J.S.D. and I.A.S. contributed to the measurement analysis. J.C.P. and J.S.D. wrote the paper.
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Petersen, J., Caswell, M., Dodge, J. et al. Nonlinear optical signatures of the tensor order in Cd2Re2O7 . Nature Phys 2, 605–608 (2006). https://doi.org/10.1038/nphys392
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DOI: https://doi.org/10.1038/nphys392
