Structural symmetry of Cd$_2$Re$_2$O$_7$ from nonlinear optics

Cd$_2$Re$_2$O$_7$, a superconducting metallic pyrochlore, undergoes a second-order structural phase transition at 200~K from a cubic to tetragonal lattice. Landau theory predicts that any second-order cubic-to-tetragonal phase transition must also possess an order parameter associated with broken inversion symmetry. By observing optical second-harmonic generation, we provide a direct demonstration that the 200~K transition in Cd$_2$Re$_2$O$_7$ involves broken inversion symmetry. Moreover, we have used the polarization dependence of SHG to refine the crystal structure. We find that the low-temperature crystal symmetry is that of the $F\overline{4}2m$ space group.