Nonlinear optical evidence for broken inversion symmetry in Cd$_2$Re$_2$O$_7$

Cd$_2$Re$_2$O$_7$ is a superconducting metallic pyrochlore that exhibits a second-order phase transition at 200 K, from a cubic to a tetragonal structure. According to Landau theory, such a phase transition must be accompanied by a loss of inversion symmetry; to date, however, the precise symmetry of the tetragonal phase remains undetermined. We observe second-order optical nonlinearity in this phase, confirming the loss of inversion symmetry. We will discuss efforts to characterize the low-temperature point group via second-harmonic polarimetry.