Phonon Thermal Transport of URu$_{2}$Si$_{2}$: Broken Translational Symmetry and Strong-Coupling of the ``Hidden Order'' to the Lattice

A dramatic increase in the total thermal conductivity ($k)$ is observed in the Hidden Order (HO) state of single crystal URu$_{2}$Si$_{2}$. Through measurements of the thermal Hall conductivity, we explicitly show that the electronic contribution to $k$ is extremely small, so that this large increase in $k$ is dominated by phonon conduction. An itinerant BCS/mean-field model describes this behavior well: the increase in $k$ is associated with the opening of a large energy gap at the Fermi Surface, thereby decreasing electron-phonon scattering. Our analysis implies that the ``Hidden Order'' parameter is strongly coupled to the lattice, suggestive of a broken symmetry involving charge degrees of freedom.