Polarizability of the low-energy excitations in disordered solids

We have attempted to measure the electric dipole moment $\mu$ of the low-energy excitations observed in glasses $T<\mathrm{}1\mathrm{}$ K. We have attacked the problem in two different ways by measuring (i) the temperature dependence of the dielectric constant of amorphous Ge${\mathrm{O}}_2$ and poly(methylmethacrylate) (PMMA) ($0.06<T<\mathrm{}0.3\mathrm{}$K), and (ii) the electric field dependence of the thermal conductivity of amorphous Si${\mathrm{O}}_2$ ($\mathcal{E}\sim {10}^5$ V/cm; $T\sim 0.5$ K). In each case we obtained a null result. We used the tunneling model proposed by Phillips and Anderson et al. to establish upper limits on $\mu$ in these glasses. A variety of free parameters are embedded in their model, and we show how the calculated limits depend on the assumptions made about these parameters. Recent dielectric measurements on Si${\mathrm{O}}_2$ by Schickfus et al. show that the tunneling systems's potential wells are approximately symmetric. Assuming that is also true for the ${\mathrm{GeO}}_2$ and PMMA systems, we are able to set upper limits on $\mu$ of 0.15 and 0.11 D, respectively. In reconciling our thermal-conductivity measurements with the $\mu$ measured by Schickfus et al., we can also conclude that the distribution in energy $\Delta$ of these tunneling systems must be nearly uniform (within 6%) over the range $1<\mathrm{}\frac{\Delta }{k_B}<3\mathrm{}$ K.