Possible glass anomalies in the shear modulus and dielectric function of solid helium

The shear modulus of solid $^4$He exhibits an anomalous change at low temperature that is qualitatively similar to a frequency change in torsional oscillator experiments. We propose that in solid $^4$He the stiffening of the shear modulus with decreasing temperature can be described with a generalized susceptibility including a glassy backaction by assuming a distribution of temperature-dependent relaxation times $\tau(T)$. The glass susceptibility captures the freezing out of glassy degrees of freedom below a characteristic crossover temperature $T_X$, when the dynamic response of the solid satisfies $\omega \tau(T_X) \sim 1$, thus leading to a viscous response. We predict that the maximum change of the amplitude of the shear modulus and the height of the dissipation peak are independent of the applied frequency $\omega$. Recent measurements of the dielectric function $\epsilon(\omega)$ by the UFL group show a similar amplitude increase. We propose that changes in $\epsilon(\omega)$ are due to the glassy dynamics of low-lying excitations and are related to the shear modulus through acousto-optical coupling. We predict a dissipation peak in the imaginary part of the dielectric function, where the change in the real part is largest.