The connection between the density of Two-level systems and density of Quasi-localised modes in glasses

The density of Two-level systems (TLS) $n_0$, which controls the low-temperature properties of glass, dramatically decreases in ultrastable glasses. Meanwhile, $A_4$, -the prefactor of the density of quasi-localised modes (QLMs)-, also dramatically decreases by several hundred times in recent numerical simulations. However, the relationship between $A_4$ and $n_0$ is unclear. In this work, I theoretically build up the relationship between $n_0$ and the density of QLMs based on the soft-potential model. I find that $n_0$ corresponds to the QLMs with a typical frequency $\omega_0$ that is about 10\% - 20\% of the Debye frequency. The decrease in $n_0$ is influenced not only by the rarefaction of the density of QLMs but also by a factor that is related to the fraction of double-well potentials. Further, I find that the estimations of $n_0$ are consistent with the order of magnitude of the change in $n_0$ found in experiment.