Toward understanding the depletion of two-level systems in glasses

 The density of Two-level systems (TLS)  controls the low-temperature thermal properties in glasses. It is recently found that TLS are almost depleted in ultrastable glasses. Although it is thought to have a close relationship with the dramatic decrease of quasi-localized modes (QLMs), it is not yet clearly formalized.  Based on the soft-potential model and the tunneling TLS model, we argue that the TLS correspond to the QLMs with being a typical frequency ω₀.  Their density n₀  is proportional to the density of QLMs  D_L(ω₀) and to the fraction of equal double-wells f(ω₀) at ω₀. We numerically estimate ω₀ and n₀ in computer glasses at different stabilities.  We find that ω₀ is about  10% to 20% of the Debye frequency.   n₀  decreases by a factor of 500 in ultrastable glasses than poorly prepared glasses with both D_L(ω₀) and f(ω₀) playing a significant role. Remarkably, the estimations of n₀  are consistent with the order of magnitude of the change in n₀ found in experiment in amorphous silicon.