Vibrational instability, two-level systems, and the boson peak in glasses

We show that the same physical mechanism is fundamental for two seemingly different phenomena such as the formation of two-level systems in glasses and the boson peak in the reduced density of low-frequency vibrational states $g\left(\omega \right)∕{\omega }^2$. This mechanism is the vibrational instability of weakly interacting harmonic modes. Below some frequency ${\omega }_c⪡{\omega }_0$ (where ${\omega }_0$ is of the order of Debye frequency), the instability, controlled by the anharmonicity, creates a new stable universal spectrum of harmonic vibrations with a boson peak feature as well as double-well potentials with a wide distribution of barrier heights. Both are determined by the strength of the interaction $I\propto {\omega }_c$ between the oscillators. Our theory predicts in a natural way a small value for the important dimensionless parameter $C=\overline{P}{\gamma }^2∕\rho v^2\approx {10}^{-4}$ for two-level systems in glasses. We show that $C\approx {(W∕\hslash {\omega }_c)}^3\propto I^{-3}$ and decreases with increasing interaction strength $I$. The energy $W$ is an important characteristic energy in glasses and is of the order of a few Kelvin. This formula relates the two-level system’s parameter $C$ with the width of the vibration instability region ${\omega }_c$, which is typically larger or of the order of the boson peak frequency ${\omega }_b$. Since $\hslash {\omega }_c\gtrsim \hslash {\omega }_b⪢W$, the typical value of $C$ and, therefore, the number of active two-level systems is very small, less than 1 per $1\times {10}^7$ of oscillators, in good agreement with experiment. Within the unified approach developed in the present paper, the density of the tunneling states and the density of vibrational states at the boson peak frequency are interrelated.

Figure 1

Distribution function $\rho \left(\kappa \right)$ calculated as ensemble average by exact diagonalization of systems of $N=2197$ oscillators with $g_0\left(\omega \right)=3{\omega }^2$ and $J=0.07$, 0.10, and 0.15 (from left to right).

Figure 2

Distribution function of the renormalized quasielastic constants, $\Phi \left(k\right)$, calculated as ensemble average by exact diagonalization of a systems of $N=2197$ oscillators with $g_0\left(\omega \right)=3{\omega }^2$ $[F\left(k\right)=3\sqrt{k}∕2]$ and $J=0.07$, 0.10, and 0.15 (solid curves, from right to left). Dotted curve: result of convolution [Eq. (2.21)] for $J=0.1$.

Figure 3

The boson peak: the reduced density of states $g_{\mathrm{tot}}\left(\omega \right)∕{\omega }^2$ given by Eq. (5.18).