Relationship between two-level systems and quasilocalized normal modes in glasses

Tunneling two-level systems (TLSs) dominate the physics of glasses at low temperatures. Yet TLSs are extremely rare, and it is thus difficult to directly observe them in silico. Developing simple structural predictors, which can provide markers for determining if a TLS is present in a given glass region, is crucial for a more efficient search. It has been speculated that vibrational quasilocalized modes (QLMs) are closely related to TLSs, and that one can extract information about TLSs from QLMs. In this work we address this possibility. In particular, we investigate the degree to which a linear or nonlinear vibrational mode analysis can predict the location of TLSs independently found by energy landscape exploration. We find that even though there is a notable spatial correlation between QLMs and TLSs, in general TLSs are strongly nonlinear, and their global properties cannot be predicted by a simple vibrational mode analysis.

Figure 1

Schematic illustration of the multidimensional minimal energy transition path between two energy minima within a DW potential. Also illustrated are the displacement vector ${\underline{r}}_{AB}$ between the two minima, the displacement vector ${\underline{r}}_{AS}$ between the first minimum and the saddle point, and the tangent vector ${\underline{r}}_{12}$ in the first minimum.

Figure 2

Participation ratio of normal modes versus their frequency, for one selected energy minimum. Modes shown in red stars are considered to be quasilocalized. In the inset, the mode frequency is shown as a function of the mode index. Data are for glasses with $T_f=0.062$.

Figure 3

Displacement fields of the linear and the nonlinear modes that have the largest overlap with the displacement field ${\underline{r}}_{AB}$ of a DW potential. The size of the particles is proportional to the particle displacement. The 10 particles that move the most in the ${\underline{r}}_{AB}$ field are shown in red. In the box faces, the corresponding projections are shown. Data are for a glass with $T_f=0.062$. The overlap coefficients defined by Eqs. (7) and (8) are $c\approx 0.95, a\approx 0.97$ for the nonlinear mode and $c\approx 0.14, a\approx 0.19$ for the linear mode.

Figure 4

Normalized scalar product $c_{\alpha }$ of the linear vibrational modes with ${\underline{r}}_{12}$ versus their participation ratio, for different number $N_b$ of images in the NEB, for a selected active TLS (energy splitting $E=8.54\times {10}^{-4}$) for $T_f=0.062$. The modes are numbered starting from 4, mode 4 being thus the softest mode.

Figure 5

Distribution over DW potentials of the largest mode projection, $c={max}_{\alpha }{c}_{\alpha }$, calculated over all modes for ${\underline{r}}_{12}, {\underline{r}}_{AS}$, and ${\underline{r}}_{AB}$. Here we used $N_b=600$, and since the NEB calculation is expensive, we have used a smaller subset of all DW potentials, selecting only TLSs with energy splitting $E<{10}^{-3}$ and participation ratio of the DW transition ${\text{PR}}_{DW}<5$ (note that the PR for DW potentials is normalized differently than for normal modes; see [30]) for $T_f=0.062$.

Figure 6

Scatter plot of the tunnel splitting $E$ versus the frequency $\omega$ in minimum $A$, for all DW potentials found at $T_f=0.062$

Figure 7

Probability distribution of overlap coefficients $c$ and $a$ for linear and nonlinear modes, obtained from a recursive procedure, starting from ${\underline{r}}_{AB}$ as initial guess. Data are for the three preparation temperatures $T_f$ and for the full set of DW potentials.

Figure 8

Overlap of the softest localized linear modes, and of the nonlinear modes that are constructed using those linear modes as an initial condition, with the displacement field ${\underline{r}}_{AB}$. Data are for $T_f=0.062$ and for the full set of DW potentials.