Figure 1

$S$ and $\tau$ spins, and their interaction with elastic strain. (a) A two-dimensional system in which the impurity can occupy one of four positions shown by the red and yellow circles; states of the same color are related by inversion symmetry and form a “$\tau$-pair”. Different colors correspond to different $S$ subspaces, between which transitions are asymmetric with respect to inversion. Distortion by a passing phonon breaks the degeneracy between states of different colors (i.e., couples to the $S$ excitations) but not between states of the same color (i.e., does not couple to the $\tau$ excitations). (b) The $\tau$-pair degeneracy is broken in a system with finite strain created by strong lattice disorder or amorphousness. This results in a finite but small $\tau$-TLS lattice interaction ${\gamma }_{\tau }$. (c) We show $2N=8$ states of an impurity, initially assumed degenerate, which are then split in a disordered crystal. The splitting between states in the same $\tau$ pair is small. (d) A specific example: the three-dimensional alkali-metal halide KBr with two CN impurities, which, in strong disorder, align along one of the three crystal axes (distorting the nearby lattice). Different $\tau$ states are connected by ${180}^{\circ }$ CN flips, whereas different $S$ states are connected by ${90}^{\circ }$ rotations.Reuse & Permissions

Figure 2

Densities of states (DOS) of $\tau$- and $S$-TLSs. (a) We show $n_{\tau }\left(E\right)$ in red (it is hardly changed by $S-\tau$ interactions) and $n_S\left(E\right)$ in blue [the noncorrelated DOS $n_S^o\left(E\right)$ is shown in green for comparison]. In (b), (c), and (d) we show the reduction factor $P\left(E\right)\equiv n_S\left(E\right)/n_S\left(\overline{E}\right)$, where $\overline{E}$ is an energy a few times larger than $g{J}_o$; essentially, $n_S\left(\overline{E}\right)$ is the $S$-spin DOS obtained if one includes $S-S$ correlations but ignores $S-\tau$ correlations, so $P\left(E\right)$ measures the effect of $S-\tau$ correlations. Numerical results are shown for $\tilde{a}=0,1.5,3,4.5,6$, for an impurity concentration $x=0.2$ and sample size ${18}^3$ cells (i.e., $\sim$4650 TLSs), in the three cases (b) $h^{\tau }=0$, (c) $h^{\tau }/\langle E_{\tau }\rangle \approx 0.3$, and (d) $h^{\tau }/\langle E_{\tau }\rangle \approx 1.$ $T_U$ is defined by the energy at which $n_{\tau }\left(E\right){\gamma }_{\tau }^2=n_S\left(E\right){\gamma }_S^2$, i.e., by $P\left(E\right)\approx 5g$. Similar results for $x=0.5$ are shown in Fig. 4. For both concentrations we find $T_U\approx 0.2\langle E_{\tau }\rangle$ for $1<a_o<6$, i.e., $T_U\approx 0.2g{J}_o$.Reuse & Permissions

Figure 3

Interaction of the symmetric TLSs with the gradient of the strain. The degeneracy of states related by inversion symmetry is broken by the second derivative of the phonon displacement.Reuse & Permissions

Figure 4

The true $S$-spin density of states $n_S\left(E\right)$, as a function of energy for different short-distance behaviors of the interaction. Details are as in Fig. 2 in the main text, only here $x=0.5$ and lattice size is ${13}^3$ cells (i.e., $\sim$4400 TLSs).Reuse & Permissions