  Amorphous solids, as well as many disordered lattices, display remarkable
universality in their low temperature acoustic properties. This universality is
attributed to the attenuation of phonons by tunneling two-level systems (TLSs),
facilitated by the interaction of the TLSs with the phonon field. TLS-phonon
interaction also mediates effective TLS-TLS interactions, which dictates the
existence of a glassy phase and its low energy properties. Here we consider
KBr:CN, the archetypal disordered lattice showing universality. We calculate
numerically, using conjugate gradients method, the effective TLS-TLS
interactions for inversion symmetric (CN flips) and asymmetric (CN rotations)
TLSs, in the absence and presence of disorder, in two and three dimensions. The
observed dependence of the magnitude and spatial power law of the interaction
on TLS symmetry, and its change with disorder, characterizes TLS-TLS
interactions in disordered lattices in both extreme and moderate dilutions. Our
results are in good agreement with the two-TLS model, recently introduced to
explain long-standing questions regarding the quantitative universality of
phonon attenuation and the energy scale of $\approx 1-3$ K below which
universality is observed.
