Excitation energies of multilevel tunneling states in the presence of large local strain fields: A possible model for the glassy state.

Abstract

The density of states, ${P}_{0}$(E), of the elementary excitation energies E is obtained for a single two-level and four-level tunneling state in the presence of a distribution ${P}_{1}$(\ensuremath{\xi}) of random local strain fields \ensuremath{\xi} and a fixed tunneling matrix element \ensuremath{\Delta}. Whereas the two-level states have low-energy excitations only when both \ensuremath{\Delta} and \ensuremath{\xi} are small, the four-level (or six-level) states have very-low-energy excitations, E\ensuremath{\propto}${\ensuremath{\Delta}}^{2}$/\ensuremath{\xi}, even for large values of \ensuremath{\Delta}, provided the strain fields \ensuremath{\xi} are sufficiently large. We thus obtain the unexpected result that the excitation energies become smaller with increasing local strain fields for the multilevel states but not for the two-level states. For the case when ${P}_{1}$(\ensuremath{\xi})\ensuremath{\propto}\ensuremath{\Vert}\ensuremath{\xi}${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}k}$ for large \ensuremath{\xi}, the density of the low-energy excitations for the four-level states is ${P}_{0}$(E)\ensuremath{\propto}${E}^{k\mathrm{\ensuremath{-}}2}$ for E\ensuremath{\ll}\ensuremath{\Delta}. For k=2, ${P}_{0}$(E) is a constant for low E for the four-level states. We also show that for a number of physically interesting probability distributions of strain fields ${P}_{0}$(E) is approximately constant for the four-level states but not for the two-level states. In particular we find that the probability for low \ensuremath{\xi} is not relevant in determining the low-temperature thermodynamic properties of the four-level (and six-level) states. The possible significance of our results to the low-temperature thermodynamic properties of glasses is discussed.