Density of states for interacting tunneling units in the absence of long-range order

We consider the effect of interactions between two-level (four-level) tunneling states. We allow the tunneling units to interact via a potential J(r)=±arⁿ, where a is a constant and r is the distance between the tunneling units, obtain the free energy F̄ up to and including the second virial coefficient, and neglect higher virial coefficients. We derive the density $P(E)$ of the elementary excitation energies E arising from F̄ exactly for a random distribution of tunneling units. We obtain that P(E)∝E^(6-2n)n for low E. The specific heat Cₚ∝T^(6-n)n and the thermal conductivity κ∝T^(4n-6)n for low temperatures T such that the product κCₚ∝T³ independent of n. In particular, for $n=3$ we find that $P(E)$ is approximately constant for low E and that Cₚ∝T and κ∝T² for low T. Our calculations also give a large T³ term in Cₚ and a somewhat flatter portion in κ for higher temperatures, both of these arising from the interacting tunneling units. The low-T dielectric susceptibility χ is predicted to have a ---lnT term in it, provided that T>T₀, where T₀ depends on the tunneling matrix element and the interaction between a pair of near-neighbor tunneling units. For T<<T₀, both Cₚ and κ are predicted to be proportional to exp(-constT) and χ approaches a constant value.