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 $J\left(r\right)=\frac{\pm a}{r^n}$, where $a$ is a constant and $r$ is the distance between the tunneling units, obtain the free energy $\overline{F}$ up to and including the second virial coefficient, and neglect higher virial coefficients. We derive the density $P\left(E\right)\mathrm{}$ of the elementary excitation energies $E$ arising from $\overline{F}$ exactly for a random distribution of tunneling units. We obtain that $P\left(E\right)\mathrm{}\propto E^{\frac{\mathrm{}(6-2n)\mathrm{}}{n}}$ for low $E$. The specific heat $C_p\propto T^{\frac{\mathrm{}(6-n)\mathrm{}}{n}}$ and the thermal conductivity $\kappa \propto T^{\frac{\mathrm{}(4n-6)\mathrm{}}{n}}$ for low temperatures $T$ such that the product $\kappa C_p\propto T^3$ independent of $n$. In particular, for $n=3\mathrm{}$ we find that $P\left(E\right)\mathrm{}$ is approximately constant for low $E$ and that $C_p\propto T$ and $\kappa \propto T^2$ for low $T$. Our calculations also give a large $T^3$ term in $C_p$ and a somewhat flatter portion in $\kappa$ for higher temperatures, both of these arising from the interacting tunneling units. The low-$T$ dielectric susceptibility $\chi$ is predicted to have a —$\ln T$ term in it, provided that $T>\mathrm{}{T}_0$, where $T_0$ depends on the tunneling matrix element and the interaction between a pair of near-neighbor tunneling units. For $T<<T_0$, both $C_p$ and $\kappa$ are predicted to be proportional to $\exp \left(\frac{-\mathrm{const}}{T}\right)$ and $\chi$ approaches a constant value.