Tunneling and Exchange in Quantum Solids

Abstract

Pairs of $^{3}\mathrm{He}$ atoms in solid bcc $^{3}\mathrm{He}$ have spatially symmetric or antisymmetric wave functions which correspond to energies ${\ensuremath{\epsilon}}^{+}$ and ${\ensuremath{\epsilon}}^{\ensuremath{-}}$, ${\ensuremath{\epsilon}}^{\ifmmode\pm\else\textpm\fi{}}=\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}\frac{1}{2}\ensuremath{\Delta}\ensuremath{\epsilon}$. The energy difference ${\ensuremath{\epsilon}}^{+}\ensuremath{-}{\ensuremath{\epsilon}}^{\ensuremath{-}}=\ensuremath{\Delta}\ensuremath{\epsilon}$ is due to a tunneling process and an interaction process. The effect of these two processes can be simulated by adding to the Hamiltonian of the solid an exchange Hamiltonian, ${\mathcal{H}}_{x}=\ensuremath{-}2\ensuremath{\Delta}\ensuremath{\epsilon}\ensuremath{\Sigma}\stackrel{\ensuremath{'}}{\mathrm{ij}}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}}_{j}=\ensuremath{-}2(\ensuremath{\Delta}{\ensuremath{\epsilon}}_{T}+\ensuremath{\Delta}{\ensuremath{\epsilon}}_{x})\ensuremath{\Sigma}\stackrel{\ensuremath{'}}{\mathrm{ij}}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}}_{j},$ where $\ensuremath{\Delta}\ensuremath{\epsilon}$ is a sum $\ensuremath{\Delta}{\ensuremath{\epsilon}}_{T}$ (due to the tunneling process) and $\ensuremath{\Delta}{\ensuremath{\epsilon}}_{x}$ (due to the interaction process). A theory of the magnitude and sign of $\ensuremath{\Delta}{\ensuremath{\epsilon}}_{x}$ and $\ensuremath{\Delta}{\ensuremath{\epsilon}}_{T}$ is given. We find $\ensuremath{\Delta}{\ensuremath{\epsilon}}_{T}<0$ and $\ensuremath{\Delta}{\ensuremath{\epsilon}}_{x}>0$. Using quite general arguments, we show that $|\ensuremath{\Delta}{\ensuremath{\epsilon}}_{T}|\ensuremath{\gtrsim}2|\ensuremath{\Delta}{\ensuremath{\epsilon}}_{x}|$. The exchange in bcc solid $^{3}\mathrm{He}$ is antiferromagnetic. Evaluation of the formulas for $\ensuremath{\Delta}{\ensuremath{\epsilon}}_{T}$ and $\ensuremath{\Delta}{\ensuremath{\epsilon}}_{x}$ using the ground-state wave function of Guyer and Sarkissian leads to $\ensuremath{\Delta}\ensuremath{\epsilon}=J$, in good agreement with experiment.