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 ${\epsilon }^{+}$ and ${\epsilon }^{-}$, ${\epsilon }^{\pm }=\epsilon \pm \frac{1}{2}\Delta \epsilon$. The energy difference ${\epsilon }^{+}-{\epsilon }^{-}=\Delta \epsilon$ 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, ${\mathcal{H}}_x=-2\mathrm{}\Delta \epsilon \Sigma \stackrel{\prime }{\mathrm{ij}}{\overrightarrow{\sigma }}_i·{\overrightarrow{\sigma }}_j=-2(\Delta {\epsilon }_T+\Delta {\epsilon }_x)\Sigma \stackrel{\prime }{\mathrm{ij}}{\overrightarrow{\sigma }}_i·{\overrightarrow{\sigma }}_j,$ where $\Delta \epsilon$ is a sum $\Delta {\epsilon }_T$ (due to the tunneling process) and $\Delta {\epsilon }_x$ (due to the interaction process). A theory of the magnitude and sign of $\Delta {\epsilon }_x$ and $\Delta {\epsilon }_T$ is given. We find $\Delta {\epsilon }_T<0\mathrm{}$ and $\Delta {\epsilon }_x>0\mathrm{}$. Using quite general arguments, we show that $\left|\Delta {\epsilon }_T\right|\gtrsim 2\left|\Delta {\epsilon }_x\right|$. The exchange in bcc solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is antiferromagnetic. Evaluation of the formulas for $\Delta {\epsilon }_T$ and $\Delta {\epsilon }_x$ using the ground-state wave function of Guyer and Sarkissian leads to $\Delta \epsilon =J$, in good agreement with experiment.