Multiple exchange inHe3and in the Wigner solid

Abstract

We use a multidimensional WKB calculation to determine the leading terms of a high-density series expansion for the multiple-exchange frequencies in solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. We also calculate, within the same formulation, the first terms of a low-density series expansion for exchange in the two-dimensional Wigner solid of electrons. The hierarchy between the exchange frequencies is dominated by the geometry of the lattice. Triple exchange is preponderant in the two-dimensional triangular lattice. This result holds for potentials as different as the ${\left(\frac{\sigma }{r}\right)}^{12}$ (solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$) and the Coulomb potential (Wigner solid). Three-particle exchange also dominates in the hcp lattice. We find for a hypothetical high-density bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ solid the same hierarchy as that deduced from the experimental results at physical densities: The planar four-particle exchange $K_P$ and the triple exchange $J_t$ dominate, the folded four-particle exchange $K_F$ is much lower. For both $K_P$ and $J_t$ the lengths of the exchange paths and the tunneling barrier heights are practically the same. We expect these results to be qualitatively valid at physical densities and understand why $K_P$ and $J_t$ depend similarly on the molar volume.