Multiple Exchange in the Quantum Crystals

We develop the generalization of the Heisenberg near-neighbor-exchange Hamiltonian necessary to incorporate the effects of multiple-exchange processes. Many-body expressions for the multiple-exchange constants (pair exchange, triple exchange, and quadruple exchange) are derived. The physics that enters these exchange constants is discussed. For the most important of the pair and triple cases, these expressions are carefully evaluated using a Monte Carlo integration scheme. We show that the exchange Hamiltonian for solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is rapidly convergent, and that the near-neighbor pair, next-near-neighbor pair, and triple-exchange processes (involving two near neighbors and a next-near neighbor) are likely to be the only important exchange processes to the low-temperature thermodynamics of bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The magnitude of the triple-exchange process is such that the "effective" next-near-neighbor pair-exchange interaction in bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is ferromagnetic. This result provides qualitative and quantitative support to the explanation of the data of Kirk and Adams made by Zane.