Cyclic four-spin exchange on a two-dimensional square lattice: Possible applications in high-Tcsuperconductors

In the planar Cu${\mathrm{O}}_2$ square lattice of new superconductors, the small energy difference of conducting holes between Cu and O sites, compared to the relatively large on-site energy for double occupancy, might favor cyclic four-hole exchange via intermediate O sites rather than superexchange processes, as in solid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The properties of the four-spin exchange and Heisenberg Hamiltonians are compared in two dimensions through exact diagonalization on small clusters up to 18 spins. Four-spin exchange could explain the large inelasticity observed in neutron and two-magnon Raman spectroscopy, and resolve some discrepancies between exchange frequencies deduced from Raman and susceptibility data.