Quantum corrections in an antiferromagnet: A systematic diagrammatical treatment of the Hubbard model.

Abstract

A generalized scrN-orbital Hubbard model, with scrN orbitals per site, is used to develop a systematic perturbation-theoretic expansion in powers of 1/scrN which preserves spin-rotational symmetry, and hence the Goldstone mode, order by order. This systematic approach is illustrated with a study of quantum spin fluctuations in the half-filled-band Hubbard model in the strong-coupling limit. Quantum corrections are obtained for spin-wave amplitude, its velocity, and the uniform perpendicular susceptibility to O(1/scrN), and for the sublattice magnetization to O(1/scrN${)}^{2}$. Order by order these corrections are in exact agreement with spin-wave theory, due to an equivalence between the generalized scrN-orbital Hubbard model in the strong-coupling limit and the spin-S Heisenberg model when scrN=2S. The O(1/scrN${)}^{2}$ correction to sublattice magnetization is shown to vanish due to an exact cancellation in bipartite, hypercubic lattices, and the correction (${\mathit{Z}}_{\mathit{c}}$-1) to spin-wave velocity is small (even in two dimensions) due to a partial cancellation. These quantum corrections appear in the exponential temperature dependence of spin-correlation length in 2D and, for the coefficient of the J/${\mathit{k}}_{\mathit{B}}$T term in the exponent, we obtain M(0)${\mathit{Z}}_{\mathit{c}}$\ensuremath{\pi}/2=1.20, in excellent agreement with recent Monte Carlo studies.