Hole dynamics in a spin background: A sum-rule-conserving theory with exact limits

A self-consistent theory is formulated for the dynamics of a hole moving in a d-dimensional, quantum-mechanical background of spins at arbitrary temperatures. The contribution of loops in the path of a hole, which are always important in dimensions d>1, is given particular attention. We first show that the Green function, thermodynamics, and dynamical conductivity can be determined exactly in the limit d→∞. On the basis of this solution, we construct an approximation scheme for the dynamics of a hole in dimensions d<∞, where loops are summed self-consistently to all orders. The resulting theory satisfies the spectral and f-sum rules and yields the exact solution for the ferromagnetic background in any dimension d. Three types of spin backgrounds are explicitly discussed: ferromagnetic, Néel, and random. In the Néel case the retraceable-path approximation by Brinkman and Rice for the Green function is found to be correct up to order 1/${\mathit{d}}^4$ for large d. Detailed calculations of the density of states D(ω) and the conductivity σ(ω) of the hole are presented for d=3 and ∞. A characteristic dependence on the particular type of spin background is found, which is especially pronounced in the case of σ(ω).