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, N\'eel, and random. In the N\'eel case the retraceable-path approximation by Brinkman and Rice for the Green function is found to be correct up to order 1/d⁴ 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}).