The Cumulant Expansion for the Anderson Lattice with Finite U: The Completeness Problem

TL;DR

This paper investigates the cumulant expansion in the Anderson lattice model with finite U, revealing that proper diagram selection ensures probability conservation and that reduced state spaces can be as effective as full spaces.

Contribution

It demonstrates how to maintain completeness in cumulant expansions by choosing appropriate diagrams, bridging the gap between finite and infinite U treatments.

Findings

Completeness is recovered when U approaches infinity via the chain approximation.

Proper diagram selection removes inconsistencies in reduced space calculations.

Reduced state space can be as effective as the full space with correct diagram choices.

Abstract

``Completeness'' (i.e. probability conservation) is not usually satisfied in the cumulant expansion of the Anderson lattice when a reduced state space is employed for $U\to \infty $. To understand this result, the well known ``Chain'' approximation is first calculated for finite $U$, followed by taking $U\to \infty $. Completeness is recovered by this procedure, but this result hides a serious inconsistency that causes completeness failure in the reduced space calculation. Completeness is satisfied and the inconsistency is removed by choosing an adequate family of diagrams. The main result of this work is that using a reduced space of relevant states is as good as using the whole space.