Linked Cluster Expansion Around Mean-Field Theories of Interacting Electrons

TL;DR

This paper develops a linked cluster expansion framework for interacting electron models, providing a systematic way to improve static mean-field theories and capturing dynamical fluctuations through self-consistent diagrammatic approaches.

Contribution

It introduces a universal linked cluster expansion scheme for electron models, unifying weak and strong coupling mean-field theories with a focus on self-consistency and dynamical effects.

Findings

The expansion captures dynamical fluctuations at weak coupling.

Self-consistent approximations ensure thermodynamic consistency.

Summing noncrossing diagrams yields a fully self-consistent theory.

Abstract

A general expansion scheme based on the concept of linked cluster expansion from the theory of classical spin systems is constructed for models of interacting electrons. It is shown that with a suitable variational formulation of mean-field theories at weak (Hartree-Fock) and strong (Hubbard-III) coupling the expansion represents a universal and comprehensive tool for systematic improvements of static mean-field theories. As an example of the general formalism we investigate in detail an analytically tractable series of ring diagrams that correctly capture dynamical fluctuations at weak coupling. We introduce renormalizations of the diagrammatic expansion at various levels and show how the resultant theories are related to other approximations of similar origin. We demonstrate that only fully self-consistent approximations produce global and thermodynamically consistent extensions of static mean field theories. A fully self-consistent theory for the ring diagrams is reached by summing the so-called noncrossing diagrams.