[Paper Review] Towards analytic description of a transition from weak to strong coupling regime in correlated electron systems. I. Systematic diagrammatic theory with two-particle Green functions

V. Janiš|arXiv (Cornell University)|Apr 9, 1997

Advanced Chemical Physics Studies11 citations

TL;DR

This paper develops a self-consistent diagrammatic theory using two-particle Green functions to analytically describe the transition from weak to strong coupling in correlated electron systems. By identifying a diverging pole in the two-particle vertex function signaling a metal-insulator transition and introducing a low-frequency factorization scheme, it achieves stable numerical solutions in the critical region, revealing the necessity of dynamical vertex renormalizations absent in simpler approximations.

ABSTRACT

We analyze behavior of correlated electrons described by Hubbard-like models at intermediate and strong coupling. We show that with increasing interaction a pole in a generic two-particle Green function is approached. The pole signals metal-insulator transition at half filling and gives rise to a new vanishing ``Kondo'' scale causing breakdown of weak-coupling perturbation theory. To describe the critical behavior at the metal-insulator transition a novel, self-consistent diagrammatic technique with two-particle Green functions is developed. The theory is based on the linked-cluster expansion for the thermodynamic potential with electron-electron interaction as propagator. Parquet diagrams with a generating functional are derived. Numerical instabilities due to the metal-insulator transition are demonstrated on simplifications of the parquet algebra with ring and ladder series only. A stable numerical solution in the critical region is reached by factorization of singular terms via a low-frequency expansion in the vertex function. We stress the necessity for dynamical vertex renormalizations, missing in the simple approximations, in order to describe the critical, strong-coupling behavior correctly. We propose a simplification of the full parquet approximation by keeping only most divergent terms in the asymptotic strong-coupling region. A qualitatively new, feasible approximation suitable for the description of a transition from weak to strong coupling is obtained.

Motivation & Objective

• To develop a systematic analytic framework for describing the crossover from weak to strong electron correlation in Hubbard-like models.
• To identify and describe the critical behavior at the metal-insulator transition using two-particle Green functions.
• To overcome numerical instabilities in strong-coupling regimes caused by diverging vertex functions.
• To demonstrate the failure of simple approximations (e.g., ring and ladder series) in capturing critical behavior.
• To establish the necessity of dynamical vertex renormalizations for accurate strong-coupling description.

Proposed method

• Formulates a linked-cluster expansion for the thermodynamic potential with electron-electron interaction as a propagator.
• Derives parquet diagrams using a generating functional to systematically include two-particle vertex contributions.
• Applies a low-frequency expansion to factorize singular terms in the vertex function, stabilizing numerical solutions near criticality.
• Identifies a pole in the two-particle Green function as a signature of metal-insulator transition and emergence of a Kondo-like scale.
• Proposes a simplified parquet approximation by retaining only the most divergent terms in the strong-coupling limit.
• Introduces a self-consistent framework that captures dynamical vertex renormalizations missing in standard perturbation theories.

Experimental results

Research questions

• RQ1 How can a continuous analytic description of the weak-to-strong coupling transition be achieved in correlated electron systems?
• RQ2 What role does the pole structure in the two-particle Green function play in signaling metal-insulator transitions?
• RQ3 Why do standard approximations like ring and ladder series fail numerically in the critical region?
• RQ4 What mechanism enables stable numerical solutions near the critical point despite vertex singularities?
• RQ5 How do dynamical vertex corrections influence the correct description of strong-coupling behavior?

Key findings

• A pole in the two-particle Green function emerges with increasing interaction, signaling the onset of a metal-insulator transition at half filling.
• The pole gives rise to a new vanishing 'Kondo' scale that invalidates weak-coupling perturbation theory.
• Numerical instabilities in simplified parquet approximations (ring and ladder series) are traced to uncontrolled vertex singularities.
• Stable solutions in the critical region are achieved through low-frequency factorization of the vertex function.
• Dynamical vertex renormalizations are essential and missing in simple approximations, necessitating a self-consistent treatment.
• A simplified, feasible approximation retaining only the most divergent terms in the strong-coupling limit provides a qualitatively new and accurate description of the crossover regime.