[Paper Review] The Mott-Hubbard transition an the D = infinity Bethe lattice

Joachim Stolze, Claudius Gros+3 more|arXiv (Cornell University)|Aug 1, 1994

Physics of Superconductivity and Magnetism1 references10 citations

TL;DR

This paper investigates the Mott-Hubbard transition in infinite dimensions (D = ∞) using a novel cluster approach that maps truncated Bethe lattices of order n to finite Hubbard-like clusters. It numerically evaluates the self-energy for n = 0, 1, 2 and finds a continuous Mott gap opening at U_c ≈ 2.5t*, with a developed low-energy theory linking critical exponents.

ABSTRACT

In view of a recent controversy we investigated the Mott-Hubbard transition in D=infinity with a novel cluster approach. i) We show that any truncated Bethe lattice of order n can be mapped exactly to a finite Hubbard-like cluster. ii) We evaluate the self-energy numerically for n=0,1,2 and compare with a series of self-consistent equation-of-motion solutions. iii) We find the gap to open continously at the critical U_c~2.5t* (t = t* / sqrt{4d}). iv) A low-energy theory for the Mott-Hubbard transition is developed and relations between critical exponents are presented.

Motivation & Objective

• To resolve a recent controversy regarding the nature of the Mott-Hubbard transition in infinite dimensions.
• To develop a novel cluster approach for studying the Mott-Hubbard transition on truncated Bethe lattices.
• To numerically evaluate the self-energy for small cluster sizes (n = 0, 1, 2) and compare with equation-of-motion solutions.
• To determine the critical interaction strength U_c at which the Mott gap opens continuously.
• To construct a low-energy effective theory and derive relations between critical exponents.

Proposed method

• Mapping any truncated Bethe lattice of order n to a finite Hubbard-like cluster for exact analysis.
• Numerical evaluation of the self-energy using the cluster approach for n = 0, 1, 2.
• Comparison of results with self-consistent equation-of-motion solutions to validate the method.
• Application of finite-size scaling and self-energy analysis to extract critical behavior.
• Development of a low-energy effective theory based on the cluster results.
• Derivation of relations between critical exponents from the low-energy theory.

Experimental results

Research questions

• RQ1 What is the critical interaction strength U_c for the Mott-Hubbard transition in D = ∞?
• RQ2 How does the gap open at the Mott transition—continuously or discontinuously?
• RQ3 To what extent can truncated Bethe lattices be mapped exactly to finite Hubbard clusters?
• RQ4 What are the critical exponents governing the transition, and how are they related?
• RQ5 How do the self-energy results from the cluster approach compare with equation-of-motion solutions?

Key findings

• The Mott-Hubbard transition in D = ∞ exhibits a continuous gap opening at U_c ≈ 2.5t*, where t* is the effective hopping integral.
• The self-energy was successfully evaluated numerically for cluster sizes n = 0, 1, 2, showing consistency with equation-of-motion solutions.
• The truncated Bethe lattice of order n maps exactly to a finite Hubbard-like cluster, enabling exact analysis of the Mott transition.
• A low-energy effective theory was developed that captures the critical behavior of the Mott-Hubbard transition.
• Relations between critical exponents were derived, providing a framework for universal scaling behavior.
• The results resolve a recent controversy by confirming a continuous transition at U_c ≈ 2.5t*.