Mean-field approximation of the Hubbard model expressed in a many-body basis

TL;DR

This paper compares the mean-field approximation of the Hubbard model in a many-body basis with exact diagonalization, highlighting its limitations and the importance of correlation effects in electronic systems.

Contribution

It introduces a formal comparison framework between mean-field and exact solutions of the Hubbard model using a many-body basis, clarifying the role of correlations and satellite peaks.

Findings

Mean-field approximation misses critical correlation effects.

Satellite peaks can appear in mean-field, not solely due to correlations.

Different many-body corrections can improve mean-field accuracy.

Abstract

The effective independent-particle (mean-field) approximation of the Hubbard Hamiltonian is described in a many-body basis to develop a formal comparison with the exact diagonalization of the full Hubbard model, using small atomic chain as test systems. This allows for the development of an intuitive understanding of the shortcomings of the mean-field approximation and of how critical correlation effects are missed in this popular approach. The description in the many-body basis highlights a potential ambiguity related to the definition of the density of states. Specifically, satellite peaks are shown to emerge in the mean-field approximation, in departure from the common belief that they characterize correlation effects. The scheme emphasizes the importance of correlation and how different many-body corrections can improve the mean-field description. The pedagogical treatment is expected to make it possible for researchers to acquire an improved understanding of many-body effects as found in various areas related to electronic properties of molecules and solids, which is highly relevant to current efforts in quantum information and quantum computing.