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

The effective independent-particle (mean-field) approximation of the Fermi-Hubbard Hamiltonian is described in a many-body basis to develop a formal comparison with the exact diagonalization of the full Fermi-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 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 the electronic properties of molecules and solids.