The atomic method for the Hubbard dimer

The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice. It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands. The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons. This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors. However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.
In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer). To obtain the Green’s function for the lattice, we employed the cumulant expansion technique, using as a ‘seed’ the exact two-site Green’s function (the atomic method) [2]. From the Green’s function, we obtained the density of states (DOS) and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].

References:
[1] J. Hubbard. Electron Correlations in Narrow Energy Bands. Proc. R. Soc. Lond. A 1963 256, doi: 10.1098/rspa.1963.0204, published 26 November 1963.
[2] T. Lobo, M. S. Figueira and M. E. Foglio, Nanotechnology. 21, 274007 (2010).
[3] Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13 - Published 1 January 1996. doi: 10.1103/RevModPhys.68.13.