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Physics of the Hubbard model solved with the cluster method

Lorenzo Fratino

Published Sep 17, 2018• 0 citations
OpenAlex

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The Variational cluster approximation (VCA) is an approximate method for finding properties the Hubbard model, the basic model used in the study of strongly correlated electron systems. Such systems include, for instance, high-temperature and organic superconductors.The method involves solving exactly the Hubbard model on a small cluster of lattice sites, after adding fields that represent the effect of the cluster's environment. We present an outline of the method and describe its implementation. In particular, we show how to efficiently carry frequency-wavevector integrals that are central to the method.
An important method in many-particle physics is related to the Hubbard model which can be used to investigate the behavior of fermions, particularly to study exchange and correlation of electron- electron interaction. For a special cluster, the molecule consisting of two atoms, the electrons interact via one hopping integral, and for a small number of N = 1; 2; 3 electrons, the model has the advantage that it can be solved exactly. Therefore, some important physical quantities like the Green's function, the spectral function and the self-energy are calculated and discussed. Exchange and correlation are fully treated. The self-energies obtained by two approximate methods, namely the Hartree-Fock and the GW approximation, are compared with the exact self-energy. The comparison allows a discussion of the validity of the approximations versus the ratio of Hubbard parameter and hopping integral.
The numerical method was developed that combines the variational cluster approximation and the exact diagonalization techniques to solve Hubbard models of various dimensionality, number of orbitals, and lattice geometry. The developed method was tested on the one-band Hubbard model; antiferromagnetic ordering and the superconducting phase were observed, in accordance with known results obtained by other techniques. The method may be successfully used for studying properties of high-temperature superconductors.