Abstract Band models with intrasite (Hubbard) and intersite Coulomb interactions are treated in a cluster approximation taking into account the interactions and the transfer within clusters of sites rigorously. Between different clusters the interaction is decoupled in a mean field approximation and the transfer term is treated as a perturbation. The ground‐state energy of the one‐dimensional Hubbard model is calculated up to fourth order in the perturbation and is compared with other approximations and the exact result. In the one‐dimensional spinless fermion model the perturbation is treated without further approximations by solving Dyson's equation for the one‐particle Green's function. The order‐disorder transition in this model is discussed.
Abstract Band models with intrasite (Hubbard) and intersite Coulomb interactions are treated in a cluster approximation taking into account the interactions and the transfer within clusters of sites rigorously. Between different clusters the interaction is decoupled in a mean field approximation and the transfer term is treated as a perturbation. The ground‐state energy of the one‐dimensional Hubbard model is calculated up to fourth order in the perturbation and is compared with other approximations and the exact result. In the one‐dimensional spinless fermion model the perturbation is treated without further approximations by solving Dyson's equation for the one‐particle Green's function. The order‐disorder transition in this model is discussed.
In a large variety of systems, like the high‐temperature superconductors, the dominant parameter of the system is the on‐site Coulomb repulsion that becomes comparable with the kinetic energy in the vicinity of the metal‐insulator transition. This implies that perturbation expansions must include this interaction in the zeroth order Hamiltonian and treat the hopping term as a perturbation. Thus, one gets a situation radically different from the usual one in which the particles are initially free and needs to develop a nonstandard perturbation theory based on a cumulant expansion. In this paper, we revisit the general formalism of the cumulant expansion approach for the single band and two‐band Hubbard model, being interested in introducing a systematic way to perform approximations which take into account cooperative phenomena. As an example we analyze the effect of the charge fluctuations on the single‐particle spectral properties.
We introduce a mean-field and a perturbative approach, based on clusters, to describe the ground state of fermionic strongly correlated systems. In the cluster mean-field approach, the ground-state wave function is written as a simple tensor product over optimized cluster states. The optimization of the single-particle basis where the cluster mean field is expressed is crucial in order to obtain high-quality results. The mean-field nature of the Ansatz allows us to formulate a perturbative approach to account for intercluster correlations; other traditional many-body strategies can be easily devised in terms of the cluster states. We present benchmark calculations on the half-filled 1D and (square) 2D Hubbard model, as well as the lightly doped regime in 2D, using cluster mean-field and second-order perturbation theory. Our results indicate that, with sufficiently large clusters or to second-order in perturbation theory, a cluster-based approach can provide an accurate description of the Hubbard model in the considered regimes. Several avenues to improve upon the results presented in this work are discussed.
A slave-spin representation of fermion operators has recently been proposed for the half-filled single and multiband Hubbard model. We show that with the addition of a gauge variable, the formalism can be extended to finite doping. We solve the resulting spin problem using the cluster mean-field approximation. This approximation takes short-range correlations into account by exact diagonalization on the cluster, whereas long-range correlations beyond the size of clusters are treated at the mean-field level. In the limit where the cluster has only one site and the interaction strength U is infinite, this approach reduces to the Gutzwiller approximation. There are some qualitative differences when the size of the cluster is finite. We first compute the critical U for the Mott transition as a function of a frustrating nearest-neighbor interaction on lattices relevant for various correlated systems, namely, the cobaltates, the layered organic superconductors and the high-temperature superconductors. For the triangular lattice, we also study the extended Hubbard model with nearest-neighbor repulsion. In addition to a uniform metallic state, we find a $\sqrt{(3)}\ifmmode\times\else\texttimes\fi{}\sqrt{(3)}$ charge density wave in a broad doping regime, including commensurate ones. We find that in the large U limit, intersite Coulomb repulsion V strongly suppresses the single-particle weight of the metallic state.
Starting from the exact solution of the Hubbard model in the atomic limit, and treating the hopping term as a perturbation by means of a diagram technique, we discuss the electronic structure of the model in some simple approximations. We show that a metal-insulator transition is obtained above a critical value of the on-site Coulomb interaction in a self-consistent evaluation of the one-particle Green's function to one-loop order. A relationship with results in infinite dimensions is briefly discussed.
We present a generalization of the recently proposed variational cluster perturbation theory to extended Hubbard models at half-filling with repulsive nearest neighbor interaction. The method takes into account short-range correlations correctly by the exact diagonalization of clusters of finite size, whereas long-range order beyond the size of the clusters is treated on a mean-field level. For one dimension, we show that quantum Monte Carlo and density-matrix renormalization-group results can be reproduced with very good accuracy. Moreover we apply the method to the two-dimensional extended Hubbard model on a square lattice. In contrast to the one-dimensional case, a first order phase transition between spin density wave phase and charge density wave phase is found as function of the nearest-neighbor interaction at onsite interactions $U\ensuremath{\geqslant}3t$. The single-particle spectral function is calculated for both the one-dimensional and the two-dimensional system.