In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2.\n\nIn the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT \nalgorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy.\nThe main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined\nFalicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase.\n\nIn the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory.\nThis model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The \nrenormalization factor q can be evaluated as a function of the filling n for 0<n<1. In the limit n->0, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2.\n\nIn the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT \nalgorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy.\nThe main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined\nFalicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase.\n\nIn the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory.\nThis model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The \nrenormalization factor q can be evaluated as a function of the filling n for 0<n<1. In the limit n->0, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.

Jan 1, 20061 citation

Symmetry breaking in the Hubbard model at weak coupling

T. Schauerte, P. G. J. van Dongen

The phase diagram of the Hubbard model is studied at weak coupling in two and three spatial dimensions. It is shown that the N\'eel temperature and the order parameter in $d=3$ are smaller than the Hartree-Fock predictions by a factor of $q=0.2599.$ For $d=2$ we show that the self-consistent (sc) perturbation series bears no relevance to the behavior of the exact solution of the Hubbard model in the symmetry-broken phase. We also investigate an anisotropic model and show that the coupling between planes is essential for the validity of mean-field-type order parameters.The phase diagram of the Hubbard model is studied at weak coupling in two and three spatial dimensions. It is shown that the N\'eel temperature and the order parameter in $d=3$ are smaller than the Hartree-Fock predictions by a factor of $q=0.2599.$ For $d=2$ we show that the self-consistent (sc) perturbation series bears no relevance to the behavior of the exact solution of the Hubbard model in the symmetry-broken phase. We also investigate an anisotropic model and show that the coupling between planes is essential for the validity of mean-field-type order parameters.

arXiv:cond-mat/0202492Physical review. B, Condensed matterFeb 7, 200215 citations

Strong coupling regime in the two-dimensional Hubbard model

Ian P. McCulloch, M. Gulácsi

Journal of Magnetism and Magnetic MaterialsMay 1, 19980 citations

The phase diagram of the Hubbard model is studied at weak coupling in two and three spatial dimensions. It is shown that the N\'eel temperature and the order parameter in

d=3

are smaller than the Hartree-Fock predictions by a factor of

q=0.2599.

For

d=2

we show that the self-consistent (sc) perturbation series bears no relevance to the behavior of the exact solution of the Hubbard model in the symmetry-broken phase. We also investigate an anisotropic model and show that the coupling between planes is essential for the validity of mean-field-type order parameters.