Since the discovery of the superconducting cuprates and their hitherto unknown electronic properties there has been a continuously growing interest in a theoretical treatment of suitable models of strongly correlated fermions which at least qualitatively reflect the essential aspects of these physical systems. The Hubbard model in two dimensions plays an important role in this context. The present thesis examines instabilities and possible symmetry broken phases of the two-dimensional repulsive Hubbard model with the help of numeric and semi analytic methods. While in the first part it is pointed out how such instabilities can be treated by a mean field theory, the second part of the work supplies the theoretical bases of the numeric procedures developed in part III. For the case of a pure nearest neighbour hopping it is shown that the second order contributions of perturbation theory generate an effective attraction between the electrons. The energy scale of the superconducting instabilities is also examined; it is shown that the vanHove singularity drastically enhances this critical scale at half filling. The investigation of a possible coexistence of two different kinds of spontaneous symmetry breaking forms the emphasis of the thesis: Lately accomplished renormalization group calculations showed (for suitably chosen parameters) the occurrence of a Pomeranchuk instability of the two-dimensional Hubbard model with nonvanishing next nearest neighbour hopping, which should give rise to a ground state that spontaneously breaks the model's lattice symmetry as well as its U(1) symmetry, i.e. the ground state could be characterized by two co-existing order parameters. For treatment of the problem of co-existing instabilities the ansatz of a renormalized perturbation theory is used, which had been suggested in order to avoid the occurrence of unphysical singularities arising within the framework of naive perturbation theory; these singularities are related to interaction-induced shifts of the Fermi surface, which cannot be captured by simply shifting the chemical potential. The attractive Hubbard model is also treated within the framework of renormalized perturbation theory; the well-known reduction of the superconducting order parameter with respect to its BCS value reported in earlier investigations can be confirmed. A spontaneously broken lattice symmetrie does not arise here, in contrast to the repulsive case.
Since the discovery of the superconducting cuprates and their hitherto unknown electronic properties there has been a continuously growing interest in a theoretical treatment of suitable models of strongly correlated fermions which at least qualitatively reflect the essential aspects of these physical systems. The Hubbard model in two dimensions plays an important role in this context. The present thesis examines instabilities and possible symmetry broken phases of the two-dimensional repulsive Hubbard model with the help of numeric and semi analytic methods. While in the first part it is pointed out how such instabilities can be treated by a mean field theory, the second part of the work supplies the theoretical bases of the numeric procedures developed in part III. For the case of a pure nearest neighbour hopping it is shown that the second order contributions of perturbation theory generate an effective attraction between the electrons. The energy scale of the superconducting instabilities is also examined; it is shown that the vanHove singularity drastically enhances this critical scale at half filling. The investigation of a possible coexistence of two different kinds of spontaneous symmetry breaking forms the emphasis of the thesis: Lately accomplished renormalization group calculations showed (for suitably chosen parameters) the occurrence of a Pomeranchuk instability of the two-dimensional Hubbard model with nonvanishing next nearest neighbour hopping, which should give rise to a ground state that spontaneously breaks the model's lattice symmetry as well as its U(1) symmetry, i.e. the ground state could be characterized by two co-existing order parameters. For treatment of the problem of co-existing instabilities the ansatz of a renormalized perturbation theory is used, which had been suggested in order to avoid the occurrence of unphysical singularities arising within the framework of naive perturbation theory; these singularities are related to interaction-induced shifts of the Fermi surface, which cannot be captured by simply shifting the chemical potential. The attractive Hubbard model is also treated within the framework of renormalized perturbation theory; the well-known reduction of the superconducting order parameter with respect to its BCS value reported in earlier investigations can be confirmed. A spontaneously broken lattice symmetrie does not arise here, in contrast to the repulsive case.
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.
Attractive non-local interactions jointly with repulsive local interaction in a microscopic modelling of electronic Fermi liquids generate a competition between an enhancement of the static charge susceptibility–ultimately signalling charge instability and phase separation–and its correlation induced suppression. We analyse this scenario through the investigation of the extended Hubbard model on a two-dimensional square lattice, using the spin rotation invariant slave-boson representation of Kotliar and Ruckenstein. The quasiparticle density of states, the renormalised effective mass and the Landau parameter F s 0 are presented, whereby the positivity of F s 0 – 1 constitutes a criterion for stability. Van Hove singularities in the density of states support possible charge instabilities. A (negative) next-nearest neighbour hopping parameter t' shifts their positions and produces a tendency towards charge instability even for low filling whereas the t' -controlled particle-hole asymmetry of the correlation driven effective mass is small. A region of instability on account of the attractive interaction V is identified, either at half filling in the absence of strong electronic correlations or, in the case of large on-site interaction U , at densities far from half filling.
Interplay of Pomeranchuk instability (spontaneous symmetry breaking of the Fermi surface) and d-wave superconductivity is studied for the repulsive Hubbard model on the square lattice with the dynamical mean-field theory combined with the fluctuation exchange approximation (FLEX+DMFT). We show that the fourfold symmetric Fermi surface becomes unstable against a spontaneous distortion into twofold near the van Hove filling, where the symmetry of superconductivity coexisting with the Pomeranchuk-distorted Fermi surface is modified from the d-wave pairing to the (d+s) wave. By systematically shifting the position of van Hove filling with varied second- and third-neighbor hoppings, we find that the transition temperature TcPI for the Pomeranchuk instability is more sensitively affected by the position of van Hove filling than the superconducting TcSC. This implies that the filling region for strong Pomeranchuk instability and that for the TcSC dome can be separated, and that Pomeranchuk instability can appear even if the peak of TcPI is lower than the peak of TcSC. An interesting finding is that the Fermi surface distortion can enhance the superconducting TcSC in the overdoped regime, which is explained with a perturbational picture for small distortions.
Salmhofer [Commun. Math. Phys. 194, 249 (1998)] recently developed a new renormalization-group method for interacting Fermi systems, where the complete flow from the bare action of a microscopic model to the effective low-energy action, as a function of a continuously decreasing infrared cutoff, is given by a differential flow equation which is local in the flow parameter. We apply this approach to the repulsive two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping amplitudes. The flow equation for the effective interaction is evaluated numerically on a one-loop level. The effective interactions diverge at a finite-energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions we extend Salmhofer's renormalization group for the calculation of susceptibilities. We compute the singlet superconducting susceptibilities for various pairing symmetries, and also charge- and spin-density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength, and band filling) we find commensurate and incommensurate antiferromagnetic instabilities or d-wave superconductivity as leading instability. We present the resulting phase diagram in the vicinity of half-filling, and also results for the density dependence of the critical energy scale.
The problem of weakly correlated electrons on a square lattice is formulated in terms of a one-loop renormalization group. Starting from the action for the entire Brillouin zone we reduce successively the cutoff \ensuremath{\Lambda} about the Fermi surface and follow the renormalization of the coupling U as a function of three energy-momenta. We calculate the intrinsic temperature scale Tco where the renormalization-group flow crosses over from the regime (\ensuremath{\Lambda}>Tco) where the electron-electron (e-e) and electron-hole (e-h) terms are equally important to the regime (\ensuremath{\Lambda}Tco) where only the e-e term plays a role. In the low-energy regime only the pairing interaction V is marginally relevant, containing contributions from all renormalization-group steps of the regime \ensuremath{\Lambda}>Tco. We identify the most attractive eigenvalue ${\ensuremath{\lambda}}_{\mathrm{min}}$ of ${\mathit{V}}_{\mathrm{\ensuremath{\Lambda}}={\mathit{T}}_{\mathit{co}}}$. At low filling, ${\ensuremath{\lambda}}_{\mathrm{min}}$ corresponds to the B2 representation (dxy symmetry), while near half filling the strongest attraction occurs in the B1 representation (${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$ symmetry). In the direction of the van Hove singularities, the order parameter shows peaks with increasing strength as one approaches half filling. We also give a possible interpretation of angle-resolved photoemission spectroscopy experiments trying to determine the symmetry of the order parameter in the high-Tc compound Bi2Sr2CaCu2O8. \textcopyright{} 1996 The American Physical Society.
High temperature superconductivity in copper oxides (cuprates) represents one of the most striking phenomena in condensed matter physics. These materials host superconducting states with the highest known critical temperatures, emerging upon doping antiferromagnetic insulator compounds. The antiferromagnetic (AF) insulating and superconducting (SC) phases defy standard descriptions in terms of band picture and conventional phonon-mediated superconductivity. In fact, these phenomena are genuine signatures of strong electronic correlations which hinder effective single-particle descriptions of the many-electron system. After 40 years since their discovery, it is widely accepted that the Hubbard model on the square lattice provides a minimal description of the low-energy electronic properties of cuprates. The model describes the competition between the electronic kinetic energy, modelled by nearest-neighbours hopping t, and the Coulomb repulsion, modelled by a purely local interaction potential U. At half-filling, this competition results in the so-called "Mott localisation" phenomenon, which distinguishes the metallic phase for U/t « 1 from an insulating phase for U/t » 1. The latter is characterised by the almost complete localisation of electrons due to the high energetic cost of hopping towards occupied sites. Interestingly, this basic competition is intimately related to AF ordering and, upon doping, to the formation of the SC state with characteristic d-wave symmetry. The fact that a purely repulsive interaction U is responsible for the formation of a SC state is remarkable, being superconductivity typically associated with an effective attractive interaction. In fact, a d-wave SC state can also be obtained, in general, by considering an attractive potential between electrons of neighbouring sites. Motivated by this observation, in this thesis, we study a modified version of the Hubbard model, dubbed as the "extended Hubbard model", in which we include a nearest-neighbours attractive potential -V. The general goal of this work is to explore the interplay between the purely local repulsion and the non-local attraction. This is not only motivated by the fact that both interactions are, in principle, responsible for d-wave superconductivity, but also by recent experimental observations which strongly suggest that the term -V needs to be included in order to accurately describe the magnetic spectra of some specific compounds. In this thesis, we study the extended Hubbard model using the Hartree-Fock method. Namely, we approximate the ground state by implementing a variational search in the subspace of the many-body Hilbert space of gaussian states. Despite its simplicity, the Hartree-Fock method provides a powerful framework to study competition among different phases, originating from different symmetry breaking mechanisms. In particular, we study the competition between three different phases: the normal phase, characterised by no symmetry breaking, the AF phase, characterised by a lowering of SU(2) symmetry, and the SC phase, characterised by the breaking of U(1) charge conservation. This thesis work led to a number of original results, which can be summarised by the general observation that the non-local attraction -V acts cooperatively with the local repulsion U for the stabilisation of the various phases hosted by the Hubbard model. For the normal phase, we find that the non-local attraction acts as a source of Mott localisation, leading to an effective reduction of the electronic bandwidth. In particular, for values of V larger than a critical value and close to half-filling, we find a strong Mott localisation characterised by the complete flattening of the electronic bands. Interestingly, this result shows an example of Mott localisation that can be described at the level of the Hartree-Fock approximation, whereas Mott localisation in the conventional Hubbard model usually requires more sophisticated approximation schemes. For the AF phase, we restrict our analysis to the half-filled case. We find that the pairing-induced Mott localisation leads to a significant enhancement of the magnetisation, primarily originating from the local repulsion U. Finally, we focus on the superconducting phase and derive self-consistency equations in both singlet and triplet Cooper pairing channels, showing how Cooper pairing can be mediated simultaneously by both the local repulsion and the non-local attraction. Therefore, we numerically investigate the singlet channel and find evidence of "symmetry mixing", namely, the formation of a superconducting gap that is neither symmetric nor anti-symmetric under rotations of π/2 in the plane. We show that symmetry mixing is connected to the emergence of a finite nematic order parameter, that signals the breaking of planar rotational symmetry. This nematic symmetry breaking results in the fact that the renormalisation of bands becomes anisotropic, producing a difference in electronic kinetics in the x and y directions. To summarise, the original results obtained in this thesis show that the non-local pairing provides a different route towards the formation of key strongly correlated phenomena, including Mott localisation and nematic superconductivity, and call for future investigation using more sophisticated methods such as, for example, the Dynamical Mean Field Theory.