The Hubbard model is studied in the vicinity of the Mott transition. Local and quantal spin fluctuations are completely included through the mapping to the Anderson model; a local Kondo temperature ${T}_{K}$ is defined as their characteristic temperature or energy scale. Antiferromagnetic spin fluctuations are included perturbatively in terms of $\frac{1}{d}$ and to leading order in $\frac{{k}_{B}{T}_{K}}{U}$ with $d$ being the spatial dimensionality, ${k}_{B}$ the Boltzmann constant, and $U$ the intrasite repulsion. The two different kinds of spin fluctuations are responsible for the formation of heavy quasiparticles. Two intersite exchange interactions are responsible for both the development of the antiferromagnetic spin fluctuations and $d\ensuremath{\gamma}$-wave Cooper pairing between the heavy quasiparticles: the superexchange interaction and an exchange interaction due to the virtual exchange of spin excitations within the heavy quasiparticle band. An experimental specific-heat coefficient of about 14 mJ/${\mathrm{K}}^{2}$ Cu${\mathrm{O}}_{2}$ mol and a Wilson ratio of about 0.4 imply that the interplay between the local and the antiferromagnetic spin fluctuations plays a crucial role in both the normal and the superconducting states of $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ with critical temperatures of ${T}_{c}\ensuremath{\simeq}90$ K. A pairing interaction deduced from the normal-state properties is strong enough to give critical temperatures as high as ${T}_{c0}\ensuremath{\simeq}200$ K in the absence of any pair breaking; there is experimental evidence that the pair-breaking effect of the antiferromagnetic spin fluctuations reduces ${T}_{c0}\ensuremath{\simeq}200$ K down to ${T}_{c0}\ensuremath{\simeq}90$ K.
The Hubbard model is studied in the vicinity of the Mott transition. Local and quantal spin fluctuations are completely included through the mapping to the Anderson model; a local Kondo temperature TK is defined as their characteristic temperature or energy scale. Antiferromagnetic spin fluctuations are included perturbatively in terms of d1 and to leading order in UkBTK with d being the spatial dimensionality, kB the Boltzmann constant, and U the intrasite repulsion. The two different kinds of spin fluctuations are responsible for the formation of heavy quasiparticles. Two intersite exchange interactions are responsible for both the development of the antiferromagnetic spin fluctuations and d\ensuremathγ-wave Cooper pairing between the heavy quasiparticles: the superexchange interaction and an exchange interaction due to the virtual exchange of spin excitations within the heavy quasiparticle band. An experimental specific-heat coefficient of about 14 mJ/K2 CuO2 mol and a Wilson ratio of about 0.4 imply that the interplay between the local and the antiferromagnetic spin fluctuations plays a crucial role in both the normal and the superconducting states of YBa2Cu3O7\ensuremath−\ensuremathδ with critical temperatures of Tc\ensuremath≃90 K. A pairing interaction deduced from the normal-state properties is strong enough to give critical temperatures as high as Tc0\ensuremath≃200 K in the absence of any pair breaking; there is experimental evidence that the pair-breaking effect of the antiferromagnetic spin fluctuations reduces Tc0\ensuremath≃200 K down to Tc0\ensuremath≃90 K.
The Hubbard model is studied for the Mott-transition region in the limit of high dimension d →+∞. Leading-order effects with respect to 1/ d are local spin fluctuations, all of which are fully included in the single-site approximation (SSA). Because the SSA is reduced to solving the Anderson model, it is argued that a normal heavy-electron liquid forms through the Kondo effect. The Fermi-liquid relations are proven within the SSA. The other leading-order effects are mean fields of magnetism caused by exchange interaction. Therefore it depends on the competition between the Kondo effect and the exchange interaction which the ground state is, magnetic or paramagnetic. However, any other effect is a higher-order effect. The 1/ d expansion from the heavy-electron liquid in the SSA is one of the most useful methods in examining lower-temperature phases in the real dimension.
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.
A microscopic theory of superconductivity in the extended Hubbard model which takes into account the intersite Coulomb repulsion and electron-phonon interaction is developed in the limit of strong correlations. The Dyson equation for normal and pair Green functions expressed in terms of the Hubbard operators is derived. The self-energy is obtained in the noncrossing approximation. In the normal state, antiferromagnetic short-range correlations result in the electronic spectrum with a narrow bandwidth. We calculate superconducting Tc by taking into account the pairing mediated by charge and spin fluctuations and phonons. We found the d-wave pairing with high-Tc mediated by spin fluctuations induced by the strong kinematic interaction for the Hubbard operators. Contributions to the d-wave pairing coming from the intersite Coulomb repulsion and phonons turned out to be small.
We examine the two existing slave-boson methods for the finite-U Hubbard model by Kotliar and Ruckenstein (KR) and by Newns et al. and Zou and Anderson (NZA) in the light of the Fermi-liquid quasiparticle picture. It is shown that to treat doubly occupied states the KR slave-boson scheme is more consistent with this picture and that the well-known erroneous gauge-symmetry breaking encountered in the mean-field treatment of the NZA double-occupancy boson is due to the presence of the hidden double-fermion operators. Then, the effective intersite antiferromagnetic interaction between quasiparticles is derived in the KR slave-boson approach by taking into account empty and double-occupancy slave-boson fluctuations to O(1/N2) in the 1/N expansion of the functional-integral formalism. The obtained effective exchange coupling constant Jeff(\ensuremath{\delta})=J(2p02-\ensuremath{\delta}/2) with J\ensuremath{\equiv}tij2/U is strongly dependent upon doping \ensuremath{\delta} and is scaled with the reduced Coulomb repulsion u\ensuremath{\equiv}U/Uc in the two-dimensional (2D) and 3D Hubbard models, where Uc is the critical value for the metal-to-insulator transition. The roles of this intersite correlation and the spin-dependent band-renormalization correlation included in the mean-field theory for the paramagnetic-to-antiferromagnetic instability are compared in the two Hubbard models. The intersite contribution is more significant in the 2D case than in the 3D case, but is generally small compared with the band effect except for small dopings \ensuremath{\delta} and smaller U values. The maximum doping ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$ for the paramagnetic-to-ferromagnetic instability in the strong-coupling limit (U\ensuremath{\rightarrow}\ensuremath{\infty}) of the 2D and 3D Hubbard models is found to be ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$\ensuremath{\sim}0.28 and ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$\ensuremath{\sim}0.38, respectively.
The principal purpose of this topical review is to single out some of the universal features of high-temperature (high-Tc) and related strongly-correlated systems, which can be compared with experiment in a quantitative manner. The description starts with the concept of exchange-interaction-mediated (real-space) pairing, combined with strong correlations among narrow band electrons, for which the reference state is that of the Mott–Hubbard insulator. The physical discussion of concrete properties relies on variational approach, starting from generalized renormalized mean-field theory (RMFT) in the form of statistically-consistent Gutzwiller approximation (SGA), and its subsequent generalization, i.e., a systematic Diagrammatic Expansion of the Variational (Gutzwiller-type) Wave Function (DE-GWF). The solution leads to the two energy scales, one involving unconventional quasiparticles close to the Fermi energy, and the other reflecting the fully correlated state, involving electrons deeper below the Fermi surface. Those two regimes are separated by a kink in the dispersion relation, which is observed in photoemission. As a result, one obtains both the doping dependent properties in the correlated state, and effective renormalized quasiparticles. The reviewed ground-state characteristics for high-Tc systems encompass high-Tc superconductivity, nematicity, charge- (and pair-) density-wave effects, as well as non-BCS kinetic energy gain in the paired state. The calculated dynamic properties are: the universal Fermi velocity, the Fermi wave-vector, the effective mass enhancement, the pseudogap; and the d-wave gap magnitude. We discuss that, within the variational approach, the minimal realistic model is represented by the so-called t-J-U Hamiltonian. Inadequacy of the t-J and Hubbard models, particularly of their RMFT versions, is discussed explicitly. For heavy fermion systems, modeled by the Anderson lattice model and with the DE-GWF approach, we discuss the phase diagram, encompassing superconducting, Kondo insulating, ferro- and anti-ferromagnetic states. The superconducting state is then a two-d-wave gap system. If the orbital degeneracy of f-electrons is included, the coexistent ferromagnetic- (spin–triplet) superconducting phases appear and match those observed for UGe2 in a semiquantitative manner. Finally, in the second part, we generalize our approach to the collective spin and charge fluctuations in high-Tc systems, starting from variational approach, defining the saddle-point state, and combining it with 1/N expansion. The present scheme differs essentially from that starting from the saddle-point Hartree–Fock approximation, and incorporating the fluctuations in the random phase approximation (RPA). The spectrum of collective spin and charge excitations is determined for the Hubbard and t-J-U models, and subsequently compared quantitatively with recent experiments. The Appendices provide formal details to make this review self-contained.
The Hubbard model in the strong-coupling regime is mainly studied by Kondo-lattice theory or 1/d expansion theory, with d the spatial dimensionality. In two dimensions and higher, the ground state within the Hilbert subspace with no order parameter is a normal Fermi liquid except for n=1 and U/W=+infinity, with n the electron density per unit cell, U the on-site repulsion, and W the bandwidth; the cooperation between the Kondo effect, which favors a local singlet on each unit cell, and a resonating-valence-bond effect, which favors a local singlet on each pair of nearest-neighbor unit cells, stabilizes the Fermi liquid, whose ground state is a singlet as a whole, in the strong-coupling regime. In the whole Hilbert space with no restriction, the normal Fermi liquid is unstable at least against a magnetic or superconducting state. This analysis confirms an early Fermi-liquid theory of high-temperature superconductivity, F. J. Ohkawa, Jpn. J. Appl. Phys. 26, L652 (1987). The ground state for n=1 and U/W=+infinity is a Mott insulator. Actual metal-insulator transitions cannot be explained within the Hubbard model. In order to explain them, the electron-phonon interaction, multi-band or multi-orbital effects, and effects of disorder should be considered beyond the Hubbard model. The crossover between local-moment magnetism and itinerant-electron magnetism corresponds to that between a localized spin and a normal Fermi liquid in the Kondo effect and it is simply a Mott metal-insulator crossover.