We present a. study of the superconducting properties of models containing the Hubbard rnpulsion term. This strnng on-site repulsion is considered as a key ingTedient of the high tempera.turn superconductivity. Though the fact that for the normal, lovv temperature superconductors, rnpulsion destroys superconducting order, it is argued in the present thesis, that for the pairing of the d-·wave symmetry in the strongly correlated electronic systems its effect is to enhance and may be to cause superconductivity. Various methods such as Variational Monte Carlo, Gutzwiller Approximation, Time Dependent Hartree-Fock and Fixed Node Approximation have been used to investigate t - U - W model, t - U - J - V and pure Hubbard models. In this thesis, by considering correlations contribution to the BCS condensation energy due to the Hubbard U it is shown that the latter lowers the total energy of a d-wave superconductor in the weak coupling limit, thus enforcing the stability of such superconductor. This effect appears to be mainly due to the enhancement of the spin fluctuations near the nesting vector Q = ( π, π). It is then studied the crossover from weak to strong coupling regimes in t - U - J - V model by increasing U. Remarkably in this model an order of magnitude growth of the superconducting order parameter is found and explained as being due to the Hubbard repulsion. vVe fi.nd also, that the pairings, originally induced by spin or charge fluctuations upon increase of U are differently renormalized, being the former enhanced, while the latter suppressed. In the fi.nal part of the thesis the superconductivity in the pure Hubbard model is carefully studied by means of Variational Monte Carlo and related numerical methods aimed to improve variational results. VVe observe the onset of strong coupling superconductivity at U/t ~ 7 within the Fixed Node Approximation in the systems of large size and compare our results for small clusters with those of Lanczos diagonalization. We show that Variational Monte Carlo, though overestimating the quasiparticle weight (ZvMc > Zexact) succeeds in reproducing the correct pairing between quasiparticles.
We present a. study of the superconducting properties of models containing the Hubbard rnpulsion term. This strnng on-site repulsion is considered as a key ingTedient of the high tempera.turn superconductivity. Though the fact that for the normal, lovv temperature superconductors, rnpulsion destroys superconducting order, it is argued in the present thesis, that for the pairing of the d-·wave symmetry in the strongly correlated electronic systems its effect is to enhance and may be to cause superconductivity. Various methods such as Variational Monte Carlo, Gutzwiller Approximation, Time Dependent Hartree-Fock and Fixed Node Approximation have been used to investigate t - U - W model, t - U - J - V and pure Hubbard models. In this thesis, by considering correlations contribution to the BCS condensation energy due to the Hubbard U it is shown that the latter lowers the total energy of a d-wave superconductor in the weak coupling limit, thus enforcing the stability of such superconductor. This effect appears to be mainly due to the enhancement of the spin fluctuations near the nesting vector Q = ( π, π). It is then studied the crossover from weak to strong coupling regimes in t - U - J - V model by increasing U. Remarkably in this model an order of magnitude growth of the superconducting order parameter is found and explained as being due to the Hubbard repulsion. vVe fi.nd also, that the pairings, originally induced by spin or charge fluctuations upon increase of U are differently renormalized, being the former enhanced, while the latter suppressed. In the fi.nal part of the thesis the superconductivity in the pure Hubbard model is carefully studied by means of Variational Monte Carlo and related numerical methods aimed to improve variational results. VVe observe the onset of strong coupling superconductivity at U/t ~ 7 within the Fixed Node Approximation in the systems of large size and compare our results for small clusters with those of Lanczos diagonalization. We show that Variational Monte Carlo, though overestimating the quasiparticle weight (ZvMc > Zexact) succeeds in reproducing the correct pairing between quasiparticles.
We present recent theoretical results on superconductivity in correlated-electron systems, especially in the two-dimensional Hubbard model and the three-band d-p model. The mechanism of superconductivity in high-temperature superconductors has been extensively studied on the basis of various electronic models and also electron-phonon models. In this study, we investigate the properties of superconductivity in correlated-electron systems by using numerical methods such as the variational Monte Carlo method and the quantum Monte Carlomethod. The Hubbard model is one of basic models for strongly correlated electron systems, and is regarded as the model of cuprate high temperature superconductors. The d-p model is more realistic model for cuprates. The superconducting condensation energy obtained by adopting the Gutzwiller ansatz is in reasonable agreement with the condensation energy estimated for YBa2Cu3O7. We show the phase diagram of the ground state using this method. We have further investigated the stability of striped and checkerboard states in the under-doped region. Holes doped in a half-filled square lattice lead to an incommensurate spin and charge density wave. The relationship of the hole density x and incommensurability δ, δ~x, is satisfied in the lower doping region, as indicated by the variationalMonte Carlocalculations for the two-dimensional Hubbard model. A checkerboard-like charge-density modulation with a roughly period has also been observed by scanning tunneling microscopy experiments in Bi2212 and Na-CCOC compounds. We have performed a variational Monte Carlo simulation on a two-dimensional t-t′-t″- U Hubbard model with a Bi-2212 type band structure and found that the period checkerboard spin modulation, that is characterized by multi Q vectors, is indeed stabilized. We have further performed an investigation by using a quantumMonte Carlomethod, which is a numerical method that can be used to simulate the behavior of correlated electron systems. We present a new algorithm of the quantum Monte Carlo diagonalization that is a method for the evaluation of expectation value without the negative sign problem. We compute pair correlation functions and show that pair correlation is indeed enhanced with hole doping.
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.
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.
In Chapter 1, we introduce the physics of the HTSC, starting with an historical overview of the problem and describing some experimental results that characterize these materials. Subsequently, we introduce the t−J model, which allows a microscopic description of the HTSC and we introduce the Resonating Valence Bond (RVB) wave function. In Chapter 2, we will describe the numerical techniques used for obtaining the results of our thesis. We start from the Lanczos method, that enable us to obtain exact results for small cluster size and then we enter in the topic of the quantum Monte Carlo technique. We describe the variational Monte Carlo method, the optimization algorithm and we will introduce the Green’s function Monte Carlo and fixed-node approximation, that improve the variational results. In Chapter 3, we will introduce our new variational wave function which generalizes the RVB state we show our results on the charge fluctuations (phase separation problem) for the two-dimensional t−J model. The main results of this chapter has been published in Physical Review B [7]. In Chapter 4, we will study the magnetic and superconducting properties of the two-dimensional t−J and t−t′−J model, trying to understand the role of the next-nearest-neighbor hopping term on the magnetic and superconducting phases. We will show a phase diagram of the magnetic and superconducting correlations, which qualitatively reproduce the actual phase diagram of HTSC and gives some indication on the origin of the electronic pairing. The main results of this chapter were submitted to Physical Review B [8].
Many theoretical approaches find d-wave superconductivity in the prototypical one-band Hubbard model for high-temperature superconductors. At strong coupling ($U\ensuremath{\ge}W$, where U is the on-site repulsion and W=8t the bandwidth) pairing is controlled by the exchange energy J=4t2/U. One may then surmise, ignoring retardation effects, that near-neighbor Coulomb repulsion V will destroy superconductivity when it becomes larger than J, a condition that is easily satisfied in cuprates, for example. Using cellular dynamical mean-field theory with an exact diagonalization solver for the extended Hubbard model, we show that pairing at strong coupling is preserved, even when $V\ensuremath{\gg}J$, as long as $V\ensuremath{\lesssim}U/2$. While at weak coupling V always reduces the spin fluctuations and hence d-wave pairing, at strong coupling, in the underdoped regime, the increase of $J=4{t}^{2}/(U\ensuremath{-}V)$ caused by V increases binding at low frequency while the pair-breaking effect of V is pushed to high frequency. These two effects compensate in the underdoped regime, in the presence of a pseudogap. While the pseudogap competes with superconductivity, the proximity to the Mott transition that leads to the pseudogap, and retardation effects, protect d-wave superconductivity from V.
We study superconductivity in the extended Hubbard model with a strong on-site repulsion U and weak attractive density-density interaction Vij. The electron correlation effects, beyond the weak-coupling regime of U, are included approximately by employing Hubbard decoupling approximations. The formulas for superconducting transition temperature Tc of the present approach are compared to those obtained with the strong-coupling expansion and slave-boson techniques. The methods share some common features. We discuss behavior of Tc for superconducting phases with s- and d-wave symmetry of the superconducting order parameter. In contrast to the mean-field treatment of U, we find that the strong on-site repulsion does not destroy superconductivity induced by ${\mathit{V}}_{\mathit{ij}}$0 even when U\ensuremath{\gg}|${\mathit{V}}_{\mathit{ij}}$|. \textcopyright{} 1996 The American Physical Society.