Superconductivity in the repulsive Hubbard model

The two-dimensional repulsive Hubbard model has been investigated by a variety of methods, from small to large U. Superconductivity with d-wave symmetry is consistently found close to half filling. After a brief review of the various methods a variational many-electron state is discussed in more detail. This trial state is a natural extension of the Gutzwiller ansatz and provides a substantial improvement thereof.Comment: Proc. Int. Conference on Quantum Phenomena in Complex Matter, Erice, July 19-25, 2010, to appear in Journal of Superconductivity and Novel Magnetism, Springer Verla

Adiabatic continuity and broken symmetry in many-electron systems: a variational perspective

Variational wave functions are very useful for describing the panoply of ground states found in interacting many-electron systems. Some particular trial states are "adiabatically" linked to a reference state, from which they borrow the essential properties. A prominent example is the Gutzwiller ansatz, where one starts with the filled Fermi sea. A simple soluble example, the anisotropic XY chain, illustrates the adiabatic continuity of this class of wave functions. To describe symmetry breaking, one has to modify the reference state accordingly. Alternatively, a quantum phase transition can be described by a pair of variational wave functions, starting at weak and strong coupling, respectively

Superconductivity and antiferromagnetism in the two-dimensional Hubbard model

En dépit de son apparente simplicité et bien qu’il ait déjà été intensivement étudié dans des contextes très variés, le modèle de Hubbard bidimensionnel est encore très loin d’avoir livré tous ses mystères. En particulier lorsqu’il décrit une interaction moyenne à fortement répulsive, c’est-à-dire un régime mal adapté aux méthodes perturbatives, son état fondamental est encore méconnu. La découverte en 1986 d’une phase supraconductrice chez les oxydes de cuivre (cuprates), apparaissant à relativement haute température, a encore avivé l’intérêt pour ce modèle puisqu’il pourrait décrire les propriétés étonnantes de ces matériaux à structures planaires. Lorsque la densité électronique correspond à une bande de valence demi remplie, le modèle de Hubbard est en effet connu pour rendre parfaitement compte de la phase antiferromagnétique obtenue expérimentalement chez les cuprates et il est probable qu’il en soit de même pour la phase supraconductive observée au-dessous d’une température critique, lorsque la densité électronique est modérément réduite en modifiant la composition du matériau (dopage). Si tel est le cas, cela signifie que seules les interactions électronique sont à l’origine de la supraconductivité, contrairement aux supraconducteurs conventionnels où l’interaction entre les électrons et les vibrations du réseau (phonons) est impliquée. Malheureusement, il existe pour l’heure peu de résultats indiscutables venant étayer cette hypothèse et la question est encore largement débattue. Dans cette thèse, la méthode variationelle est mise à profit pour tenter d’apporter une réponse claire concernant la possibilité d’un état fondamental supraconducteur pour le modèle de Hubbard répulsif. Bien que la fiabilité de ses résultats soit conditionnée par le choix d’une fonction variationelle adéquate, cette méthode est tout spécialement adaptée au régime approprié pour les cuprates, c’est-à-dire à une interaction intermédiaire. L’optimisation d’une fonction variationelle élaborée permet d’approcher de très près l’état fondamental exact, ce qui n’était pas le cas des fonctions variationelles connues jusqu’ici. En fait, la considération de petits systèmes rend accessible une solution exacte, qui indique que l’erreur commise dans notre calcul est trop faible pour influencer qualitativement nos résultats. Ceux-ci montrent qu’une phase antiferromagnétique est favorable au demi remplissage et d’une phase supraconductrice émerge lorsque la densité électronique diminue, avec un paramètre d’ordre possédant une symétrie onde-d. Toutes les caractéristiques principales propres aux oxydes de cuivre sont donc retrouvées pour l’état fondamental variationel du modèle de Hubbard répulsif. Des similitudes remarquables sont aussi observées au niveau quantitatif lorsque les propriétés de cet état fondamental sont comparées aux données expérimentales obtenues pour les cuprates.Despite its apparent simplicity and even if it has been studied intensively in various contexts, the two-dimensional Hubbard model is not yet completely understood. In particular when it describes a system with an intermediate interaction, out of range for perturbative methods, its ground state is still not well known. The discovery in 1986 of a superconducting phase in the copper oxides (cuprates), which appears at rather high temperature, has further enhanced the interest for this model, as it may describe the amazing poperties of these layered materials. Indeed, when the electronic density corresponds to a half-filled band, the Hubbard model is known to nicely account for the antiferromagnetic phase obtained experimentally for the cuprates and it is possible, as first proposed by P. W. Anderson, that it describes equally well the superconducting phase observed below a critical temperature, when the electronic density is moderatly reduced by modifying the compound composition (doping). In this case, the superconductivity originates from purely electronic interactions, contrary to conventional superconductivity where the interaction between electrons and lattice vibrations (phonons) is involved. Unfortunately, very few stringent results are available in order to support this statement and this question is still largely debated. In this thesis, the variational method is used in order to scrutinize the possibility of a superconducting ground state for the repulsive Hubbard model. Although its reliability is based on the adequate choice of the variational wave function, this method is especially suited for treating intermediate interactions, which is the appropriate regime of the cuprates. The optimization of a refined wave function allows us to obtain a variational ground state which is much closer to the exact ground state than those obtained so far using less elaborate wave functions. Actually, the exact study of small systems indicates that the error of our calculation is too small to allow for a qualitatively different behavior. Our results show that an antiferromagnetic phase is dominant at half-filling, while a superconducting phase with a d-wave symmetry of the order parameter emerges at moderate doping. The key features of the copper oxides are therefore found in the variational ground state of the repulsive Hubbard model. Some amazing similarities are also observed at the quantitative level when the properties of this ground state are compared to the experimental data obtained for the cuprates

Superconductivity and antiferromagnetism in the two-dimensional Hubbard model: a variational study

A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength (U=8t) and for moderate sizes (from 6×6 to 10×10). Our ansatz is superior to other widely used variational wave functions. The results for order parameters and correlation functions provide insight into the antiferromagnetic state at half-filling as well as strong evidence for a superconducting phase away from half-filling

Order parameter, correlation functions, and fidelity susceptibility for the BCS model in the thermodynamic limit

The exact ground state of the reduced BCS Hamiltonian is investigated numerically for large system sizes and compared with the BCS ansatz. A “canonical” order parameter is found to be equal to the largest eigenvalue of Yang's reduced density matrix in the thermodynamic limit. Moreover, the limiting values of the exact analysis agree with those obtained for the BCS ground state. Exact results for the ground-state energy, level occupations, and a pseudospin-pseudospin correlation function are also found to converge to the BCS values already for relatively small system sizes. However, discrepancies persist for a pair-pair correlation function, for interlevel correlations of occupancies and for the fidelity susceptibility, even for large system sizes where these quantities have visibly converged to well-defined limits. Our results indicate that there exist nonperturbative corrections to the BCS predictions in the thermodynamic limit

Superconductivity from repulsion: Variational results for the two-dimensional Hubbard model in the limit of weak interaction

The two-dimensional Hubbard model is studied for small values of the interaction strength (U of the order of the hopping amplitude t), using a variational ansatz well suited for this regime. The wave function, a refined Gutzwiller ansatz, has a BCS mean-field state with d-wave symmetry as its reference state. Superconducting order is found for densities n <1 (but not for n=1). This resolves a discrepancy between results obtained with the functional renormalization group, which do predict superconducting order for small values of U, and numerical simulations, which did not find superconductivity for U<4t. Both the gap parameter and the order parameter have a dome-like shape as a function of n with a maximum for n about 0.8. Expectation values for the energy, the particle number and the superconducting order parameter are calculated using a linked-cluster expansion up to second order in U. In this way large systems (millions of sites) can be readily treated and well converged results are obtained. A big size is indeed required to see that the gap becomes very small at half filling and probably tends to zero in the thermodynamic limit, whereas away from half filling a finite asymptotic limit is reached. For a lattice of a given size the order parameter becomes finite only above a minimal coupling strength U_c. This threshold value decreases steadily with increasing system size, which indicates that superconductivity exists for arbitrarily small U for an infinite system. For moderately large systems the size dependence is quite irregular, due to variations in level spacings at the Fermi energy. The fluctuations die out if the gap parameter spans several level spacings.Comment: 20 pages, 13 figure

Fermion quartets on the square lattice

We study a microscopic model for four spinless fermions on the square lattice which exhibits a quartet bound state in the strong coupling regime. The four-particle quantum states are analyzed using symmetry arguments and by introducing a zoo of relevant lattice animals. These considerations, as well as variational and exact diagonalization calculations demonstrate the existence of a narrow quartet band at small hopping and a first order transition to delocalized fermions at a critical hopping parameter, in qualitative contrast to, e. g., the BCS-BEC crossover in the attractive Hubbard model. In the case of pure attraction, an intermediate phase is found, in which a more extended and presumably more mobile hybrid quartet dominates the ground state. We comment on the relevance of the spin degree of freedom and on the reasons why electron quartetting is rarely observed in real materials

Electron doping and superconductivity in the two-dimensional Hubbard model

An elaborate variational wave function is used for studying superconductivity in the (repulsive) two-dimensional Hubbard model, including both nearest- and next-nearest-neighbor hoppings. A marked asymmetry is found between the “localized” hole-doped region and the more itinerant electron-doped region. Superconductivity with d-wave symmetry turns out to be restricted to densities where the Fermi surface crosses the magnetic zone boundary. A concomitant peak in the magnetic structure factor at (π,π) clearly points to a magnetic mechanism