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.
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.
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 $U\ensuremath{\lesssim}4t$. Both the gap parameter and the order parameter have a domelike shape as a function of
n with a maximum for $n\ensuremath{\approx}0.8$. 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
Uc. 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.
The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function
Ap(\ensuremath{\omega}) is momentum dependent. Using
Ap(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.
The deformation of the Fermi surface induced by Coulomb interactions is investigated in the $t\ensuremath{-}{t}^{\ensuremath{'}}$-Hubbard model. The interplay of the local U and extended V interactions is analyzed. It is found that the interactions V enhance small anisotropies producing deformations of the Fermi surface that break the point group symmetry of the square lattice at the Van Hove filling. This Pomeranchuk instability competes with magnetic instabilities and is suppressed at a critical value
U(V). The interaction V effectively renormalizes the ${t}^{\ensuremath{'}}$ parameter to smaller values, which favors nesting. It also induces changes in the topology of the Fermi surface, which can go from holelike to electronlike. This may explain the results of recent angle-resolved photoemission spectroscopy experiments.
While it is over 30 years since the high-T c superconductor was discovered, the cuprates have attracted great interests in many scientific fields.Understanding high-T c is one of the most challenging issues in the condensed matter theory.The mechanism of the superconductivity in these materials is still not fully understood, and continues to be discussed intensely.Specifically, the two-dimensional repulsive Hubbard model, which is a representative model for the cuprates, still has many questions, and many analytical and numerical methods have been proposed for studying the superconductivity, magnetism and other instabilities.In the present thesis, we propose to combine the non-perturbative dynamical mean field theory (DMFT) with the perturbative diagrammatic expansion techniques to treat the local correlation effects ("Mott physics") and momentum dependent pairing interaction (that produces the d-wave pairing superconductivity) to explore the superconductivity and various instabilities in the Hubbard model.First, we formulate the combination of the fluctuation exchange (FLEX) approximation and DMFT in a self-consistent manner in terms of the Luttinger-Ward functional for superconducting phases.The resultant phase diagram exhibits a clear T c dome structure against the band filling as observed experimentally, while in the ordinary FLEX, T c almost monotonically increases toward the half-filling.We trace the origin of the T c -dome back to a combined effect of the filling dependence of the local vertex correction and the filling dependence of the pairing interaction.As another effect inherent in the strongly correlated systems, we study a phase transition for the Pomeranchuk instability, where the four-fold rotational symmetry in the electronic structure on the tetragonal lattice is spontaneously broken into two-fold due to an interaction effect.We find that the Pomeranchuk instability is remarkably more sensitive to the Fermi surface warping than the superconducting T c dome.An interesting question is the effect of the distorted Fermi surface due to the Pomeranchuk instability on the superconductivity, and we have revealed that the gap function is slightly changed from ordinary d-wave pairing to (d+s)-wave.Furthermore, we find that the Fermi surface distortion can enhance the superconducting T c in frustrated systems.i Finally, we consider the extension of the DMFT vertex to study the vertex correction effect on nonlocal self-energy and the pairing interaction.For this purpose, we employ the DΓA (dynamical vertex approximation) formalism, which is extended here to accommodate superconducting phases.For the Hubbard model we obtain the T c -dome due to a combined effect as in the FLEX+DMFT case, but the transition temperature is reduced by the vertex correction.We also find that the pairing interaction is strongly screened by the vertex correction effect.ii