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arXiv:0901.1760 · Physical Review B

Dynamical mean-field theory and numerical renormalization group study of superconductivity in the attractive Hubbard model

Johannes Bauer, Alex C. Hewson, N. Dupuis

Published Jun 16, 2009• 59 citations
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Abstract

We present a study of the attractive Hubbard model based on the dynamical mean-field theory (DMFT) combined with the numerical renormalization group (NRG). For this study, the NRG method is extended to deal with self-consistent solutions of effective impurity models with superconducting symmetry breaking. We give details of this extension and validate our calculations with DMFT results with antiferromagnetic ordering. We also present results for static and integrated quantities for different filling factors in the crossover from weak to strong-coupling superfluidity. We study the evolution of the single-particle spectra throughout the crossover regime. Although the DMFT does not include the interaction of the fermions with the Goldstone mode, we find strong deviations from the mean-field theory in the intermediate and strong-coupling regimes. In particular, we show that low-energy charge fluctuations induce a transfer of spectral weight from the Bogoliubov quasiparticles to a higher-energy incoherent hump.

DOI
10.1103/physrevb.79.214518

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We present a study of the attractive Hubbard model based on the dynamical mean-field theory (DMFT) combined with the numerical renormalization group (NRG). For this study, the NRG method is extended to deal with self-consistent solutions of effective impurity models with superconducting symmetry breaking. We give details of this extension and validate our calculations with DMFT results with antiferromagnetic ordering. We also present results for static and integrated quantities for different filling factors in the crossover from weak to strong-coupling superfluidity. We study the evolution of the single-particle spectra throughout the crossover regime. Although the DMFT does not include the interaction of the fermions with the Goldstone mode, we find strong deviations from the mean-field theory in the intermediate and strong-coupling regimes. In particular, we show that low-energy charge fluctuations induce a transfer of spectral weight from the Bogoliubov quasiparticles to a higher-energy incoherent hump.
We study dynamic ground-state properties in the crossover from weak-(BCS) to strong-coupling (BEC) superfluidity. Our approach is based on the attractive Hubbard model which is analyzed by the dynamical mean-field theory (DMFT) combined with the numerical renormalization group (NRG). We present an extension of the NRG method for effective impurity models to self-consistent calculations with superconducting symmetry breaking. In the one-particle spectra we show quantitatively how the Bogoliubov quasiparticles at weak coupling become suppressed at intermediate coupling. We also present results for the spin and charge gap. The extension of the NRG method to self-consistent superconducting solutions opens the possibility to study a range of other important applications.
We study the normal (nonsuperconducting) phase of the attractive Hubbard model within the dynamical mean field theory (DMFT) using the numerical renormalization group (NRG) as an impurity solver. A wide range of attractive potentials U is considered, from the weak-coupling limit, where superconducting instability is well described by the BCS approximation, to the strong-coupling region, where the superconducting transition is described by Bose condensation of compact Cooper pairs, which are formed at temperatures much exceeding the superconducting transition temperature. We calculate the density of states, the spectral density, and the optical conductivity in the normal phase for this wide range of U, including the disorder effects. We also present the results on superconducting instability of the normal state dependence on the attraction strength U and the degree of disorder. The disorder influence on the critical temperature T c is rather weak, suggesting in fact the validity of Anderson’s theorem, with the account of the general widening of the conduction band due to disorder.
We study the normal (non-superconducting) phase of attractive Hubbard model within dynamical mean field theory (DMFT) using numerical renormalization group (NRG) as impurity solver. Wide range of attractive potentials UUU is considered, from the weak-coupling limit, where superconducting instability is well described by BCS approximation, up to the strong-coupling region, where superconducting transition is described by Bose-condensation of compact Cooper pairs, which are formed at temperatures much exceeding superconducting transition temperature. We calculate density of states, spectral density and optical conductivity in the normal phase for this wide range of UUU, including the disorder effects. Also we present the results on superconducting instability of the normal state dependence on the attraction strength UUU and the degree of disorder. Disorder influence on the critical temperature TcT_cTc​ is rather weak, suggesting in fact the validity of Anderson theorem, with the account of the general widening of the conduction band due to disorder.
We calculate the renormalized parameters for the quasiparticles and their interactions for the Hubbard model in the paramagnetic phase as deduced from the low-energy Fermi-liquid fixed point using the results of a numerical renormalization-group calculation (NRG) and dynamical mean-field theory (DMFT). Even in the low-density limit there is significant renormalization of the local quasiparticle interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}$, in agreement with estimates based on the two-particle scattering theory of J. Kanamori [Prog. Theor. Phys. 30, 275 (1963)]. On the approach to the Mott transition we find a finite ratio for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{U}/\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}$, where $2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}$ is the renormalized bandwidth, which is independent of whether the transition is approached by increasing the on-site interaction UUU or on increasing the density to half filling. The leading ${\ensuremath{\omega}}^{2}$ term in the self-energy and the local dynamical spin and charge susceptibilities are calculated within the renormalized perturbation theory (RPT) and compared with the results calculated directly from the NRG-DMFT. We also suggest, more generally from the DMFT, how an approximate expression for the $\mathbf{q},\ensuremath{\omega}$ spin susceptibility $\ensuremath{\chi}(\mathbf{q},\ensuremath{\omega})$ can be derived from repeated quasiparticle scattering with a local renormalized scattering vertex.
We study the normal (non-superconducting) phase of attractive Hubbard model within dynamical mean field theory (DMFT) using numerical renormalization group (NRG) as impurity solver. Wide range of attractive potentials UUU is considered, from the weak-coupling limit, where superconducting instability is well described by BCS approximation, up to the strong-coupling region, where superconducting transition is described by Bose-condensation of compact Cooper pairs, which are formed at temperatures much exceeding superconducting transition temperature. We calculate density of states, spectral density and optical conductivity in the normal phase for this wide range of UUU, including the disorder effects. Also we present the results on superconducting instability of the normal state dependence on the attraction strength UUU and the degree of disorder. Disorder influence on the critical temperature TcT_cTc​ is rather weak, suggesting in fact the validity of Anderson theorem, with the account of the general widening of the conduction band due to disorder.
We study the attractive Hubbard model within the dynamical mean-field theory, to elucidate how the pseudogap and superconductivity at strong attractive interaction are related to those found in the repulsive Hubbard model, and thereby to bridge cold fermions and cuprate high-temperature superconductors from a microscopic point of view. We propose that a unified understanding is obtained by investigating single-particle excitation dynamics, in which emergent and hidden fermions coupled to quasiparticles consistently account for the numerical results in both attractive and repulsive cases. In the attractive case, the quasiparticle dynamics is observable by virtually breaking a tightly bound pair, where we find two qualitatively different regions crossing over each other within the strong-coupling superconductivity phase. Among them, the region close to the critical temperature shares characteristic dynamics with the repulsive interaction case, where the normal and anomalous parts of the self-energy show strong low-energy peaks while they are hidden in the quasiparticle spectral weight. These prominent self-energy peaks are understood by the coupling of the quasiparticle to the hidden fermionic excitation, emergent from a strong-coupling effect. The pseudogap above the critical temperature is also accounted for by the same hidden fermion.