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 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.
The BCS-BEC crossover is studied in a systematic way in the broken-symmetry phase between zero temperature and the critical temperature. This study bridges two regimes where quantum and thermal fluctuations are, respectively, important. The theory is implemented on physical grounds, by adopting a fermionic self-energy in the broken-symmetry phase that represents fermions coupled to superconducting fluctuations in weak coupling and to bosons described by the Bogoliubov theory in strong coupling. This extension of the theory beyond mean field proves important at finite temperature, to connect with the results in the normal phase. The order parameter, the chemical potential, and the single-particle spectral function are calculated numerically for a wide range of coupling and temperature. This enables us to assess the quantitative importance of superconducting fluctuations in the broken-symmetry phase over the whole BCS-BEC crossover. Our results are relevant to the possible realizations of this crossover with high-temperature cuprate superconductors and with ultracold fermionic atoms in a trap.
We use the variational cluster approximation to study the superconducting ground state in the two-dimensional attractive Hubbard model, putting particular emphasis on the significance of quantum fluctuations of the system. We first show that the order parameter is suppressed in comparison with that obtained using the mean-field theory owing to the effects of spatial fluctuations in two-dimensional systems. We then show that the calculated Bogoliubov quasiparticle spectra and condensation amplitude clearly exhibit the character of Cooper pairs in momentum space and that the pair coherence length
ξ evaluated from the condensation amplitude demonstrates a smooth crossover in real space from a weakly paired BCS state (
ξ≫a) to a BEC state of tightly bound pairs (
ξ≪a), where
a is the lattice constant. The calculated kinetic and potential energies in the superconducting and normal ground states indicate that the superconducting state in the weak-coupling region is driven by the gain in potential energy, while that in the strongcoupling region is driven by the gain in kinetic energy.
We investigate by means of Dynamical mean-field theory the crossover from BCS superconductivity to Bose-Einstein (BE) condensation of preformed pairs which occurs in the attractive Hubbard model by increasing the attraction strength. Fixing the density to a generic value
n=0.75, we follow the evolution of the two energy scales underlying the superconducting phenomenon, the gap ${\ensuremath{\Delta}}_{0}$, and the superfluid stiffness
DS, which controls the phase coherence. The BCS--BE crossover is clearly mirrored in a change in the hierarchy of these two scales, the smallest of the two controlling the critical temperature. In the whole intermediate-to-strong-coupling region
Tc scales with
DS, while
TC is proportional to ${\ensuremath{\Delta}}_{0}$ only in the BCS regime. This evolution as a function of the interaction qualitatively resembles what happens in the cuprates when the doping is decreased towards the Mott insulator. This continuous change reflects also in the energetic balance at the superconducting transition. While, as it is well known, superconductivity is stabilized by a potential energy gain in the BCS regime, the strong-coupling superconductivity is made stable by a reduction of kinetic energy. Interestingly the intermediate-coupling region, where the maximum
Tc is achieved, behaves similarly to the strong-coupling regime, and its gain in kinetic energy is the largest as a function of the coupling. Since the integral of the optical conductivity is proportional to the kinetic energy, the above finding implies that the attractive Hubbard model can account qualitatively for the anomalous behavior of optical spectra around
Tc, where an increase of spectral weight is observed in under and optimally doped cuprates, while the overdoped samples have a more standard behavior. This qualitative agreement is lost in the normal phase, specifically at strong coupling, calling for the inclusion of strong correlation effects in the theoretical description.
The BCS-BEC crossover within the two-dimensional attractive Hubbard model is studied by using the Cellular Dynamical Mean-Field Theory, both in the normal and superconducting ground states. Short-range spatial correlations incorporated in this theory remove the normal-state quasiparticle peak and the first-order transition found in the Dynamical Mean-Field Theory, rendering the normal state crossover smooth. For
U smaller than the bandwidth, pairing is driven by the potential energy, while in the opposite case it is driven by the kinetic energy, resembling a recent optical conductivity experiment in cuprates. Phase coherence leads to the appearance of a collective Bogoliubov mode in the density-density correlation function and to the sharpening of the spectral function.