We investigate the two-dimensional attractive Hubbard model with quantum Monte Carlo techniques to reveal the crossover from a BCS-type superconductivity in the weak-coupling regime to a superconductivity properly described by a Bose-Einstein condensation (BEC) of local, preformed pairs. The crossover from BCS to BEC is particularly well exposed in the temperature dependence of both the spin susceptibility and the double occupancy, as well as by the appearance of a pseudogap in the density of states far above
Tc. These features are also mirrored in the shape of the specific-heat peak around
Tc, the separation of the temperature regimes where pair formation and their condensation occur, and in the transfer of spectral weight from the single-particle excitation branch to a pair band in the normal state. \textcopyright{} 1996 The American Physical Society.
We present an effective Hamiltonian based real-space approach for studying the weak-coupling BCS to the strong-coupling Bose-Einstein condensate crossover in the two-dimensional attractive Hubbard model at finite temperatures. We introduce and justify an effective classical Hamiltonian to describe the thermal fluctuations of the relevant auxiliary fields. Our results for
Tc and phase diagrams compare very well with those obtained from more sophisticated and CPU-intensive numerical methods. We demonstrate that the method works in the presence of disorder and can be a powerful tool for a real-space description of the effect of disorder on superconductivity. From a combined analysis of the superconducting order parameter, the distribution of auxiliary fields, and the quasiparticle density of states, we identify the regions of metallic, insulating, superconducting, and pseudogapped behavior. Our finding of the importance of phase fluctuations for the pseudogap behavior is consistent with the conclusions drawn from recent experiments on NbN superconductors. The method can be generalized to study superconductors with nontrivial order-parameter symmetries by identifying the relevant auxiliary variables.
Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak [Bardeen-Cooper-Schrieffer, (BCS)] and strong [Bose-Einstein condensation (BEC)] couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice from which several different quantities have been calculated and should be useful as a road map to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature
Tc in terms of the band filling $\ensuremath{\langle}n\ensuremath{\rangle}$ and interaction strength
U from which we pinpoint a somewhat wide region $|U|/t\ensuremath{\approx}5\ifmmode\pm\else\textpm\fi{}1$ (
t is the hopping amplitude) and $\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath{\approx}0.79\ifmmode\pm\else\textpm\fi{}0.09$ leading to a maximum ${T}_{c}\ensuremath{\approx}0.16t$. Two additional temperature scales, namely, pairing
Tp and degeneracy
Td have been highlighted: The former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), whereas the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.
In this paper we address the behavior of the superfluid transition temperature
Tc in the attractive Hubbard model. The attractive Hubbard model can be regarded as the generalization of the BCS to Bose-Einstein condensation (BEC) crossover to a lattice and, as such, may have implications for future optical lattice studies. Nevertheless, the BEC limit of the Hubbard model is very different from that of the better-studied Fermi gases, owing to the strong intersite repulsion between pairs. Here we address systematically the effects of pairing fluctuations for all filling fractions over the entire range of attractive interaction strength. A central conclusion of our work is that in a lattice, around half filling, the smooth evolution from the BCS to the BEC limits is interrupted:
Tc vanishes when the system approaches the bosonic regime with increasing interaction strength. We suggest that this interruption of crossover may signal a quantum critical transition to another form of superfluid not continuously connected to a BCS-like phase. A simple variational ansatz for an alternate ground state in this more strongly coupled superfluid is presented. A generalization of the (
s-wave) Hubbard model to
d-wave pairing allows us to address issues of relevance to high-
Tc superconductivity. The phase diagram (representing the pairing or pseudogap onset temperature
T∗ and
Tc) shows that here too one observes a vanishing of
Tc when
T∗ becomes sufficiently large. Given this predicted breakdown of the crossover, and given the striking similarity to features of the cuprate phase diagram, we suggest that future experiments on ultracold fermions in optical lattices should not be exclusively limited to the repulsive Hubbard model, but should address the attractive model in order to elucidate features of high-temperature superconductivity.
The two-dimensional attractive Hubbard model is studied in the weak-to-intermediate-coupling regime by employing a nonperturbative approach. It is shown that this approach is in quantitative agreement with Monte Carlo calculations for both single-particle and two-particle quantities. Both the density of states and the single-particle spectral weight show a pseudogap at the Fermi energy below some characteristic temperature
T∗, also in good agreement with quantum Monte Carlo calculations. The pseudogap is caused by critical pairing fluctuations in the low-temperature renormalized classical regime $(\ensuremath{\Elzxh}\ensuremath{\omega}<{k}_{B}T)$ of the two-dimensional system. With increasing temperature the spectral weight fills in the pseudogap instead of closing it, and the pseudogap appears earlier in the density of states than in the spectral function. Small temperature changes around
T∗ can modify the spectral weight over frequency scales much larger than temperature. Several qualitative results for the s-wave case should remain true for d-wave superconductors.
The finite-temperature phase diagram of the attractive Hubbard model is studied by means of the dynamical mean-field theory. We first consider the normal phase of the model by explicitly frustrating the superconducting ordering. In this case, we obtain a first-order pairing transition between a metallic phase and a paired phase formed by strongly coupled incoherent pairs. The transition line ends in a finite temperature critical point, but a crossover between two qualitatively different solutions still occurs at higher temperature. Comparing the superconducting- and the normal-phase solutions, we find that the superconducting instability always occurs before the pairing transition in the normal phase, i.e. T c > T pairing . Nevertheless, the high-temperature phase diagram at T > T c is still characterized by a crossover from a metallic phase to a preformed pair phase. We characterize this crossover by computing different observables that can be used to identify the pseudogap region, like the spin susceptibility, the specific heat and the single-particle spectral function.