Two-dimensional Attractive Hubbard Model Research Articles

Comment on “BCS to Bose-Einstein crossover phase diagram at zero temperature for adx2−y2order parameter superconductor: Dependence on the tight-binding structure”

The work by Soares et al. [Phys. Rev. B 65, 174506 (2002)] investigates the BCS to Bose-Einstein crossover for d-wave pairing in the two-dimensional attractive Hubbard model. Contrary to their claims, we found that a nonpairing region does not exist in the density vs coupling phase diagram. The gap parameter at $T=0,$ as obtained by solving analytically as well as numerically the BCS equations, is in fact finite for any nonzero density and coupling, even in the weak-coupling regime.

Reply to “Comment on ‘BCS to Bose-Einstein crossover phase diagram at zero temperature for adx2−y2order parameter superconductor: Dependence on the tight-binding structure’ ”

In a recent Comment, Perali, Pieri, and Strinati [Phys. Rev. B 68, 066501 (2003)] came forward with a correction to the work of Soares et al. [Phys. Rev. B 65, 174506 (2002)], in the sense that they correctly found no nonpairing region in the density vs coupling diagram of the two-dimensional attractive Hubbard model for a d-wave symmetry for the order parameter. But, the BCS-Bose-Einstein crossover diagram also changes, a point missed by Perali, Pieri, and Strinati in their Comment. Furthermore, they use an analytical treatment of the gap equation to find the order parameter as C function of interaction strength, namely, $\ensuremath{\Delta}/4t$ vs $V/4t.$ We point out that this treatment is not completely correct for the evaluation of the chemical potential, $\ensuremath{\mu}/4t$ vs $V/4t,$ when we include the Debye frequency, ${\ensuremath{\omega}}_{D}/4t,$ and their cutoff frequency, ${\ensuremath{\omega}}_{o}.$ As a consequence, the phase diagram, namely, n vs $V/4t,$ strongly depends on the value of ${\ensuremath{\omega}}_{D}/4t,$ i.e., for small values of ${\ensuremath{\omega}}_{D}/4t$ the phase diagram moves to higher values of $V/4t.$

Phase diagram for the attractive Hubbard model in two dimensions in a conserving approximation

We calculate the superconducting transition temperature as a function of interaction strength and density for a two-dimensional attractive Hubbard model in the fluctuation exchange approximation (FEA). Significant corrections to mean-field theory are apparent, including a maximum in ${T}_{c}$ as a function of U and a fluctuation-broadened specific heat peak. At $n=1,$ competing pair fluctuations and charge-density fluctuations fail to drive ${T}_{c}$ to zero. This failure of the FEA reflects an inadequate account of the coupling among fluctuations. Because ${T}_{c}$ is determined from self-consistent calculations of thermodynamic potentials, the calculations are extremely sensitive to numerical errors, requiring accuracy up to 1 part in ${10}^{6}.$ We outline a procedure for systematically reducing numerical errors associated with the finite-frequency cutoff in numerical many-body calculations such as ours.

Pairing fluctuations and pseudogaps in the attractive Hubbard model

The two-dimensional attractive Hubbard model is studied in the weak to intermediate coupling regime by employing a non-perturbative approach. It is first 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 $\omega < T$ 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.

Pair-fluctuation-induced pseudogap in the normal phase of the two-dimensional attractive Hubbard model at weak coupling

One-particle spectral properties in the normal phase of the two-dimensional attractive Hubbard model are investigated in the weak-coupling regime using the non-self-consistent T-matrix approximation. The corresponding equations are evaluated numerically directly on the real-frequency axis. For temperatures sufficiently close to the superconducting transition temperature, a pseudogap in the one-particle spectral function is observed, which can be assigned to the increasing importance of pair fluctuations.

Superfluid density at finite temperatures in two-dimensional attractive Hubbard model: crossover from BCS to Bose–Einstein condensation region

Effects of two corrections to the vertex part of the current–current correlation function for the two-dimensional attractive Hubbard model are examined to see the change of the stiffness or superfluid density N s at temperature T KT, whose dependence on the strength of the interaction displays crossover from BCS to Bose–Einstein condensation. The ladder correction gives no new transverse part, so it does not change N s from non-gauge invariant calculation. On the contrary, the Aslamazov–Larkin-type correction gives rise to an additional term to the transverse part, improving our previous work on T KT– U relation especially in the region of Bose–Einstein condensation.

Excitation spectrum of the two-dimensional attractive Hubbard model

We calculate the one-particle spectral functions above the superconducting transition temperature T C, in the framework of a functional integral approach. The coupling of the electronic self-energy to pair fluctuations, which are treated by means of a time-dependent Ginzburg–Landau equation, yields a double-peak structure, around the Fermi wavenumber. The peak separation is essentially temperature-independent, but the structure sharpens when T C is approached.

Pseudogap in the one-electron spectral functions of the attractive Hubbard model

We calculate the one-electron Green's function of the two-dimensional (2D) attractive Hubbard model by coupling the electrons to pair fluctuations. The latter are approximated by homogeneous amplitude fluctuations and phase correlations corresponding to the XY-model. The electronic density of states shows a pseudogap at temperatures well above the transition temperature T c. For a quasi-3D system, a superconducting gap emerges out of the pseudogap below T c.

The pseudogap and photoemission spectra in the attractive Hubbard model

Angle-resolved photoemission spectra are calculated microscopically for the two-dimensional attractive Hubbard model. A system of self-consistent T-matrix equations are solved numerically in the real-time domain. The single-particle spectral function has a two-peak structure resulting from the presence of bound states. The spectral function is suppressed at the chemical potential, leading to a pseudogap-like behaviour. At high temperatures and densities the pseudogap diminishes and finally disappears; these findings are similar to the experimental observations for the cuprates.

ATTRACTIVE HUBBARD MODEL AND SINGLE-PARTICLE PSEUDOGAP CAUSED BY CLASSICAL PAIRING FLUCTUATIONS IN TWO-DIMENSIONS

It is shown that in the two-dimensional attractive Hubbard model, the mean-field phase transition is replaced by a renormalized classical regime of fluctuations where a pseudogap opens up in the single-particle spectral weight. It is argued that this pseudogap and precursors of the ordered state quasiparticles can occur only in strongly anisotropic quasi two-dimensional materials. This precursor phenomenon differs from preformed local pairs. Further, while critical antiferromagnetic fluctuations would also lead to a pseudogap in the repulsive model, there are some important differences with the superconducting case.

Density-Induced Breaking of Pairs in the Attractive Hubbard Model

A conserving T-matrix approximation is applied to the two-dimensional attractive Hubbard model in the low-density regime. A set of self-consistent equations is solved in the real-frequency domain to avoid the analytic continuation procedure. By tuning the chemical potential the particle density was varied in the limits 0.01 < n < 0.18. For the value of the attractive potential U=8t the binding energy of pairs monotonically decreases with increasing n, from its zero-density limit 2.3t and vanishes at a critical density n=0.19. A pairing-induced pseudogap in the single-particle density of states is found at low densities and temperatures.

Coupled Electrons and Pair Fluctuations in Two Dimensions: A Transition to Superconductivity in a Conserving Approximation

We report a superconducting transition in the fully conserving fluctuation exchange approximation for a two-dimensional attractive Hubbard model. The transition temperature agrees well with quantum Monte Carlo calculations, but the superfluid density and specific heat near ${T}_{c}$ have anomalous temperature dependences for lattice sizes up to $64\ifmmode\times\else\texttimes\fi{}64$.

From BCS-like superconductivity to condensation of local pairs: A numerical study of the attractive Hubbard model.

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 ${\mathit{T}}_{\mathit{c}}$. These features are also mirrored in the shape of the specific-heat peak around ${\mathit{T}}_{\mathit{c}}$, 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.

Excitation spectrum of the attractive Hubbard model.

We study excitation-spectrum and normal-state properties of the two-dimensional attractive Hubbard model using the conserving, self-consistent {ital T}-matrix formalism in the intermediate coupling regime and at low electron concentration. Numerical results are presented for one-particle and two-particle excitation spectra, the one-particle momentum distribution, the chemical potential, and the static spin susceptibility. For a coupling strength of {ital U}/{ital t}=4.0, the one-particle spectral function, {ital A}(k,{omega}), shows two peaks of different weights. One peak can be associated with pair formation, whereas the other corresponds to renormalized quasiparticle excitation. It turns out that the two-band feature is reasonably well described by an ansatz for {ital A}(k,{omega}), which satisfies the first four frequency moments.

Superfluid density in the two-dimensional attractive Hubbard model: Quantitative estimates.

A nonzero superfluid density is equivalent to the occurrence of a Meissner effect and therefore signals superconductivity. A recent theorem shows that in the case of a spectrum with a gap the superfluid density is equivalent to the Drude weight. This theorem is employed to compare approximate calculations of the superfluid density in the two-dimensional attractive Hubbard model using the Hartree-Fock approximation with exact diagonalization calculations of the Drude weight. Direct comparison of the approximate results with recent finite-temperature quantum Monte Carlo calculations is also made. The approximate results are found to be quantitatively accurate for all fillings, except close to half-filling.