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Physical review. B, Condensed matter

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

P. J. H. Denteneer

Published Mar 1, 1994• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.49.6364

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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.
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∗,{T}^{*},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∗{T}^{*}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 critical temperature for superconductivity for the two-dimensional attractive (negative- U ) Hubbard model on a square lattice is calculated as a function of the strength of the interaction and of the density of electrons. The critical temperature is obtained by calculating the helicity modulus associated with a wavelike distortion of the gap parameter in the BCS approximation as a function of temperature and subsequent comparison with the Kosterlitz-Thouless relation between critical temperature and helicity modulus. The adequacy of the approximations is discussed and a comparison with recent Monte Carlo calculations is made.
The attractive Hubbard model for normal and superconducting ground states is examined for arbitrary electron concentration by the use of the slave-boson mean-field approximation (SBMFA) technique. This approach, at a saddle-point level, is equivalent to the Gutzwiller approximation. Several superfluid characteristics of the model are shown for hypercubic lattices of all dimensions, including , and a comparison with the Bardeen-Cooper-Schrieffer Hartree-Fock approximation (BCS-HFA) calculations and exact results is made. Our results show quantitative and qualitative corrections of the SBMFA to the HFA. The improvement of the SBMFA over the HFA diminishes with increasing lattice dimensionality. We have also evaluated the energy difference between the superconducting and the normal states; this is compared with available results for the superconducting critical temperatures in various dimensions and Uemura-type plots are obtained. The results confirm and substantiate the assertion of a continuous evolution of the superfluid properties of the model from the weak-coupling (BCS-like) to the strong-coupling (composite-boson superconductivity) limit.
We present results for the zero- and finite-temperature Drude weight D(T)D(T)D(T) and for the Meissner fraction of the attractive and the repulsive Hubbard model, as well as for the model with next-nearest-neighbor repulsion. They are based on quantum Monte Carlo studies and on the Bethe ansatz. We show that the Drude weight is well defined as an extrapolation on the imaginary frequency axis, even for finite temperature. The temperature, filling, and system size dependence of D is obtained. We find counterexamples to a conjectured connection of dissipationless transport and integrability of lattice models.
Based on extensions of the grand-canonical quantum Monte Carlo algorithm to incorporate magnetic fields, we provide numerical data confirming the existence of a Kosterlitz-Thouless transition in the attractive Hubbard model. Here, we calculate the temperature derivative of the superfluid density, [\ensuremath{\partial}\ensuremath{\beta}Ds{\mathit{D}}_{\mathit{s}}Ds​(\ensuremath{\beta})]/\ensuremath{\partial}\ensuremath{\beta}, to pin down the transition. The latter quantity is obtained from the difference in internal energy of systems which differ by a phase twist \ensuremath{\pi}/2 in the boundary condition along one lattice direction. Away from half-band filling, [\ensuremath{\partial}\ensuremath{\beta}Ds{\mathit{D}}_{\mathit{s}}Ds​(\ensuremath{\beta})]/\ensuremath{\partial}\ensuremath{\beta} shows a response which increases with lattice size at the transition temperature. In contrast, such a signal is not observed at half-band filling.