The helicity modulus, which is the stiffness associated with a twisted order parameter, for the two-dimensional Hubbard model is calculated for the equivalent cases of (i) attractive on-site interaction (negative U) with arbitrary strength, arbitrary electron density, and zero magnetic field and (ii) repulsive on-site interaction (positive U) with arbitrary strength, at half-filling and in an arbitrary magnetic field. An explicit formula for the helicity modulus is derived using the Bogoliubov-Hartree-Fock approximation. An improved value for the helicity modulus is obtained by performing variational Monte Carlo calculations using a Gutzwiller projected trial wave function. To within a small correction term the helicity modulus is found to be given by -1/8 the average kinetic energy. The variational Monte Carlo calculation is found to increase the value of the helicity modulus by a small amount (about 5% for intermediate values of the interaction strength) compared to the results from the Bogoliubov-Hartree-Fock approximation. In the case of attractive interaction, from a comparison with the Kosterlitz-Thouless relation between critical temperature and helicity modulus, the critical temperature for a Kosterlitz-Thouless transition is calculated and a phase diagram is obtained. An optimal critical temperature is found for an intermediate value of U. We discuss connections of our results with results in the literature on the Hubbard model using the random-phase approximation and quantum Monte Carlo calculations.
The critical temperature for the attractive Hubbard model on a square lattice is determined from the analysis of two independent quantities, the helicity modulus ${\ensuremath{\rho}}_{s}$ and the pairing correlation function
Ps. These quantities have been calculated through quantum Monte Carlo simulations for lattices up to $18\ifmmode\times\else\texttimes\fi{}18,$ and for several densities, in the intermediate-coupling regime. Imposing the universal-jump condition for an accurately calculated ${\ensuremath{\rho}}_{s},$ together with thorough finite-size scaling analyses (in the spirit of the phenomenological renormalization group) of
Ps, suggests that
Tc is considerably higher than hitherto assumed.
We have considered the half-filled disordered attractive Hubbard model on a square lattice, in which the on-site attraction is switched off on a fraction
f of sites, while keeping a finite
U on the remaining ones. Through quantum Monte Carlo simulations for several values of
f and
U and for system sizes ranging from $8\ifmmode\times\else\texttimes\fi{}8$ to $16\ifmmode\times\else\texttimes\fi{}16$, we have calculated the configurational averages of the equal-time pair structure factor
Ps and, for a more restricted set of variables, the helicity modulus ${\ensuremath{\rho}}_{s}$, as functions of temperature. Two finite-size scaling Ans\"atze for
Ps have been used: one for zero temperature and the other for finite temperatures. We have found that the system sustains superconductivity in the ground state up to a critical impurity concentration
fc, which increases with
U, at least up to
U=4 (in units of the hopping energy). Also, the normalized zero-temperature gap as a function of
f shows a maximum near $f\ensuremath{\sim}0.07$ for $2\ensuremath{\lesssim}U\ensuremath{\lesssim}6$. Analyses of the helicity modulus and of the pair structure factor led to the determination of the critical temperature as a function of
f for
U=3, 4, and 6: they also show maxima near $f\ensuremath{\sim}0.07$, with the highest
Tc increasing with
U in this range. We argue that, overall, the observed behavior results from both the breakdown of charge-density-wave-superconductivity degeneracy and the fact that free sites tend to ``push'' electrons toward attractive sites; the latter effect being more drastic at weak couplings.
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.
We study the superconducting Kosterlitz-Thouless (KT) transition of the attractive Hubbard model on a two-dimensional triangular lattice using auxiliary field quantum Monte Carlo method for system sizes up to $14\ifmmode\times\else\texttimes\fi{}14$ sites. Combining three methods to analyze the numerical data, we find, for the attractive interaction of $U=\ensuremath{-}4t$, that the transition temperature is broadly maximized around the band filling of
1.2<n<1.6, while it is found to be rather low in the range of
n<1 and
n>1.8. We discuss that this band filling dependence may be explained in terms of the band filling dependence of the superfluid density and the effective mass. Our analysis also teaches us that the estimation of the KT transition temperature requires some caution when the density of states near the Fermi level is small and the calculation is restricted to small system sizes.
We study the phase diagram for the attractive (i.e., negative-U) Hubbard model on a simple-cubic lattice, through Monte Carlo simulations. We obtain the critical temperature
Tc for superconductivity from a finite-size scaling analysis of the data for the pairing correlations. For fixed on-site attraction U,
Tc displays a maximum near the filling factor 0.9, roughly independent of U. For fixed filling we estimate the crossover temperature ${\mathit{T}}^{\ifmmode\times\else\texttimes\fi{}}$(U), separating the normal states: metallic and spin gap. There is also a critical value
Up for pair formation, the magnitude of which seems to be independent of doping. The relevance of these results to the high-
Tc oxides is discussed.