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.
To distinguish a metal from an insulator, the Drude weight (D, zero-frequency component of conductivity) is a useful measure, as Kohn pointed out long ago: D > 0(=0) for a metal (insulator). Later, Millis and Coppersmith showed that a variational wave function ΨQ in which the key ingredient for a Mott transition (a doublon-holon binding effect) is introduced exhibits D > 0 (metallic) even for sufficiently large correlation strength, namely, a Mott transition is absent from. ΨQ In contrast, variational Monte Carlo studies using ΨQ confirmed, by studying relevant quantities such as doublon density d and a superconducting correlation function, Pd∞ that ΨQ undoubtedly raises a Mott transition. This contradiction has been a long-standing perplexing problem. We definitely settle this problem by adding ΨQ to a configuration-dependent phase factor Pθ, which has been hitherto overlooked. This factor appropriately picks out a negative counterpart in D for insulators, so that D vanishes. Because Pθ does not affect the quantities such as d and Pd∞, the previous results of ΨQ on the Mott transition remain intact for PθΨQ.
The question of conductivity is revisited. Using the total momentum shift operator to construct the perturbed many-body Hamiltonian and ground state wave function the second derivative of the ground state energy with respect to the perturbing field is expressed in terms of the one and two-body momentum densities. The distinction between the adiabatic and envelope function derivatives, hence that between the Drude and superfluid weights, can be introduced in a straightforward manner. It is shown that a discontinuity in the momentum density leads to a contribution to the Drude weight, but not the superfluid weight, however a δ-function contribution in the two-body momentum density (such as in the BCS wave-funtion) contributes to both quantities. The connection between the discontinuity in the momentum density and localization is also demonstrated.
The question of conductivity is revisited. Using the total momentum shift operator to construct the perturbed many-body Hamiltonian and ground state wave function the second derivative of the ground state energy with respect to the perturbing field is expressed in terms of the one and two-body momentum densities. The distinction between the adiabatic and envelope function derivatives, hence that between the Drude and superfluid weights can be introduced in a straightforward manner. It is shown that a discontinuity in the momentum density leads to a contribution to the Drude weight, but not the superfluid weight, however a
δ-function contribution in the two-body momentum density (such as in the BCS wave-funtion) contributes to both quantities. The connection between the discontinuity in the momentum density and localization is also demonstrated.
Based on extensions of quantum Monte Carlo algorithms to incorporate magnetic fields, two criteria to detect superconductivity in two-dimensional Hubbard models are investigated. In order to provide such criteria, we calculate both the internal energy E(\ensuremath{\Phi},T) as well as the ground-state energy,
E0(\ensuremath{\Phi}), for Hubbard models on a cylinder geometry threaded by a flux \ensuremath{\Phi}. The temperature derivative of the superfluid density, \ensuremath{\partial}\ensuremath{\beta}
Ds(\ensuremath{\beta})/\ensuremath{\partial}\ensuremath{\beta}, 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. In the framework of a Kosterlitz-Thouless transition, \ensuremath{\partial}\ensuremath{\beta}
Dx(\ensuremath{\beta})/\ensuremath{\partial}\ensuremath{\beta} scales to a Dirac \ensuremath{\delta} function at the transition temperature. On finite-sized lattices, \ensuremath{\partial}\ensuremath{\beta}
Ds(\ensuremath{\beta})/\ensuremath{\partial}\ensuremath{\beta} shows a response which increases with lattice size. Flux quantization is a T=0 method. From the functional form of
E0(\ensuremath{\Phi}), superconducting or nonsuperconducting ground states may be identified. In both approaches, superconductivity may be detected without prior knowledge of the symmetry and nature of the pairing correlations. For single-band Hubbard models, our main quantum Monte Carlo results include numerical data which (a) confirm the existence and pin down the transition temperature of a Kosterlitz-Thouless-type transition in the attractive Hubbard model away from half-band filling and (b) show that the quarter-filled repulsive Hubbard model is not superconducting. For the three-band Hubbard model we consider two parameter sets which take into account the differences in static magnetic structure and Fermi surfaces between La-Sr-Cu-O and Y-Ba-Cu-O materials. For both parameter sets, the finite-temperature approach showed no sign of a Kosterlitz-Thouless-type transition up to inverse temperatures \ensuremath{\beta}=17.5, in units of the Cu-O hopping, and hole doping \ensuremath{\delta}=0.25. Flux quantization results for the Y-Ba-Cu-O parameters on clusters up to 8\ifmmode\times\else\texttimes\fi{}8 unit cells equally showed no sign of superconductivity at a hole doping \ensuremath{\delta}=0.25.
The Drude weight for the one-dimensional Hubbard model is investigated at finite temperatures by using the Bethe ansatz solution. Evaluating finite-size corrections to the thermodynamic Bethe ansatz equations, we obtain the formula for the Drude weight as the response of the system to an external gauge potential. We perform low-temperature expansions of the Drude weight in the case of half-filling as well as away from half-filling, which clearly distinguish the Mott-insulating state from the metallic state.