The two-dimensional Hubbard model near half-filling is studied using a configuration-interaction method. Superconducting order-parameter and spin-spin correlation functions are calculated as functions of distance to probe for superconductivity and antiferromagnetism. Consistent with our previous results, these more extensive calculations point toward d-wave superconductivity in the ground state of the two-dimensional repulsive Hubbard model. The d-wave correlations tend to increase with interaction strength, size of the state space used in the calculation, and doping away from half-filling. The superconducting correlations on the Fermi surface are interpreted as d-wave pairing between spin density-wave quasiparticles.
In the first part of this paper we explore a configuration-interaction approach to the two-dimensional Hubbard model near half filling. To reproduce the result of degenerate second-order perturbation theory it is necessary to include Fock states having kinetic energy within 4t of the lowest kinetic-energy states. Using the ground state from the largest configuration-interaction calculation, we calculate the extended s- and d-wave superconducting correlation functions as a function of distance. There are indications of a nondecaying long-distance tail in the d-wave correlation function, that is, d-wave superconductivity in the ground state of the repulsive Hubbard model. In the second part of the paper, we reexamine our effective Hamiltonian for the repulsive Hubbard model near half filling. Simple considerations lead us to a more intuitively plausible form consistent with our previous results and spin-rotational invariance.
In the first part of this paper we explore a configuration-interaction approach to the two-dimensional Hubbard model near half filling. To reproduce the result of degenerate second-order perturbation theory it is necessary to include Fock states having kinetic energy within 4{ital t} of the lowest kinetic-energy states. Using the ground state from the largest configuration-interaction calculation, we calculate the extended {ital s}- and {ital d}-wave superconducting correlation functions as a function of distance. There are indications of a nondecaying long-distance tail in the {ital d}-wave correlation function, that is, {ital d}-wave superconductivity in the ground state of the repulsive Hubbard model. In the second part of the paper, we reexamine our effective Hamiltonian for the repulsive Hubbard model near half filling. Simple considerations lead us to a more intuitively plausible form consistent with our previous results and spin-rotational invariance.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.
A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength
(U=8t) and for moderate sizes (from $6\ifmmode\times\else\texttimes\fi{}6$ to $10\ifmmode\times\else\texttimes\fi{}10$). Our ansatz is superior to other widely used variational wave functions. The results for order parameters and correlation functions provide insight into the antiferromagnetic state at half-filling as well as strong evidence for a superconducting phase away from half-filling.
The two-dimensional Hubbard model is studied for small values of the interaction strength (
U of the order of the hopping amplitude
t), using a variational ansatz well suited for this regime. The wave function, a refined Gutzwiller ansatz, has a BCS mean-field state with
d-wave symmetry as its reference state. Superconducting order is found for densities
n<1 (but not for
n=1). This resolves a discrepancy between results obtained with the functional renormalization group, which do predict superconducting order for small values of
U, and numerical simulations, which did not find superconductivity for $U\ensuremath{\lesssim}4t$. Both the gap parameter and the order parameter have a domelike shape as a function of
n with a maximum for $n\ensuremath{\approx}0.8$. Expectation values for the energy, the particle number, and the superconducting order parameter are calculated using a linked-cluster expansion up to second order in
U. In this way large systems (millions of sites) can be readily treated and well converged results are obtained. A big size is indeed required to see that the gap becomes very small at half filling and probably tends to zero in the thermodynamic limit, whereas away from half filling a finite asymptotic limit is reached. For a lattice of a given size the order parameter becomes finite only above a minimal coupling strength
Uc. This threshold value decreases steadily with increasing system size, which indicates that superconductivity exists for arbitrarily small
U for an infinite system. For moderately large systems the size dependence is quite irregular, due to variations in level spacings at the Fermi energy. The fluctuations die out if the gap parameter spans several level spacings.