The Cray X1 in the Center for Computational Sciences at Oak Ridge National Laboratory as well as algorithmic improvements over the past decade enable significant new science in the simulation of high-temperature cuprate superconductors. We describe the method of dynamic cluster approximation with quantum Monte Carlo, along with its computational requirements. We then show the unique capabilities of the X1 for supporting this method and delivering near optimal performance. This allows us to study systematically the cluster size dependence of the superconductivity in the conventional two-dimensional Hubbard model, which is commonly believed to describe high-temperature superconductors. Due to the non-locality of the d-wave superconducting order parameter, the results on small clusters show large size and geometry effects. In large enough clusters, converged results are found that display a finite temperature instability to d-wave superconductivity. The results we report here demonstrate for the first time that superconductivity is possible in a system of strongly correlated electrons without the need of a phonon mediated attractive interaction.
The superconducting instabilities of the doped repulsive 2D Hubbard model are studied in the intermediate to strong coupling regime with the help of the dynamical cluster approximation. To solve the effective cluster problem we employ an extended noncrossing approximation, which allows for a transition to the broken symmetry state. At sufficiently low temperatures we find stable d-wave solutions with off-diagonal long-range order. The maximal T(c) approximately 150 K occurs for a doping delta approximately 20% and the doping dependence of the transition temperatures agrees well with the generic high- T(c) phase diagram.
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.
Dynamic cluster quantum Monte Carlo calculations for a doped two-dimensional extended Hubbard model are used to study the stability and dynamics of
d-wave pairing when a nearest-neighbor Coulomb repulsion
V is present in addition to the on-site Coulomb repulsion
U. We find that
d-wave pairing and the superconducting transition temperature
Tc are only weakly suppressed as long as
V does not exceed
U/2. This stability is traced to the strongly retarded nature of pairing that allows the
d-wave pairs to minimize the repulsive effect of
V. When
V approaches
U/2, large momentum charge fluctuations are found to become important and to give rise to a more rapid suppression of
d-wave pairing and
Tc than for smaller
V.
Using a dynamical cluster quantum Monte Carlo approximation, we investigate the
d-wave superconducting transition temperature
Tc in the doped two-dimensional repulsive Hubbard model with a weak inhomogeneity. The inhomogeneity is introduced in the hoppings ${t}^{\ensuremath{'}}$ and
t in the form of a checkerboard pattern where
t is the hopping within a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette, and ${t}^{\ensuremath{'}}$ is the hopping between the plaquettes. We find inhomogeneity suppresses
Tc. The characteristic spin excitation energy and the strength of
d-wave pairing interaction decrease with decreasing
Tc, suggesting a strong correlation between these quantities.
We present results of three studies done using a dynamical cluster quantum Monte Carlo approximation.First, we investigate the d-wave superconducting transition temperature T c in the doped 2D repulsive Hubbard model with a weak inhomogeneity in hopping in the form of checkerboard pattern or a lattice of 2 × 2 plaquettes.Near neighbor hoppings within a plaquette is t and that of between the plaquettes is t ′ .We investigate T c in the weak inhomogeneous limit 0.8t < t ′ < 1.2t.We find inhomogeneity (t ′ = t) suppresses T c .The characteristic spin excitation energy (effective exchange energy) and the strength of d-wave pairing interaction decrease with decreasing T c .The latter observations suggest a strong correlation among effective exchange interaction, T c and the d-wave pairing interaction of the system.Second 1 , we further find that enhancement of effective exchange interaction causes a slight increase in T c of a weakly disordered system with low impurity concentration, compared to the homogeneous system.Here the disorder is introduced to homogeneous repulsive 2D Hubbard model as a weak local potential disorder.Third, we present an improved maximum entropy method to analytically continue quantum Monte Carlo data with a severe sign problem.1 A result from a collaborative study done with A. Kemper of