Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of
U/t from 2 to 8, for a range of densities including half-filling and
n=0.3,0.5,0.6,0.75, and
0.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.
Fixed-node Green's-function Monte Carlo calculations have been performed for very large 16x16 two-dimensional Hubbard lattices, large interaction strengths U=10,20, and 40, and many (15 similar to 20) densities between empty and half-filling. The nodes were fixed by a simple Slater-Gutzwiller trial wave function. For each value of U we obtained a sequence of ground-state energies which is consistent with the possibility of a phase separation close to half-filling, with a hole density in the hole-rich phase which is a decreasing function of U. The energies suffer, however, from a fixed-node bias: more accurate nodes are needed to confirm this picture. Our extensive numerical results and their test against size, shell, shape, and boundary-condition effects also suggest that phase separation is quite a delicate issue, on which simulations based on smaller lattices than considered here are unlikely to give reliable predictions. RI Capone, Massimo/A-7762-2008; Guidoni, Leonardo/G-3759-2011
Fixed-node Green's-function Monte Carlo calculations have been performed for very large 16\ifmmode\times\else\texttimes\fi{}16 two-dimensional Hubbard lattices, large interaction strengths
U=10,20, and
40, and many $(15\ensuremath{\sim}20)$ densities between empty and half-filling. The nodes were fixed by a simple Slater-Gutzwiller trial wave function. For each value of U we obtained a sequence of ground-state energies which is consistent with the possibility of a phase separation close to half-filling, with a hole density in the hole-rich phase which is a decreasing function of
U. The energies suffer, however, from a fixed-node bias: more accurate nodes are needed to confirm this picture. Our extensive numerical results and their test against size, shell, shape, and boundary-condition effects also suggest that phase separation is quite a delicate issue, on which simulations based on smaller lattices than considered here are unlikely to give reliable predictions.
We investigate the stability with respect to phase separation or charge-density-wave formation of the two-dimensional Hubbard model for various values of the local Coulomb repulsion and electron densities using Green-function Monte Carlo techniques. The well known sign problem is particularly serious in the relevant region of small hole doping. We show that the difference in accuracy for different doping makes it very difficult to probe the phase separation instability using only energy calculations, even in the weak-coupling limit
(U=4t) where reliable results are available. By contrast, the knowledge of the charge correlation functions allows us to provide clear evidence of a spatially homogeneous ground state up to
U=10t.We report on a numerical study of the two-dimensional Hubbard model and describe two new algorithms for the simulation of many-electron systems. These algorithms allow one to carry out simulations within the grand canonical ensemble at significantly lower temperatures than had previously been obtained and to calculate ground-state properties with fixed numbers of electrons. We present results for the two-dimensional Hubbard model with half- and quarter-filled bands. Our results support the existence of long-range antiferromagnetic order in the ground state at half-filling and its absence at quarter-filling. Results for the magnetic susceptibility and the momentum occupation along with an upper bound to the spin-wave spectrum are given. We find evidence for an attractive effective d-wave pairing interaction near half-filling but have not found evidence for a phase transition to a superconducting state.
Monte Carlo results for a two-dimensional Hubbard model in the intermediate-coupling regime U=4t are compared with a diagrammatic spin-fluctuation approximation. Simulations on an 8\ifmmode\times\else\texttimes\fi{}8 lattice doped away from half filling were carried out down to temperatures of order 140 of the bandwidth. Results for the spin susceptibility \ensuremath{\chi}(q,i\ensuremath{\omega}
m), the electron self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}
n), various pair-field susceptibilities, and the irreducible particle-particle scattering vertex \ensuremath{\Gamma}(p\ensuremath{'},i\ensuremath{\omega}${}_{\mathrm{n}\ensuremath{'}}$|p,i\ensuremath{\omega}
n) were obtained. A random-phase approximation for \ensuremath{\chi}(q,i\ensuremath{\omega}
m) with a renormalized Coulomb coupling U\ifmmode\bar\else\textasciimacron\fi{} is shown to provide a fit to the Monte Carlo data. A similar approximation for the Berk-Schrieffer spin-fluctuation interaction also provides a reasonable fit to the self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}
n) in the region explored by the Monte Carlo data. However, a similar approximation for the irreducible particle-particle interaction failed to reproduce the Monte Carlo results. Higher-order vertex corrections were calculated, but significant discrepancies with Monte Carlo results for \ensuremath{\Gamma} remain.