Systems of fermions described by the three-dimensional (3D) repulsive Hubbard model on a cubic lattice have recently attracted considerable attention due to their possible experimental realization via cold atoms in an optical lattice. Analytical and numerical results are limited away from half-filling. We study the ground state of the doped system from weak to intermediate interaction strengths within the generalized Hartree-Fock approximation. The exact solution to the self-consistent-field equations in the thermodynamic limit is obtained and the ground state is shown to exhibit antiferromagnetic order and incommensurate spin-density waves (SDW). At low interaction strengths, the SDW has unidirectional character with a leading wavevector along the [100]-direction, and the system is metallic. As the interaction increases, the system undergoes a simultaneous structural and metal-to-insulator transition to a unidirectional SDW state along the [111]-direction, with a different wavelength. We systematically determine the real and momentum space properties of these states. The crossover from 3D to two dimensions (2D) is then studied by varying the inter-plane hopping amplitude, which can be experimentally realized by tuning the distance between a stack of square-lattice layers. Detailed comparisons are made between the exact numerical results and predictions from the pairing model, a variational ansatz based on the pairing of spins in the vicinity of the Fermi surface. Most of the numerical results can be understood quantitatively from the ansatz, which provides a simple picture for the nature of the SDW states.
Systems of fermions described by the three-dimensional (3D) repulsive Hubbard model on a cubic lattice have recently attracted considerable attention due to their possible experimental realization via cold atoms in an optical lattice. Analytical and numerical results are limited away from half-filling. We study the ground state of the doped system from weak to intermediate interaction strengths within the generalized Hartree-Fock approximation. The exact solution to the self-consistent-field equations in the thermodynamic limit is obtained and the ground state is shown to exhibit antiferromagnetic order and incommensurate spin-density waves (SDW). At low interaction strengths, the SDW has unidirectional character with a leading wavevector along the [100]-direction, and the system is metallic. As the interaction increases, the system undergoes a simultaneous structural and metal-to-insulator transition to a unidirectional SDW state along the [111]-direction, with a different wavelength. We systematically determine the real and momentum space properties of these states. The crossover from 3D to two dimensions (2D) is then studied by varying the inter-plane hopping amplitude, which can be experimentally realized by tuning the distance between a stack of square-lattice layers. Detailed comparisons are made between the exact numerical results and predictions from the pairing model, a variational ansatz based on the pairing of spins in the vicinity of the Fermi surface. Most of the numerical results can be understood quantitatively from the ansatz, which provides a simple picture for the nature of the SDW states.
One of the most interesting and challenging problems in physics is understanding strongly correlated many-body systems, where strong interactions can yield many remarkable phenomena such as superfluidity in 4He, high-temperature superconductivity, etc. In order to attack these problems, we often need to reduce the complexity of the systems to simple models in hopes of getting better insights into the properties of the systems. The Hubbard model, the focus of this dissertation, is one of the most famous examples of such model, which describes a tunneling of electrons between nearest neighbor sites of a lattice with on-site interactions. This simple model is an important concept in condensed matter physics and provides rich understandings of electronic and magnetic properties of materials. Despite its simplicity, there is no general analytical solution to the Hubbard model beyond 1D.;The discovery of ultracold atoms and optical lattices opens up the possibility of emulating the Hubbard model in experiments. Optical lattices provide an ideal realization of the Hubbard model where relevant parameters can be tuned systematically. It makes theoretical studies of the Hubbard model increasingly attractive since a direct comparison between theoretical calculations and experimental results becomes more and more possible.;In this dissertation, the ground-state properties of the repulsive Hubbard model for weak to intermediate interaction strengths in two, three dimensions and their dimensional crossover are studied within the mean field theory. We show that the system exhibits unidirectional spin-density wave (SDW) order with antiferromagnetic correlations and a long wavelength modulation. The modulating wave is along the [0011-direction at low interaction strength U/t and along the [1111-direction at higher U/t. The evolution of the wavelength of the SDW is determined as a function of U/t, the density, and t⊥/t. With an analysis of the pairing of spins based on nesting and deformation of the Fermi surface, we discuss how these results can be rationalized and how a simple, predictive model can be constructed for the properties of the SDW states.
The ground states of the two-dimensional repulsive Hubbard model are studied within the unrestricted Hartree-Fock (UHF) theory. Magnetic and charge properties are determined by systematic, large-scale, exact numerical calculations, and quantified as a function of electron doping, h. In the solution of the self-consistent UHF equations, multiple initial configurations and simulated annealing are used to facilitate convergence to the global minimum. New approaches are employed to minimize finite-size effects in order to reach the thermodynamic limit. At low to moderate interacting strengths and low doping, the UHF ground state is a linear spin-density wave (l-SDW), with antiferromagnetic order and a modulating wave. The wavelength of the modulating wave is 2/h. Corresponding charge order exists but is substantially weaker than the spin order, hence holes are mobile. As the interaction is increased, the l-SDW states evolve into several different phases, with the holes eventually becoming localized. A simple pairing model is presented with analytic calculations for low interaction strength and small doping, to help understand the numerical results and provide a physical picture for the properties of the SDW ground state. By comparison with recent many-body calculations, it is shown that, for intermediate interactions, the UHF solution provides a good description of the magnetic correlations in the true ground state of the Hubbard model.
The realization of antiferromagnetic (AF) correlations in ultracold fermionic atoms on an optical lattice is a significant achievement. Experiments have been carried out in one, two, and three dimensions, and have also studied anisotropic configurations with stronger tunneling in some lattice directions. Such anisotropy is relevant to the physics of cuprate superconductors and other strongly correlated materials. Moreover, this anisotropy might be harnessed to enhance AF order. Here we numerically investigate, using the determinant quantum Monte Carlo method, a simple realization of anisotropy in the three-dimensional (3D) Hubbard model in which the tunneling between planes, ${t}_{\ensuremath{\perp}}$, is unequal to the intraplane tunneling t. This model interpolates between the three-dimensional isotropic (${t}_{\ensuremath{\perp}}=t$) and two-dimensional (2D; ${t}_{\ensuremath{\perp}}=0$) systems. We show that at fixed interaction strength to tunneling ratio (U/t), anisotropy can enhance the magnetic structure factor relative to both 2D and 3D results. However, this enhancement occurs at interaction strengths below those for which the N\'eel temperature ${T}_{\mathrm{N}\stackrel{\ifmmode \acute{}\else \'{}\fi{}}{\mathrm{e}}\mathrm{el}}$ is largest, in such a way that the structure factor cannot be made to exceed its value in isotropic 3D systems at the optimal U/t. We characterize the 2D-3D crossover in terms of the magnetic structure factor, real space spin correlations, number of doubly occupied sites, and thermodynamic observables. An interesting implication of our results stems from the entropy's dependence on anisotropy. As the system evolves from 3D to 2D, the entropy at a fixed temperature increases. Correspondingly, at fixed entropy, the temperature will decrease going from 3D to 2D. This suggests a cooling protocol in which the dimensionality is adiabatically changed from 3D to 2D.
We present a numerically exact study of the Hubbard model with spin-dependent anisotropic hopping on the square lattice using auxiliary-field quantum Monte Carlo method. At half filling, the system undergoes Ising phase transitions on cooling, leading to the formation of Ising-type antiferromagnetic order for repulsive interactions and charge-density wave order for attractive interactions at finite temperatures. By elegantly implementing the sign-problem-free condition and Hubbard-Stratonovich transformations, we achieve significant improvements in precision control of the numerical calculations and obtain highly accurate results of the transition temperatures from weak to strong interactions across representative anisotropies. We further characterize the system by examining the temperature dependence of various thermodynamic properties, including the energy, double occupancy, specific heat, and charge susceptibility. Specifically, we provide unbiased numerical results of the entropy map on temperature-interaction plane, the critical entropy, and the spin, singlon and doublon correlations, all of which are directly measurable in optical lattice experiments. Away from half filling, we explore the behavior of the sign problem and investigate the possible emergence of stripe spin-density wave order in the system with repulsive interaction.
We determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near 1/8 doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about 1/5 doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.
The Hubbard model on a square lattice in the ground state is investigated. Various aspects of Mott transition at half-filling are clarified from the quantum Monte Carlo study. Critical exponents of the transition for charge and spin correlations are estimated. For the doping concentration δ, the charge susceptibility is proportional to δ -1 indicating the divergence of charge mass for δ→0, while the spin susceptibility is always finite and not singular near δ=0. Several theoretical consequences of the above results are discussed. Incommensurate peak amplitude of spin correlation scales as (δ-δ c ) -1 with δ c <0.01 in disagreement with RPA results ln (δ-δ c ). The antiferromagnetic order at the half-filling also shows strong correlation character and is not sensitive to the shape of the fermi surface, while the incommensurate peak observed away from the half-filling is sensitively suppressed by the loss of partial nesting consistently with the weak coupling picture. Comparisons with experimental indications of high- T c oxides are made.