Determining the properties of the two-dimensional Hubbard model is an outstanding problem in physics. Applying recent advances in constrained path auxiliary-field quantum Monte Carlo techniques and simulating large rectangular supercells, we characterize the magnetic and charge properties in the ground state as a function of doping. At intermediate interaction strengths, an incommensurate spin density wave (SDW) state is found, with antiferromagnetic order and essentially homogeneous charge correlation. The wavelength of the collective mode decreases with doping, as does its magnitude. The SDW order vanishes beyond a critical doping. As the interaction is increased, the holes go from a wavelike (delocalized) to a particlelike (localized) state, and charge ordering develops which eventually evolves into stripelike states.
Determining the properties of the two-dimensional Hubbard model is an outstanding problem in physics. Applying recent advances in constrained path auxiliary-field quantum Monte Carlo techniques and simulating large rectangular supercells, we characterize the magnetic and charge properties in the ground state as a function of doping. At intermediate interaction strengths, an incommensurate spin density wave (SDW) state is found, with antiferromagnetic order and essentially homogeneous charge correlation. The wavelength of the collective mode decreases with doping, as does its magnitude. The SDW order vanishes beyond a critical doping. As the interaction is increased, the holes go from a wavelike (delocalized) to a particlelike (localized) state, and charge ordering develops which eventually evolves into stripelike states.
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 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.
Using a path-integral formalism and decomposing the Hubbard interaction into charge and spin order parameters, we develop a 1/S loop expansion of the two-dimensional Hubbard model. At the saddle-point level, we present a detailed study of the energetics of twisted antiferromagnetic order. Doping the antiferromagnetic state leads to a rich mean-field phase diagram at zero temperature, which includes spiral phases, a column phase, and ferromagnetism. The on-site magnetization, the density of states, the plasma frequency, and the quasiparticle weights are evaluated for the twisted spin-density-wave (SDW) states as a function of doping and U/t. Doping-induced frustration remarkably leads to a closure of the Mott-Hubbard band gap for any U/t. At one-loop order, we delineate the collective spin and charge fluctuations of the mean-field SDW state. The action for the two transverse spin modes at half-filling maps to a nonlinear \ensuremath{\sigma} model, which tends to disorder upon doping. Twisted magnetic ground states exhibit a third Goldstone mode due to the complete breaking of rotational invariance. For the commensurate SDW, the charge fluctuations couple only to longitudinal spin fluctuations, whereas, for the incommensurate metallic SDW states, the charge modes are coupled nontrivially to the massless spin waves. This suggests a metallic state quite unlike traditional Fermi liquids.
We perform extensive auxiliary-field quantum Monte Carlo (AFQMC) calculations for the three-band Hubbard (Emery) model in order to study the ground-state properties of copper-oxygen planes in the cuprates. Employing cutting-edge AFQMC techniques with a self-consistent gauge constraint in auxiliary-field space to control the sign problem, we reach supercells containing $\ensuremath{\approx}500$ atoms to capture collective modes in the charge and spin orders and characterize the behavior in the thermodynamic limit. The self-consistency scheme interfacing with generalized Hartree-Fock calculations allows high accuracy in AFQMC to resolve small energy scales, which is crucial for determining the complex candidate orders in such a system. We present detailed information on the charge order, spin order, momentum distribution, and localization properties as a function of charge-transfer energy for the the underdoped regime. In contrast with the stripe and spiral orders under hole doping, we find that the corresponding 1/8 electron-doped system exhibits purely antiferromagnetic order in the three-band model, consistent with the asymmetry between electron and hole doping in the phase diagram of cuprates.
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.
In a 1/S loop expansion of the two-dimensional Hubbard model, doping leads to incommensurate twist spin-density-wave (SDW) states at the saddle point. Further doping reduces the on-site magnetization and increases the SDW pitch, conspiring to close the effective Mott-Hubbard gap at \ensuremath{\delta}\ensuremath{\sim}0.07--0.30 for U/t\ensuremath{\sim}2.5--10. At the one-loop level, the two massless fluctuations map to the nonlinear \ensuremath{\sigma} model at zero doping while for the twist SDW states, three massless spin-wave modes arise which are strongly coupled to the charge fluctuations, therefore implying nontrivial charge-spin response functions for the metal.