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Physical review. B, Condensed matter

Charged domain walls in the two-dimensional Hubbard model

J. Yang, W. P. Su

Published Oct 1, 1991• 10 citations
Publisher

Abstract

In this paper we numerically study the single-band Hubbard model on a two-dimensional square lattice using a mean-field iteration approach. We calculate the energies and configurations of finite-size systems for various doping densities up to 20%. The results show that there exists an attractive interaction among the holes, which forces the holes to form domain walls. The most stable configuration of domain-wall structure is determined, and its stability against transverse spin fluctuations is studied. When the nearest-neighbor repulsion V is considered, the system becomes richer in phases. The binding energy is reduced and vanishes completely at ${\mathit{V}}_{\mathit{c}}$=0.17t (for U/t=5, with t and U being the hopping integral and the on-site Hubbard repulsion, respectively). For small value (but larger than ${\mathit{V}}_{\mathit{c}}$) of V, we find that the holes form well-defined spin bags, which are separated from each other for a very small doping density. For a doping density larger than a critical value, they form domain walls. The physical implications of these results and their relation to experiment are discussed.

DOI
10.1103/physrevb.44.6838

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In this paper we numerically study the single-band Hubbard model on a two-dimensional square lattice using a mean-field iteration approach. We calculate the energies and configurations of finite-size systems for various doping densities up to 20%. The results show that there exists an attractive interaction among the holes, which forces the holes to form domain walls. The most stable configuration of domain-wall structure is determined, and its stability against transverse spin fluctuations is studied. When the nearest-neighbor repulsion V is considered, the system becomes richer in phases. The binding energy is reduced and vanishes completely at Vc{\mathit{V}}_{\mathit{c}}Vc​=0.17t (for U/t=5, with t and U being the hopping integral and the on-site Hubbard repulsion, respectively). For small value (but larger than Vc{\mathit{V}}_{\mathit{c}}Vc​) of V, we find that the holes form well-defined spin bags, which are separated from each other for a very small doping density. For a doping density larger than a critical value, they form domain walls. The physical implications of these results and their relation to experiment are discussed.
We study, within the Hartree-Fock approximation, the relative stability of different classes of states for the single-band Hubbard model on a square lattice in the presence of a nearest-neighbor Coulomb repulsion V and away from half-filling. We consider (i) homogeneous metallic states (including spiral phases), (ii) states that exhibit phase separation into hole-rich and antiferromagnetic regions, and (iii) insulating (longitudinally polarized) spin-density-wave states with domain walls that trap holes. For V=0, the spiral states have the lowest energy for hole concentration \ensuremath{\delta}>${\mathrm{\ensuremath{\delta}}}_{\mathit{s}}$. At ${\mathrm{\ensuremath{\delta}}}_{\mathit{s}}$\ensuremath{\sim}0.25--0.3, there is a transition to domain walls for small to intermediate U/t. At larger U/t the transition is to a phase-separated state, with ${\mathrm{\ensuremath{\delta}}}_{\mathit{s}}$ decreasing slowly with increasing U/t. We find that even a small V destabilizes the domain walls relative to the spiral states; e.g., for V/t\ensuremath{\sim}0.2, the domain-wall states are unstable for all U/t and all \ensuremath{\delta}\ensuremath{\ge}0.05. The effect of V on the phase-separated state is much less drastic; its region of stability relative to the spiral states also shrinks to smaller \ensuremath{\delta}, but rather slowly. The last results are in agreement with the findings from a Schwinger-boson treatment of the t-J model.
The Hubbard model in a two-dimensional square lattice is studied within the unrestricted Hartree-Fock approximation. A wide variety of solutions are found, as a function of filling and value of the Coulomb repulsion. Their properties range from those of magnetic polarons to a behavior similar to electrons in an external magnetic field (flux phases). The main features of the vortex solution are studied. It is shown that its energy increases logarithmically with system size, and that it becomes more stable for high values of the Coulomb repulsion and doping levels. Also, multivortex configurations are shown to exist. The implications of our results in the qualitative behavior of the Hubbard model as function of doping and strength of the interaction are discussed.
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 1/81/81/8 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 1/51/51/5 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 interactions between holes in the Hubbard model, in the low density, intermediate- to strong-coupling limit, are investigated by systematically improving mean field calculations. The configuration-interaction basis set is constructed by applying to local unrestricted Hartree-Fock configurations all lattice translations and rotations. It is shown that this technique reproduces, correctly, the properties of the Heisenberg model, in the limit of large U. Upon doping, dressed spin polarons in neighboring sites have an increased kinetic energy and an enhanced hopping rate. Both effects are of the order of the hopping integral and lead to an effective attraction at intermediate couplings. The numerical results also show that when more than two holes are added to the system, they do not tend to cluster, but rather hole pairs remain far apart. Hole-hole correlations are also calculated and shown to be in qualitative agreement with exact calculations for $4\ifmmode\times\else\texttimes\fi{}4$ clusters. In particular our results indicate that for intermediate coupling the hole-hole correlation attains a maximum when the holes are in the same sublattice at a distance of 2\sqrt{2}2​ times the lattice spacing, in agreement with exact results and the $t\ensuremath{-}J$ model. The method is also used to derive known properties of the quasiparticle band structure of isolated spin polarons.
We consider a Hubbard-like model of strongly interacting spinless fermions and hardcore bosons on a square lattice, such that nearest neighbor occupation is forbidden. Stripes (lines of holes across the lattice forming antiphase walls between ordered domains) are a favorable way to dope this system below half filling. The problem of a single stripe can be mapped to fermions in one dimension, which allows understanding of its elementary excitations and calculation of the stripe's effective mass for transverse vibrations. Using Lanczos exact diagonalization, we investigate the excitation gap and dispersion of a hole on a stripe, and the interaction of two holes. We also study the interaction of a few stripes, finding that they repel and the interaction energy decays with stripe separation as if they are hardcore particles moving in one (transverse) direction. To determine the stability of an array of stripes against phase separation into a particle-rich phase and a hole-rich liquid, we evaluate the liquid's equation of state, finding that the stripe-array is not stable for bosons but is possibly stable for fermions.
Understanding the physics of the two-dimensional Hubbard model is widely believed to be a key step in achieving a full understanding of high- T c cuprate superconductors. In recent years, progress has been made by large-scale numerical simulations at finite doping and, on the other hand, by microscopic theories able to capture the physics of individual charge carriers. In this work, we study single pairs of dopants in a cylindrical system using the density-matrix renormalization group algorithm. We identify two coexisting charge configurations that couple to the spin environment in different ways: a tightly bound configuration featuring (next-)nearest-neighbor pairs and a stripelike configuration of dopants on opposite sides of the cylinder, accompanied by a spin domain wall. Thus, we establish that the interplay between stripe order and uniform pairing, central to the models’ phases at finite doping, has its origin at the single-pair level. By interpolating between the Hubbard and the related t − J model, we are able to quantitatively understand discrepancies in the pairing properties of the two models through the three-site hopping term usually omitted from the t − J Hamiltonian. This term is closely related to a next-nearest-neighbor tunneling t ′ , which we observe to upset the balance between the competing stripe and pair states on the two-dopant level.