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Physical review. B, Condensed matter

Hole pairs in the one-dimensional Hubbard model in a staggered field

Janez Jaklič, Peter Prelovšek

Published Mar 1, 1993• 5 citations
Publisher

Abstract

The one-dimensional Hubbard model in a staggered magnetic field h, introduced recently to simulate planar correlated electronic systems, is studied within the unrestricted Hartree-Fock approximation. The emphasis is on solutions for a few holes added to the reference antiferromagnet. Both analytical results, available at U,ht, and numerical results confirm the polaronic character of a single-hole state for all U,h, while bipolarons (bound pairs of holes) are stable for 0U${\mathit{U}}_{\mathit{c}}$(h). Since under the same conditions we do not find any evidence for stable hole clusters and the phase separation, the latter regime remains a candidate for hole pairing.

DOI
10.1103/physrevb.47.6142

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The one-dimensional Hubbard model in a staggered magnetic field h, introduced recently to simulate planar correlated electronic systems, is studied within the unrestricted Hartree-Fock approximation. The emphasis is on solutions for a few holes added to the reference antiferromagnet. Both analytical results, available at U,ht, and numerical results confirm the polaronic character of a single-hole state for all U,h, while bipolarons (bound pairs of holes) are stable for 0UUc{\mathit{U}}_{\mathit{c}}Uc​(h). Since under the same conditions we do not find any evidence for stable hole clusters and the phase separation, the latter regime remains a candidate for hole pairing.
We examine the influence of a staggered magnetic field on states involving a few holes in the one-dimensional t-J and Hubbard models. We apply exact diagonalizations of small systems, an analytical treatment of an anisotropic model, and a high-field perturbation expansion. Our results suggest that even modest staggered fields, h\ensuremath{\ll}t, induce the formation of bound hole pairs in a broad range of parameters, in particular for J/t1 in the t-J model and for U/t>1 in the Hubbard model. In the same regions, by studying the density-density correlations and the compressibility, we argue that many holes form a paired state and that phase separation does not occur.
The influence of the staggered magnetic field h on properties of the one-dimensional t-J and Hubbard models with nearly half-filled bands are studied. Using the exact diagonalization method for small systems as well as analytical approaches possible in the anisotropic model and for h\ensuremath{\gg}t, we determine the phase diagram of the t-J-h model at low doping, which displays three different regimes: the phase with repelling holes for JJc{\mathit{J}}_{\mathit{c}}Jc​, the phase with paired holes for Jc{\mathit{J}}_{\mathit{c}}Jc​JJs{\mathit{J}}_{\mathit{s}}Js​, and the phase separation for J>Js{\mathit{J}}_{\mathit{s}}Js​. Employing for h\ensuremath{\ne}0 the analogy with the universal behavior of one-dimensional interacting fermions within the Luther-Emery regime, we calculate the charge exponent ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, which can be ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$>1 in the pairing regime, indicating dominant superconducting fluctuations. An analogous analysis, in particular the perturbation expansion in t/h, performed for the Hubbard t-U-h model shows a phase of stable hole pairs for UUc{\mathit{U}}_{\mathit{c}}Uc​. Again ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$ seems to approach a nonuniversal value at half-filling.
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.
The single-hole wave function of the one-band Hubbard model on finite clusters of the square lattice, is written as a linear combination of Hartree-Fock (HF) magnetic polarons localized on each cluster site, for clusters of up to 12\ifmmode\times\else\texttimes\fi{}12 sites. This procedure restores the translational symmetry of the solutions, broken at the HF level. The formation energy of a hole is reduced due to the increased delocalization. This correction brings the results closer to exact calculations, when available, and is more important at intermediate couplings (U\ensuremath{\approxeq}4-10t), and tends to zero as U\ensuremath{\rightarrow}\ensuremath{\infty}. It is shown that holes tend to hop between sites in the same sublattice. The contribution to the hole-formation energy associated with zero-point fluctuations is also calculated.
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.
The Hartree-Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated in the μ−Uμ-Uμ−U plane, restricted to phases with no charge density modulation, extending the results presented in cond-mat/9511116. This allows antiferromagnetism, saturated ferromagnetism, spiral spin density waves and a collinear structure with unit cell ↑↑↓↓\uparrow \uparrow \downarrow \downarrow↑↑↓↓. The spiral phase is unstable against phase separation near quarter-, half- and three-quarter-filling. For large UUU this occurs at hole (or electron) doping of (3t/π2U)1/3(3t/π^{2}U)^{1/3}(3t/π2U)1/3 from half filling.