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Physical review. B, Condensed matter

Multiple-polaron description of the wave function of a single hole in Hubbard clusters of the square lattice

Enrique Louis, G. Chiappe, F. Guinea, José Antonio Vergés, E. V. Anda

Published Oct 1, 1993• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.48.9581

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Physical review. B, Condensed matter•Nov 1, 1991•13 citations
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 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 wave-function renormalization constant Z has been calculated for the one-band Hubbard model on a square lattice. Near half-filling the Hamiltonian has been solved on finite clusters (up to 16\ifmmode\times\else\texttimes\fi{}16) by means of the unrestricted Hartree-Fock (UHF) approximation. Some checks of the UHF results have been made on 4\ifmmode\times\else\texttimes\fi{}4 clusters by means of the Lanczos method for U=\ensuremath{\infty}. At high dilutions (two electrons) exact calculations have been carried out. The ground-state wave functions and energies for the different cases here considered are discussed. The system shows nonconventional behavior (Z=0) near half-filling. For a given value of U, Z vanishes at low dopings, and becomes finite as doping is increased. For very large U the value of Z remains finite for all dopings, excluding half-filling. These results are cast into a plausible phase diagram.
We solved numerically the integral equation for the selfenergy which describes the motion of a single hole in a two-dimensional quantum antiferromagnet (AF) within the selfconsistent Born approximation. This formulation stresses the similarity of the AF-spin polaron with the standard polaron problem. We confine our calculation to finite cluster geometries and compare with results from previous exact diagonalization studies. The spectral function is characterized by a narrow quasiparticle (qp) peak at the low energy side of the spectra, which appears to be well separated from the incoherent band part for large enough J values. For small J we find a reduced width of ~7t for the incoherent band. The bottom of the coherent qp band always occurs at (±π/2, ±π/2). Its bandwidth initially increases with J until J≃t and then decreases as 2t2/J. A complete characterization of our solution is given, including the dispersion relation and effective masses of this quasiparticle. The comparison with exact diagonalization studies for a 4×4 cluster is remarkably good. From our results we see that a lattice of 16×16 sites describes adequately the thermodynamic limit. We conclude that the simple Born approximation is a valuable scheme to characterize the dynamics of one hole in the t-J model. both in the perturbative and the strong coupling regimes.
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.
The effective Hamiltonian, obtained from the Hubbard model in the strong-coupling limit, is diagonalized exactly for a periodic two-dimensional square lattice of ten sites. We obtain the ground state of the system for any filling of the lattice and the KKK-space excitation spectrum for a single hole. Within our finite-size system, it is found that for very small coupling ratio tU\frac{t}{U}Ut​, each hole creates a local ferromagnetic environment at least of the size of our system, whereas for increasing tU\frac{t}{U}Ut​, the holes tend to form clusters, induced by antiferromagnetic spin correlations. A range of tU\frac{t}{U}Ut​ exists for which pairing of the holes may be possible.
The ground-state wave function and energy are calculated for the one-band Hubbard Hamiltonian on a 4\ifmmode\times\else\texttimes\fi{}4 cluster of the square lattice up to first and second order of the on-site Coulomb repulsion U, respectively. Changes induced by a deviation of the number of electrons from half-filling (one electron per lattice site) are closely followed. Charge-charge and spin-spin correlation functions are directly obtained from the ground-state wave function and show the existence of antiferromagnetic correlations that extend over the whole cluster at half-filling. Correlations among sites separated by more than a lattice site diminish very rapidly as holes are introduced into the cluster. The analysis is extended to the negative-U case. Ground-state properties change qualitatively.