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arXiv:cond-mat/9811298 · Physical review. B, Condensed matter

Configuration-interaction approach to hole pairing in the two-dimensional Hubbard model

Enrique Louis, F. Guinea, María P. López-Sancho, José Antonio Vergés

Published Jun 1, 1999• 21 citations
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Abstract

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 $\sqrt{2}$ 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.

DOI
10.1103/physrevb.59.14005

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Physical review. B, Condensed matter•Jul 1, 1988•66 citations
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×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 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−Jt\ensuremath{-}Jt\ensuremath−J model. The method is also used to derive known properties of the quasiparticle band structure of isolated spin polarons.
The interactions between holes in the Hubbard model, in the low-density, inter mediate-to-strong coupling limit, are investigated. 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. Our results are derived by systematically improving mean-field calculations. The method can also be used to derive known properties 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.
We study the ``t-J'' model with strongly correlated electrons (infinite-U Hubbard-model kinetic energy plus superexchange J) by exact diagonalization on small two-dimensional clusters of six, eight, and twelve sites on a square lattice with periodic boundary conditions. We find an attractive pairing potential for two holes when J/t\ensuremath{\ge}(J/t)c{)}_{\mathit{c}})c​\ensuremath{\simeq}0.2, with t the hopping matrix element. Ground-state calculations show that extended s-wave and d-wave pairing susceptibilities and equal-time pairing correlations grow weakly with increasing J and system size at fixed filling n. Pairing correlations may be enhanced for (J/t)>(J/t)c{)}_{\mathit{c}})c​\ensuremath{\approxeq}0.2. We compare equal-time pairing correlations to a mean-field theory for J=0 and note potential difficulties with interpreting numerical results for such small systems. The widths of one- and two-hole energy bands in low-spin states are of order J; the one-hole effective mass is larger than the two-hole effective mass in such states. Spin-charge correlation functions for a single hole in the eight- and twelve-site lattices are consistent with the spin-charge decoupling scenario for resonating-valence-bond states.
The Holstein-Hubbard model is studied in two dimensions to explore the nature of polarons as a function of hole concentration in the strong correlation regime. A series of canonical transformations is performed to obtain an effective extended Hubbard model which, in the strong on-site electron-electron correlation regime, transforms into a t-J model and is solved using the mean-field Hartree-Fock approximation (HFA). It is shown that the hopping parameter which is related to the bandwidth of the polaron is very small for low (x ≈ 0) and high (x ≈ 1) hole concentrations whereas the hopping parameter is much larger for the intermediate hole concentrations.
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 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.