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arXiv:cond-mat/9302020 · Physical review. B, Condensed matter

Effect of three-particle correlations in low-dimensional Hubbard models

Theodore C. Hsu, Benoît Douçot

Published Jul 15, 1993• 4 citations
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Abstract

A simple approximation which captures some nonperturbative aspects of the one-electron Green function of strongly interacting Fermion systems is developed. It provides a way to go one step beyond the usual dilute limit since particle-particle scattering as well as particle-hole scattering is treated on the same footing. Intermediate states are constrained to contain only one particle-hole excitation besides the incoming particle. The Faddeev equations resulting from an exact treatment of this three-body problem are investigated. In one dimension the method is able to show spin and charge decoupling, but does not reproduce the exact nature of power-law singularities.

DOI
10.1103/physrevb.48.2131

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A simple approximation which captures some nonperturbative aspects of the one-electron Green function of strongly interacting Fermion systems is developed. It provides a way to go one step beyond the usual dilute limit since particle-particle scattering as well as particle-hole scattering is treated on the same footing. Intermediate states are constrained to contain only one particle-hole excitation besides the incoming particle. The Faddeev equations resulting from an exact treatment of this three-body problem are investigated. In one dimension the method is able to show spin and charge decoupling, but does not reproduce the exact nature of power-law singularities.
We propose an approximate version of the three-body scattering theory as an extension of local approximation. We solve the three-body problem by limiting the number of three-body states, instead of solving the Faddeev equation, in the calculation of the self-energy. Applying this simple extension to the antiferromagnetic state at half-filling in the two-dimensional Hubbard model, we demonstrate that it gives a reasonable spectral function for the single-particle Green's function in comparison with the results of the exact diagonalization for finite clusters and the quantum Monte Carlo method. We also show that it gives the ground state energy and the sublattice magnetization comparable to previous variational Monte Carlo results.
The Hubbard I decoupling approximation [J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963)] is extended by introducing a k-dependent self-energy which describes a mobile correlation hole of opposite spin propagating along with the electron in its way around the lattice. The theory, despite its simplicity, (1) becomes exact in the strong-coupling limit; and (2) reproduces with great accuracy the ground-state energy, the double occupancy, the on-site correlation functions, and the effective hopping for all the values of the Hubbard Coulomb repulsion U. It also gives reasonable spectral functions leading to (3) a single-particle Green function showing the well-known spectral-weight transfer between the Hubbard bands as soon as the electron density deviates from half-filling, and (4) a momentum distribution as well as magnetic and charge structure factors in qualitative agreement with quantum Monte Carlo simulations, slave-boson calculations, and conserving approximations like fluctuation exchange and parquet. On the other hand, as in most decoupling schemes, liftime effects and the near-Fermi-level additional structure characteristic of strongly correlated electron systems are missing.
A three-body scattering theory previously proposed by one of the present authors is developed to be applied to the saturated ferromagnetic state in the two-dimensional Hubbard model. The single-particle Green's function is calculated by taking account of the multiple scattering between two electrons and one hole. Several limiting cases are discussed and the relation to the variational principle is examined. The importance of the three-body correlation is demonstrated in comparison with the results of the ladder approximation. A possible phase boundary for the Nagaoka ground state is presented for the square lattice, which improves the previous variational results.
The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable of the many-body problem. A new approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ is proposed which is based on its scaling properties and which recovers exactly the limit of strong electron correlations at half-band filling. In this way, a more accurate description of W is obtained throughout the domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}},$ including the crossover from weak to strong correlations. As examples of applications results are given for the ground-state energy, charge-excitation gap, and charge susceptibility of the Hubbard model in one-, two-, and three-dimensional lattices. The performance of the method is demonstrated by comparison with available exact solutions, with numerical calculations, and with LDFT using a simpler dimer ansatz for W. Goals and limitations of the different approximations are discussed.
An exact expression for the Green's function G(k,\ensuremath{\omega}) of one hole in the U\ensuremath{\rightarrow}\ensuremath{\infty} one-dimensional Hubbard model is derived from the known Bethe-ansatz solution of the model. Due to the spin-charge decoupling of the one-hole excitation spectrum and the Fermi-like statistics of the spin excitations, G shows branch-cut singularities at \ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}2 sink and a nontrivial dependence on the momentum of the hole. As a result, the hole can propagate although the quasiparticle weight vanishes.
We analyse the behaviour of correlated electrons described by Hubbard-like models at intermediate coupling. We argue that with increasing interaction a pole in a generic two-particle Green function is approached. The pole signals condensation of electron-hole pairs and a metal-insulator transition at half-filling. The two-particle singularity calls for a sophisticated renormalization of the interaction strength. A self-consistent diagrammatic technique with renormalized two-particle Green functions is developed. The theory is based on a linked-cluster expansion for the thermodynamic potential with electron-electron interaction as the propagator. The simplest theory with full vertex renormalization, summing self-consistently multiple scatterings from two electron-hole channels, is proposed. We obtain an approximation with a generating functional in closed form enabling us to handle appropriately singularities in two-particle Green functions. The approximation is shown to be asymptotically exact in an external magnetic field close to the fully polarized ferromagnetic state at half-filling and zero temperature.