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arXiv:cond-mat/9902221 · Journal of the Physical Society of Japan

Electron Correlations in the Nagaoka State on the Square Lattice -Three-Body Correlations-

Jun-ichi Igarashi, Manabu Takahashi, Tatsuya Nagao

Published Nov 15, 1999• 8 citations
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Abstract

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.

DOI
10.1143/jpsj.68.3682

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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.
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.
Electron correlation is studied in the special case of a single reversed spin in an otherwise fully aligned band in the one-dimensional Hubbard model for the purpose of examining our previous theory which exactly takes account of the three-body correlation. The importance of the three-body correlation is demonstrated through a comparison of our results with those of exact theories. The ground-state energy and the single particle spectra are in good agreement with McGuire-Lieb-Wu's exact solution and also with exact numerical solutions for finite size systems over a wide range of the intraatomic Coulomb integral and the electron number. The properties of the ground state are studied in detail through the calculation of pair correlation functions.
We investigate the instability of the saturated ferromagnetic ground state (Nagaoka state) in the Hubbard model on various lattices in dimensions d=2d=2d=2 and d=3d=3d=3. A variational resolvent approach is developed for the Nagaoka instability both for $U=\ensuremath{\infty}$ and for $U<\ensuremath{\infty}$ which can easily be evaluated in the thermodynamic limit on all common lattices. Our results significantly improve former variational bounds for a possible Nagaoka regime in the ground-state phase diagram of the Hubbard model. We show that a pronounced particle-hole asymmetry in the density of states and a diverging density of states at the lower band edge are the most important features in order to stabilize Nagaoka ferromagnetism, particularly in the low-density limit.
By introducing multiple-site correlation functions, we propose a hierarchical Green’s function approach, and apply it to study the characteristic properties of a two-dimensional (2D) square lattice Hubbard model by solving the equation of motions of a one-particle Green’s function and related multiple-site correlation functions. Under a cut-off approximation and taking the Fourier representation of multiple-site correlation functions, we obtain an analytical expression of one-particle Green’s function with static correlation functions. Then we calculate the spectral density function of electrons, and obtain that besides two main peaks corresponding to the lower and upper Hubbard bands in the spectral density function, there emerge some novel states between these two main peaks, and the total spectral weight of these emerged states is proportional to the hole doping concentration [Formula: see text]. Meanwhile, there also emerge some collective modes related to possible charge/spin density wave and/or electronic pairing density wave ordering states. This calculation is completely consistent with the spectroscopy observations of the cuprate superconductors in normal states. On the other hand, the appearance of the static correlation functions in the one-particle Green’s function can be used to describe the intertwined orders observed in the normal state of the cuprate superconductors.
Abstract The instability of the fully polarized ferromagnetic ground state (Nagaoka state) of the Hubbard model on the square lattice is investigated. We use single spin flip variational wave functions including majority spin correlation effects and calculate spin flip energies in the thermodynamic limit. With very local wave functions and with moderate numbers of variational parameters we reproduce the best known estimate for the critical hole density δcr = 0.29 and we obtain an estimate of Ucr = 63 t for the critical coupling which is considerably better than the best estimate of Ucr = 42 t previously known. The simplicity of our wave functions makes the physical origin of the various aspects of the instability particularly transparent.