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Europhysics Letters (EPL)

Quasi-Particles in the Strong-Coupling Limit of the 2D Hubbard Model

W. Stephan, K. J. von Szczepanski, M Ziegler, Peter Horsch

Published Apr 1, 1990• 7 citations
Publisher

Abstract

The single-particle Green's function for the strong-coupling limit of the single-band Hubbard model ( t-J model) at half-filling is calculated numerically using a Lanczos algorithm. Two-dimensional systems of up to 18 sites with periodic boundary conditions are examined, and evidence is presented for a coherent quasi-particle band of width J which emerges at the low-energy edge of the incoherent spectrum. We find the quasi-particle dispersion to be consistent with ε k = ( J /2)[cos ( k x ) + cos ( k y )] 2 , corresponding to effective next-nearest-neighbour hopping.

DOI
10.1209/0295-5075/11/7/016

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The single-particle Green's function for the strong-coupling limit of the single-band Hubbard model ( t-J model) at half-filling is calculated numerically using a Lanczos algorithm. Two-dimensional systems of up to 18 sites with periodic boundary conditions are examined, and evidence is presented for a coherent quasi-particle band of width J which emerges at the low-energy edge of the incoherent spectrum. We find the quasi-particle dispersion to be consistent with ε k = ( J /2)[cos ( k x ) + cos ( k y )] 2 , corresponding to effective next-nearest-neighbour hopping.
Results for the single-particle Green's function for the strong-coupling limit of the single-band Hubbard model ($t\ensuremath{-}J$ model) obtained by numerical and analytical methods are discussed. The numerical calculation of the spectral function is performed for small systems of up to 18 sites using the Lanczos method. A detailed analysis of the spectral functions for the $t\ensuremath{-}J$ model in one and two dimensions is given. We also investigate the effect of the next-nearest-neighbor hopping terms, which appear in order t2U\frac{{t}^{2}}{U}Ut2​ and are omitted in the standard $t\ensuremath{-}J$ model. The role of quantum fluctuations on the shape of the spectra and the fundamental difference between one and two dimensions is elucidated. In 2D we find evidence for a coherent quasiparticle band of width 2J2J2J with dispersion $^{\ensuremath{\omega}}\mathrm{k}\ensuremath{\sim}(\frac{J}{2}){(cos{k}_{x}+cos{k}_{y})}^{2}$ and spin S=12S=\frac{1}{2}S=21​, which emerges at the low-energy edge of the incoherent spectrum. This form of dispersion is due to effective next-nearest-neighbor processes related to spin flips and agrees with our extension of the approach of Brinkman and Rice to finite JJJ.
An analytic approach is presented of electronic one-particle spectra of the one-band Hubbard model at half filling in the antiferromagnetic phase. Starting from the strong-coupling regime U\ensuremath{\gg}:t, a projection technique is used to set up self-consistent coupled equations for the electron Green function, which are valid down to values U\ensuremath{\approx}t. The self-consistent equation for the hole propagator is a direct generalization of the one found from the t-J model. This gives further support to the ``string'' picture, where propagation of holes creates strings of overturned spins with which the holes interact. In the present work hopping of holes (or electrons) with up spin on the down sublattice is also taken into acount, as well as transitions between the lower and upper Hubbard bands. These are shown to change significantly the incoherent part of the t-J model spectra, by smearing out the shake-off peaks, reminiscent of higher bound string states due to multispin scattering. Coherent (quasiparticle) peaks exist at the band edges, on both sides of the insulating gap. We show that with decreasing U the quasiparticle concept loses its meaning for wave vectors at the center of the magnetic Brillouin zone (MBZ). For large values of U the dispersion of the quasiparticle is found to scale with its band width, which is of order J. Extrema are always found at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2). The weight of the quasiparticle at this k value decreases logarithmically with increasing U. In the strong-coupling limit the spectrum tends to be symmetric, i.e., to become an even function of the frequency around the chemical potential, for any wave vector. For small values of U the dispersion at the edge of the MBZ flattens away, as expected when approaching the noninteracting limit. The spectral function in this regime, for wave vectors away from the edge of the MBZ, is concentrated mainly on one side of the chemical potential.
We present an analytic solution of the self consistent random phase approximation equations for the half-filled Hubbard model, in the strong coupling limit. In the half-filled limit the Hubbard model shows antiferromagnetic ordering. The self energy is calculated by considering processes whereby the electrons emit and absorb low frequency spin waves, which are the Goldstone modes of the broken symmetry phase. We examine the electronic spectrum, which is dominated by the broad incoherent portions of the spectrum associated with the upper and lower Hubbard bands. The quasi-particle spectrum has a divergent wave function renormalization, indicating a divergent renormalization of the quasi-particle mass and zero quasi-particle weight, as in a non-Fermi liquid. The self consistency equation determined from the Goldstone modes in the susceptibility shows that the antiferromagnetism is prohibited in one and two dimensions but not in three. The magnetic ordering temperature is determined by dimensionality and the magnitude of the Anderson kinetic exchange of the effective Heisenberg Hamiltonian.
Abstract Equations for the electron Green's function of the two‐dimensional Hubbard model, derived using the strong coupling diagram technique, are self‐consistently solved for different electron concentrations n and tight‐binding dispersions. Comparison of spectral functions calculated for the ratio of Hubbard repulsion to the nearest neighbor hopping with Monte Carlo data shows not only qualitative, but in some cases quantitative agreement in position of maxima. General spectral shapes, their evolution with momentum and filling in the wide range are also similar. At half‐filling and for the next nearest neighbor hopping constant the Mott transition occurs at , where is the initial bandwidth. This value is close to those obtained in the cases of the semi‐elliptical density of states and for . In the case and the Mott gap reaches maximum width at , and it is larger than that at for half‐filling. In all considered cases positions of spectral maxima are close to those in the Hubbard‐I approximation.