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arXiv:cond-mat/0601331 · Physical Review B

One-loop approximation for the Hubbard model

A. Sherman

Published Apr 7, 2006• 29 citations
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Abstract

The diagram technique for the one-band Hubbard model is formulated for the case of moderate to strong Hubbard repulsion. The expansion in powers of the hopping constant is expressed in terms of site cumulants of electron creation and annihilation operators. For the Green's function, an equation of the Larkin type is derived and solved in a one-loop approximation for the case of two dimensions, nearest-neighbor hopping, and half-filling. The obtained four-band structure of the spectrum and the shapes of the spectral function are close to those observed in Monte Carlo calculations. The four-band structure has its origin in the regions of large damping which separate the bands. It is shown that the maxima forming the bands are of a dissimilar origin in different regions of the Brillouin zone.

DOI
10.1103/physrevb.73.155105

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The diagram technique for the one-band Hubbard model is formulated for the case of moderate to strong Hubbard repulsion. The expansion in powers of the hopping constant is expressed in terms of site cumulants of electron creation and annihilation operators. For the Green's function, an equation of the Larkin type is derived and solved in a one-loop approximation for the case of two dimensions, nearest-neighbor hopping, and half-filling. The obtained four-band structure of the spectrum and the shapes of the spectral function are close to those observed in Monte Carlo calculations. The four-band structure has its origin in the regions of large damping which separate the bands. It is shown that the maxima forming the bands are of a dissimilar origin in different regions of the Brillouin zone.
The diagram technique for the one-band Hubbard model is formulated for the case of moderate to strong Hubbard repulsion.The expansion in powers of the hopping constant is expressed in terms of site cumulants of electron creation and annihilation operators.For Green's function an equation of the Larkin type is derived and solved in a one-loop approximation for the case of two dimensions and nearest-neighbor hopping.With decreasing the electron concentration in addition to the four bands observed at half-filling, a narrow band arises near the Fermi level.The dispersion of the new band, its bandwidth and the variation with filling are close to those of the spin-polaron band in the t-J model.
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.
The equation for the electron Green's function of the fermionic Hubbard model, derived using the strong coupling diagram technique, is solved self-consistently for the near-neighbor form of the kinetic energy and for half-filling. In this case the Mott transition occurs at the Hubbard repulsion Uc ≈ 6.96t, where t is the hopping constant. The calculated spectral functions, density of states (DOS) and momentum distribution are compared with results of Monte Carlo simulations. A satisfactory agreement was found for U > Uc and for temperatures, at which magnetic ordering and spin correlations are suppressed. For U < Uc and lower temperatures the theory describes qualitatively correctly the positions and widths of spectral continua, variations of spectral shapes and occupation numbers with changing wave vector and repulsion. The locations of spectral maxima turn out to be close to the positions of δ-function peaks in the Hubbard-I approximation.
Electronic states of a two-dimensional Hubbard model near half filling is investigated by use of the composite-operator method, which is improved by including two-site correlations in combined use with the resolvent method. The densities of states obtained for the upper and lower Hubbard bands have three-peek structures at half filling and give shapes above a certain temperature similar to the ones obtained in the numerical simulation by Dagotto, Ortolani, and Scalapino [Phys. Rev. B 46, 3183 (1992)], but show narrowing of the bands at low temperature due to the formation of spin-singlet states among the nearest-neighbor electrons. A doped case shows a coherent peak at the Fermi level and the feature of the density of states is very similar to the one from numerical simulation. The one-particle spectra in the (k,k) direction is also presented. The results show that, at half filling, the spectral weight is not evenly distributed between two bands; most of the weight of the lower Hubbard band appears between (0,0) and (\ensuremath{\pi}/2a,\ensuremath{\pi}/2a), while that of the upper Hubbard band lies in the region between (\ensuremath{\pi}/2a,\ensuremath{\pi}/2a) and (\ensuremath{\pi}/a,\ensuremath{\pi}/a), keeping roughly the original band distribution. With hole doping, a flat band develops at the Fermi level with a large Fermi surface and the lower Hubbard band extends through the whole Brillouin zone.
The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].