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arXiv:1501.03587 · physica status solidi (b)

The Hubbard model in the strong coupling theory at arbitrary filling

A. Sherman

Published May 8, 2015• 7 citations
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Abstract

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.

DOI
10.1002/pssb.201552026

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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.
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.
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 three-dimensional repulsive Hubbard model is investigated using the strong-coupling diagram technique. For half-filling, the boundary between paramagnetic and antiferromagnetic states is determined for the range of the Hubbard repulsion 4t≤U≤12t4t\leq U\leq12t4t≤U≤12t, where ttt is the hopping integral between neighboring sites. Along this boundary, the density of states is calculated, and it demonstrates the Mott transition at U≈9tU\approx9tU≈9t. For U≥6tU\geq6tU≥6t and half-filling, the density of states has the shape inherent in the strong electron repulsion, while for U=4tU=4tU=4t, its shape points to weak coupling. The dependence of the Néel temperature TNT_{\rm N}TN​ on the electron concentration nˉ\bar{n}nˉ is investigated for the cases U=4tU=4tU=4t and 12t12t12t. In the former case, TNT_{\rm N}TN​ decreases monotonously with nˉ\bar{n}nˉ, while in the latter case, there is a plateau in the dependence near nˉ=0.87\bar{n}=0.87nˉ=0.87. The plateau is connected to a reconstruction of the density of states caused by an effective weakening of electron coupling due to electron depopulation. For U=12tU=12tU=12t and half-filling, the magnetic critical exponent γ≈1.4γ\approx1.4γ≈1.4, which is close to the value in the Heisenberg model. Some features resembling the first-order phase transition, revealing themselves in a finite crystal, are discussed.