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METALLIC STATES IN THE HUBBARD-MODEL AT HALF-FILLING

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Published Sep 10, 2026• 0 citations
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Abstract

Using second-order perturbation theory in U for the self-energy, we solve the self-consistent equations for the local Green function of the infinite-dimensional Hubbard model. We consider the cases of Gaussian and semicircular free densities of states (DOS), and calculate the spectral densities rho(omega) for various U. In the semicircular case we investigate the existence of a Mon transition at half-filling. Even for large U we obtain a metallic solution. Treating the free Dos as a variable parameter, the method also gives an approximate solution of the finite-dimensional Hubbard model. We consider the case of a (regularized) logarithmic Dos appropriate for the two-dimensional case.

DOI
10.82444/warw.33545779

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Using second-order perturbation theory in U for the self-energy, we solve the self-consistent equations for the local Green function of the infinite-dimensional Hubbard model. We consider the cases of Gaussian and semicircular free densities of states (DOS), and calculate the spectral densities rho(omega) for various U. In the semicircular case we investigate the existence of a Mon transition at half-filling. Even for large U we obtain a metallic solution. Treating the free Dos as a variable parameter, the method also gives an approximate solution of the finite-dimensional Hubbard model. We consider the case of a (regularized) logarithmic Dos appropriate for the two-dimensional case.
Using second-order perturbation theory in U for the self-energy, we solve the self-consistent equations for the local Green function of the infinite-dimensional Hubbard model. We consider the cases of Gaussian and semicircular free densities of states (DOS), and calculate the spectral densities rho ( omega ) for various U. In the semicircular case we investigate the existence of a Mott transition at half-filling. Even for large U we obtain a metallic solution. Treating the free DOS as a variable parameter, the method also gives an approximate solution of the finite-dimensional Hubbard model. We consider the case of a (regularized) logarithmic DOS appropriate for the two-dimensional case.
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 density of states (DOS) of the two-dimensional Hubbard model is calculated in second-order perturbation with respect to the on-site Coulomb interaction U . The direct analytic continuation formula is presented to calculate the self-energy on the real-frequency axis. The momentum as well as the energy integral in the self-energy is seriously carried out with use of the Fast-Fourier-Transformation algorithm. Near the half-filling, the imaginary part of the self-energy becomes proportional to the energy in the vicinity of the Fermi level in our weak-coupling calculation. In the interacting case, the van Hove singularity (VHS) in DOS is smeared by the self-energy effect, but there still exists a sharp peak structure due to VHS.
Antiferromagnetic and metal to insulator transitions in the ground state of two-dimensional half-filled and nearly half-filled Hubbard models with the nearest neighbor and the second nearest neighbor transfers (- t and t ′ , respectively) are studied by using the Green's function approach with the self-energy up to the second order in the coupling constant U . For a choice of the parameter value t ′ / t =0.2 the half-filled model undergoes the first order phase transition between paramagnetic metallic (PM) phase and antiferromagnetic insulator (AFI) phase at U / t =4.0 while the first order transition between PM and antiferromagnetic metallic (AFM) phases takes place for nearly half-filled band. The one-electron spectral density is calculated for the PM, AFM, and AFI phases near the phase boundaries. In the PM phase the quasi-particle dispersion indicates the band narrowing from the Hartree-Fock results and the flat band behavior as was indicated by previous stadies and was observed in high- T c cuprates. The quasi-particle dispersion in the AFM phase is seen to be close to the Hartree-Fock results.
We study the Hubbard model in dimension d= infinity , within an approximation that simultaneously reproduces the atomic limit and the weak-coupling limit up to second order in the interaction strength U. This method, which is a slightly modified version of an approximation suggested by Edwards and Hertz, is applied to the calculation of the self-energy and of the spectral function in the paramagnetic regime for different U, band fillings n, and temperatures T. We find Fermi-liquid behaviour for small U and (for half-filling n=0.5) a metal-insulator (Mott) transition at a critical value U cr , but also a metallic non-Fermi-liquid phase for intermediate values U<U cr . We compare our results with results obtained within the second-order perturbation theory in U and (at finite temperature) with the (essentially exact) quantum Monte Carlo (QMC) results. Qualitatively, the approximation reproduces the correct tendencies obtained by QMC, but in detail there are still discrepancies, in particular for intermediate values of U.
We study the metal-insulator transition of the d-dimensional Hubbard model by treating the hopping term between adjacent 1-d chains in the frame of a Clifford linearization scheme, and keeping the full model along the chains. A general equation for the critical point is worked out in terms of the correlation functions of the one-dimensional model, assuming that the transition is of the second order. The equation, which holds at any temperature, is here investigated especially at T=0, where the latter condition holds true. The solution shows the existence of an insulating ground state in any d only at half-filling for U strictly positive, as in the exact 1-d case. The transition is found to be related to a parameter reminiscent of entropy.