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physica status solidi (b)

On the density of states at the fermi level for normal systems

Ulrich Behn, H. Zinke

Published Dec 1, 1977• 2 citations
Publisher

Abstract

Abstract In the single‐site and the nearest‐neighbour approximation, respectively, the density of electronic states at the Fermi level for normal systems at T = 0 is considered as a functional of the unperturbed density of states. For electron–hole symmetric systems, e.g. the half‐filled‐band Hubbard model, the density of states at the Fermi level in a normal single‐site approximation is either equal to the unperturbed value or vanishes. The expected gradual change in the density of states can be obtained in the nearest‐neighbour approximation.

DOI
10.1002/pssb.2220840229

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physica status solidi (b)•Mar 1, 1981•5 citations
Abstract In the single‐site and the nearest‐neighbour approximation, respectively, the density of electronic states at the Fermi level for normal systems at T = 0 is considered as a functional of the unperturbed density of states. For electron–hole symmetric systems, e.g. the half‐filled‐band Hubbard model, the density of states at the Fermi level in a normal single‐site approximation is either equal to the unperturbed value or vanishes. The expected gradual change in the density of states can be obtained in the nearest‐neighbour approximation.
We investigate the ground state (T = 0 K) of the one-dimensional symmetrical (n = 1) Hubbard model formalized in terms of the system of integral equations, which we previously obtained using the method of the generating functional of Green’s functions with the subsequent Legendre transformation. In a wide range of variations in the parameter of Coulomb interaction U, the following characteristics of the system have been calculated: the electron density of states, the electron band spectrum, the number of doubly occupied lattice sites, the localized magnetic moment, the correlator of the square of the longitudinal component of spin at a site,<S Z 2>, and the internal energy of the system. It has been shown that, for all U > 0, the model yields two solutions, i.e., an antiferromagnetic insulator and a paramagnetic insulator, in which there are no single-electron quasi-particles at the Fermi level. The energy of the paramagnetic solution in the region of U < 1.1 is considerably less than that of the antiferromagnetic solution for the case of U > 1.1, we have the opposite situation.
The problem of the ground state of the electronic system in the Hubbard model for U=∞ is discussed. The author investigates the normal (singlet or nonmagnetic) N state of the electronic system over the entire range of electron densities n⩽1. It is shown that the energy of the N state ɛ 0 (1) (n) in a one-particle approximation, such as (e.g.) the extended Hartree-Fock approximation, is lower than the energy of the saturated ferromagnetic FM state ɛ FM(n) for all n. The dynamic magnetic susceptibility is calculated in the random phase approximation, and it is shown that the N state is stable over the entire range of electron densities: The static susceptibility (ω=0) does not have a band singularity in the zero-wave vector limit q→0. A formally exact representation is obtained for the mass operator of the one-particle Green’s function, and an approximation of this operator is proposed: M k(E)⋍λF(E), where λ=n(1−n)/(1−n/2)z is the kinematic interaction parameter, z is the number of nearest neighbors, and F(E) is the total single-site Green’s function. For an elliptical density of states the integral equation for F(E) is solved exactly, ad it is shown that the spectral intensity rigorously satisfies the sum rule. The calculated energy of the strongly correlated N state ɛ 0(n)<ɛ FM(n) for all n, and in light of this relationship the author discusses the hypothesis that the ground state of the system is the normal (singlet) state in the thermodynamic limit. The electron distribution function at T=0 differs significantly from the Fermi step; it is “smeared” along the entire energy spectrum, and discontinuities do not occur in the region of the chemical potential m.
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.
Abstract In the framework of the Poisson approximation to a functional scheme for the Hubbard model the effect of the spatial correlation of the discrete stochastic potential Uvi(t) simulating the spin‐↓ subsystem is considered. For a half‐filled band in the paramagnetic case at zero temperature the density of states is calculated in a nearest‐neighbour approximation. For strong Coulomb repulsion the subbands occur sharper and the minimum in between is deeper compared with the single‐site approximation. The influence of the spatial correlation on the possible appearance of a gap within a dynamic approximation is discussed.