The sum rule for the moments of the spectral density is discussed for the single-band Hubbard model. It is shown that respecting the sum rule up to the order m = 3 is conceptually important for a qualitatively correct description of the quasi-particle band structure in the strong-correlation regime. Different analytical approximations for the self-energy are analyzed with respect to their compatibility with the moment sum rule. To estimate the practical usefulness of the sum rule, correlation functions and dynamical quantities are determined. The results obtained within the various approximation schemes of different complexity are compared with each other and also with essentially exact results available for infinite-dimensional lattices. It turns out that the m = 3 moment is rather unimportant for the paramagnetic phase on the hyper-cubic lattice. Contrary, it decisively influences the magnetic phase boundary as well as the critical temperature for the ferromagnetic phase on an f.c.c.-type lattice.
The sum rule for the moments of the spectral density is discussed for the single-band Hubbard model. It is shown that respecting the sum rule up to the order m = 3 is conceptually important for a qualitatively correct description of the quasi-particle band structure in the strong-correlation regime. Different analytical approximations for the self-energy are analyzed with respect to their compatibility with the moment sum rule. To estimate the practical usefulness of the sum rule, correlation functions and dynamical quantities are determined. The results obtained within the various approximation schemes of different complexity are compared with each other and also with essentially exact results available for infinite-dimensional lattices. It turns out that the m = 3 moment is rather unimportant for the paramagnetic phase on the hyper-cubic lattice. Contrary, it decisively influences the magnetic phase boundary as well as the critical temperature for the ferromagnetic phase on an f.c.c.-type lattice.
Exact relations are derived for the Fermi Hubbard spectral weight function for infinite U at zero temperature in the thermodynamic limit for any dimension and any lattice structure. These relations involve moments of the spectral weight function but differ from similar work by (a) restricting the moments over the interesting low energy (lower Hubbard band) spectrum and (b) without any of the usual approximations (e.g. decoupling) for the requisite higher order correlation functions. The latter are determined exactly as explicit functions of the single particle correlation function in the infinite U limit. The methods behind these results have the additional virtue of simplicity - cumbersome, but entirely straightforward. In a companion paper, the relations offer rich insights into the model behavior and offer severe tests - a computer lab - of any proposed solution to the Hubbard model in the strong correlation (large finite U) region. As the electron density approaches one electron per site, then the correlation functions become local, so are trivial to compute exactly. In that limit and for the paramagnetic case, the second central moment of the spectral weight function is not zero, but quite large.
Exact relations are derived for the Fermi Hubbard spectral weight function for infinite U at zero temperature in the thermodynamic limit for any dimension,any lattice structure and general hopping matrix. These relations involve moments of the spectral weight function but differ from similar work by (a) restricting the moments over the interesting low energy (lower Hubbard band) spectrum and (b) without any of the usual approximations (e.g. decoupling) for the requisite higher order correlation functions. The latter are determined exactly as explicit functions of the single particle correlation function in the infinite U limit. The methods behind these results have the additional virtue of simplicity - cumbersome, but entirely straightforward. In a companion paper, the relations offer rich insights into the model behavior and offer severe tests - a computer lab - of any proposed solution to the Hubbard model in the strong correlation (large finite U) region. As the electron density approaches one electron per site, then the correlation functions become local, so are trivial to compute exactly. In that limit and for the paramagnetic case, the second central moment of the spectral weight function is not zero, but quite large.
A companion paper derived exact relations for the Fermi Hubbard spectral weight function for infinite U at zero temperature in the thermodynamic limit for any dimension and any lattice structure. These relations involve moments of the spectral weight function but differ from earlier work by (a) restricting the moments over the interesting low energy (lower Hubbard band) spectrum and (b) without any of the usual approximations (e.g. decoupling) for the requisite higher order correlation functions. The latter are determined exactly as explicit functions of the single particle equal time correlation function in the infinite U limit. Here the relations offer rich insights into the model behavior and offer exacting tests - a computer lab - of any proposed solution to the Hubbard model in the strong correlation (large finite U) region. Specific tests are applied to mean field theory solutions to the Hubbard model as well as some one dimensional results. In each case, substantial problems are exposed. The relations are also used to determine a best(moment-preserving) solution subject only to the one assumption of a dominant single peak in the lower Hubbard band spectral weight function. As shown in the companion paper, the second central moment of the lower Hubbard band spectral weight function is not zero, as might be expected, as the electron density approaches one electron per site (paramagnetic case). That moment is large, offering a significant test for any approximate solution.
In the general frame of a modified many-body perturbation theory, the single-band Hubbard model is treated in the limit of weak electron-electron interaction. In this modification the decomposition of the Hamiltonian into an effective free system and an effective perturbation is rearranged iteratively in each order of the formalism, resulting in a successive reduction of the interaction part. In comparison with conventional perturbation techniques, this means an extended range of validity. Imposing thermodynamic self-consistency, we calculate the wave-vector-dependent electronic self-energy up to second order and analyze the influence of the Coulomb correlations. Several well-known physical properties of the Hubbard model are recovered in characteristic single-particle quantities. Therefore, the effect of the band occupation n and the electron-electron repulsion U on spectral densities, band structures, densities of states, magnetization, etc. is investigated in detail. Some of the most remarkable results are: first, a splitting of the free Bloch band into two quasiparticle subbands when the coupling strength reaches moderate values. These subbands are linked by a small damping-induced background in the density of states and are separated roughly by an energy of the order of U. In addition, the upper subband is shifted to higher energies in the spectrum proportionally to U while the lower one is nearly fixed. Second, we found a paramagnetic-ferromagnetic phase diagram that has a critical coupling strength Uc and exhibits strong indications of a lower critical occupation number nc. There is no spontaneous magnetization in the system for parameters smaller than these values. Compared with the Stoner model, one thus obtains striking improvements. In many cases our conclusions for the weak-coupling limit are qualitatively in good agreement with well-known results in the strong-correlation regime.
We explore the effect of the self-energy, , having a single pole, , with spectral weight and quasi-particle lifetime , on the density of states. We obtain the set of parameters , , and by means of the moment approach (exact sum rules) of Nolting. Due to our choice of self-energy, the system is not a Fermi liquid for any value of the interaction, a result which also holds in the moment approach of Nolting without lifetime effects. Our self-energy satisfies the Kramers - Kronig relationships since it is analytic in one of the complex half-planes. By increasing the value of the local interaction, , at half-filling , there is a transition from a paramagnetic metal to a paramagnetic insulator (a Mott metal - insulator transition) for values of of the order of ( W is the bandwidth) which is in agreement with numerical results for finite lattices and for an infinite number of dimensions . These results expose the main weakness of the spherical approximation of Nolting: a finite gap for any finite value of the interaction, i.e., an insulator for any finite value of . Lifetime effects are absolutely indispensable to making our scheme work better than that based on improving the narrowing band factor, , beyond that obtained from the spherical approximation of Nolting.
We study the possibility and stability of band ferromagnetism in the single-band Hubbard model for the simple cubic (sc) lattice. A nonlocal self-energy is derived within a modified perturbation theory. Results for the spectral density and quasiparticle density of states are shown with special attention to the effects of k dependence. The importance of nonlocal correlations for the fulfillment of the Mermin-Wagner theorem is our main result. A phase diagram showing regions of ferromagnetic order is calculated for the three-dimensional lattice. Besides, we show results for the optical conductivity and prove that the renormalized one-loop contribution to the conductivity already cancels the Drude peak exactly in case of a local self-energy which is not true anymore for a nonlocal self-energy.