Abstract We have calculated the ground state energy of the Hubbard model in the approximation of Hubbard's first paper1 . For the neutral model with nearest neighbour interaction the energy resulting from the selfconsistent paramagnetic solution is compared with those ones following from the (ferromagnetic) Hartree-Fock and an (antiferromagnetic) single particle theory. The energy of the latter one turns out to be the best approximation of the true ground state energy of the model for all values of the coupling constant V0 , but the energy derived from Hubbard's approximation, in spite of the absence of magnetic ordering, is a reasonable approximation at least for sufficiently large values of V0.
Abstract We have calculated the ground state energy of the Hubbard model in the approximation of Hubbard's first paper1 . For the neutral model with nearest neighbour interaction the energy resulting from the selfconsistent paramagnetic solution is compared with those ones following from the (ferromagnetic) Hartree-Fock and an (antiferromagnetic) single particle theory. The energy of the latter one turns out to be the best approximation of the true ground state energy of the model for all values of the coupling constant V0 , but the energy derived from Hubbard's approximation, in spite of the absence of magnetic ordering, is a reasonable approximation at least for sufficiently large values of V0.
The variational approach initiated by Gutzwiller is extended to include antiferromagnetism. A simplified assumption corresponding to that adopted by him is used in order to evaluate the ground state energy of the Hubbard model and to draw the phase diagram similar to those constructed by Penn on the basis of the Hartree-Fock approximation. It is found that the antiferromagnetic state in the phase diagram is restricted to a narrow region bounded by threshold values for both the coupling constant and electron density. For sufficiently large intraatomic repulsion paramagnetic state becomes more stable than antiferromagnetic state except for the case of one electron per site. The phase transition between paramagnetic and antiferromagnetic states is of first order and the coexistence of these two states with different density occurs.
We discuss, in connection with the problem of the ground state in the Hubbard model with U =∞, the normal (nonmagnetic) N -state of a system over the entire range of electron concentrations n ≤1. It is found that in a one-particle approximation, e.g., in the generalized Hartree-Fock approximation, the energy ε 0 ( n ) of the N -state is lower than the energy ε FM ( n ) of a saturated ferromagnetic state for all values of n . Using the random phase approximation we calculate the dynamical magnetic susceptibility and show that the N -state is stable for all values of n . A formally exact representation is derived for the mass operator of the one-particle electron Green’s function, and its expression in the self-consistent Born approximation is obtained. We discuss the first Born approximation and show that when correlations are taken into account, the attenuation vanishes on the Fermi surface and the electron distribution function at T =0 acquires a Migdal discontinuity, whose magnitude depends on n . The energy of the N -state in this approximation is still lower than ε FM ( n ) for n <1. We show that the spin correlation functions are isotropic, which is a characteristic feature of the singlet states of the system. We calculate the spin correlation function for the nearest neighbors in the zeroth approximation as a function of n . Finally, we conclude that the singlet state of the system in the thermodynamic limit is the ground state.
We prove rigorously that the Hartree-Fock approximation to the ground state of the d-dimensional Hubbard model leads to saturated ferromagnetism when the particle density (more precisely, the chemical potential µ) is small and the coupling constant U is large, but finite. As in the usual Hartree-Fock theory we restrict attention to Slater determinants that are eigenvectors of the z-component of the total spin, Sz = ∑ x nx, ↑ − nx,↓, and we find that the choice 2Sz = N = particle number gives the lowest energy at fixed 0 < µ < 4d. 1 BLT, Version of 27-Jun-2005 2 1
As a contribution to the study of the Hartree–Fock theory, we prove rigorously that the Hartree–Fock approximation to the ground state of the d-dimensional Hubbard model leads to saturated ferromagnetism when the particle density (more precisely, the chemical potential μ) is small and the coupling constant U is large, but finite. This ferromagnetism contradicts the known fact that there is no magnetization at low density, for any U, and thus shows that HF theory is wrong in this case. As in the usual Hartree–Fock theory, we restrict attention to Slater determinants that are eigenvectors of the z-component of the total spin, 𝕊z = ∑x nx,↑ - nx,↓, and we find that the choice 2𝕊z = N = particle number gives the lowest energy at fixed 0 < μ < 4d.
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.
The Hubbard model was originally introduced as the simplest system which may exhibit an insulating (Mott) state. This state is the result of strong electron–electron interactions. In this chapter we consider the Hubbard model at half-filling. The main goal here is the study of the magnetic properties of its ground state. Apart from an exact solution in one dimension, no exact results are available for this problem. This leads to the use of several approximations. The most popular one, and the oldest, is the mean-field theory (MFT). In the MFT one has the bias that the ground state does have some sort of magnetic order (i.e. ferromagnetic, Néel antiferromagnetic, etc.). The problem is then usually solved by means of a variational ansatz. However, one is usually interested in more than just the ground-state energy, which, after all, is not directly measurable and depends very sensitively on the properties at short distances. Most often we wish to evaluate the long-distance, low-frequency, properties of the correlation and response functions of this theory. Moreover, in some cases, such as in one dimension, the fluctuations overwhelm the MFT predictions. In this chapter we will consider the standard MFT (i.e. Hartree–Fock), which is expected to become accurate at weak coupling. We will consider both ferromagnetic and antiferromagnetic states. We will also rederive these results using path integrals. As a byproduct, we will also have a theory of the fluctuations: the non-linear sigma model.