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Physical review. B, Condensed matter

Ground state of theU=∞ Hubbard model with infinite-range hopping

José Antonio Vergés, F. Guinea, J. Galán, P. G. J. van Dongen, G. Chiappe, Enrique Louis

Published Jun 1, 1994• 13 citations
Publisher

Abstract

The ground-state wave function and energy of the Hubbard model with unlimited hopping can be obtained for arbitrary fillings using the transformation recently introduced by Brandt and Giesekus. A simple linear behavior is obtained for the ground-state energy dependence on the number of particles: ${\mathit{E}}_{\mathrm{GS}}$=-2t(L-N), where -t is the hopping energy (t>0), L is the number of sites, and N is the number of electrons. The ground state shows a very large total spin degeneracy. The system is shown to behave as a Fermi liquid with a spectral weight Z=1-N/L for all fillings less than 1 (N\ensuremath{\le}L).

DOI
10.1103/physrevb.49.15400

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Physics of the Solid State•Feb 1, 1997•3 citations
The ground-state wave function and energy of the Hubbard model with unlimited hopping can be obtained for arbitrary fillings using the transformation recently introduced by Brandt and Giesekus. A simple linear behavior is obtained for the ground-state energy dependence on the number of particles: EGS{\mathit{E}}_{\mathrm{GS}}EGS​=-2t(L-N), where -t is the hopping energy (t>0), L is the number of sites, and N is the number of electrons. The ground state shows a very large total spin degeneracy. The system is shown to behave as a Fermi liquid with a spectral weight Z=1-N/L for all fillings less than 1 (N\ensuremath{\le}L).
We present the exact solution of the simplified Hubbard model in which only one kind of electrons can hop and this quantum mechanical hopping of electrons is assumed to be unconstrained. It is shown that the model still behaves non-trivially, although it no longer depends on the lattice structure and the dimensionality of the system. For this case we find: (i) a gap in the ground state energy always exists at the half-filled band point (n = 1), (ii) a preferred magnetic state at n = 1 and large U is a total spin singlet, (iii) U-dependence of the ground state energy has qualitatively the same form as one of the conventional Hubbard model with the (t2/U)-behavior at large U. A phase diagram of the model is discussed.
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 t - J model with constant t and J between any pair of sites is studied by exploiting the symmetry of the Hamiltonian with respect to site permutations. For a given number of electrons and a given total spin the exchange term simply yields an additive constant. Therefore the real problem is to diagonalize the ` t model', or equivalently the infinite U Hubbard Hamiltonian. Using extensively the properties of the permutation group, we are able to find explicitly both the energy eigenvalues and eigenstates, labelled according to spin quantum numbers and Young diagrams. As a corollary we also obtain the degenerate ground states of the finite U Hubbard model with infinite-range hopping -t>0 .
The ground-state properties of a simple model for transition metal oxide compounds in ferromagnetic phase have been studied. This model only contains the orbital degree of freedom. The two-orbital spinless Hubbard model in the limit of strong on-site coulomb interaction (U) has been used to describe the motion of election in e[subscript g] orbital. In the large-U and half-filling limits, the effective Hamilto-nian has been derived from the Hubbard model by the canonical transformation. After the transformation, the doubly occupied states have been projected out and the size of the Hillbert space has been reduced. For this effective Hamiltonian, the hopping amplitudes have been defined according to the managanite system which consists of three axes: x-, y- and z-axis. Since the ground-state properties of the effective Hamiltonian have been calculated only in one-dimension, then the results of this work have been separated in three cases; the electrons hopping in x, y, and z axes respectively. The ground-state wavefunction of this effective Hamiltionian has been calculated by infinite density matrix renormalization group algorithm. This algorithm leads to the ground state at the thermodynamic limit when the number of sties is large. It has been found that the ground-state energies are equivalent in all hopping axes. Furthermore, the nearest neighbor orbital correlations at the ground state for all axes are antiferro-orbital.
We analyze the ground-state properties of the U=\ensuremath{\infty} Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 cluster using a Lanczos technique. The quantum numbers of the ground-state, its momentum distribution, the magnetic structure factor, some 〈nˆk{\mathrm{\^{}}}_{\mathbf{k}}ˆk​n${\mathrm{\^{}}}_{\mathbf{k}\ensuremath{'}}$〉 dynamic correlation functions, and the overlap of the ground state with the noninteracting one (both before and after a Gutzwiller projection) are given for all fillings. As the number of electrons approaches half-filling the system undergoes a transition from a conventional ground state that overlaps appreciably with the noninteracting one to a new ground state that shows local ferromagnetic correlations and negligible overlap with the corresponding U=0 ground state. A detailed analysis of the wave function corresponding to the ground state of 14 electrons (two holes) proves the existence of a finite shift of the Fermi surface. Our results support the existence of a stable magnetic phase at small but finite hole doping. This new magnetic phase is locally ferromagnetic although the total spin is not maximum.
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.