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Physical review. B, Condensed matter

Spectral properties of the large-negative-U Hubbard model

Henk J. Eskes, Andrzej M. Oleś

Published Jan 15, 1997• 7 citations
Publisher

Abstract

Using the formalism introduced by Harris and Lange [Phys. Rev. 157, 295 (1967)] explicit expressions for partial sum rules of the individual Hubbard bands in the large-negative-U limit are derived. The one-particle spectrum, optical spectrum, as well as charge- and spin-response functions are considered. The approach gives a transparent description of the main features of these spectra. The main sum rules depend on only two independent nearest-neighbor expectation values which are estimated in the ground state using numerical calculations, exact results, and linear-spin-wave theory. Simple expressions for the intensities of the upper and lower Hubbard bands in the optical conductivity show that this spectrum is extremely sensitive on details of the ground state. The charge-density wave and superconducting phases are clearly distinguishable and even the transition to the broken-symmetry state may have a detectable influence on the conductivity.

DOI
10.1103/physrevb.55.2032

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Physical review. B, Condensed matter•Feb 1, 1992•17 citations
Using the formalism introduced by Harris and Lange [Phys. Rev. 157, 295 (1967)] explicit expressions for partial sum rules of the individual Hubbard bands in the large-negative-U limit are derived. The one-particle spectrum, optical spectrum, as well as charge- and spin-response functions are considered. The approach gives a transparent description of the main features of these spectra. The main sum rules depend on only two independent nearest-neighbor expectation values which are estimated in the ground state using numerical calculations, exact results, and linear-spin-wave theory. Simple expressions for the intensities of the upper and lower Hubbard bands in the optical conductivity show that this spectrum is extremely sensitive on details of the ground state. The charge-density wave and superconducting phases are clearly distinguishable and even the transition to the broken-symmetry state may have a detectable influence on the conductivity.
Using strong-coupling perturbation theory for the Hubbard model, explicit expressions are obtained for the integrated weights, energy positions, and widths of the upper and lower Hubbard bands separately, both for the optical and photoemission spectrum. In one dimension all expressions can be explicitly evaluated using the large-U Bethe-ansatz wave function. The k\ensuremath{\rightarrow}-dependent moments for the one-particle spectrum are compared with the closely related two-pole ansatz. The strong momentum dependence of the spectra demonstrates the importance of the k\ensuremath{\rightarrow}-dependent correlation functions in the spectral moments, which are often neglected. In order to estimate corrections due to the neglect of higher-order terms (in t/U), all results are compared with numerical-diagonalization data for intermediate-U values. The sum-rule results show a rapid spectral-weight transfer from the high- to the low-energy regime upon doping, similar to what is observed experimentally in one-particle and optical spectra of Cu (high-Tc{\mathit{T}}_{\mathit{c}}Tc​) and Ni compounds. Such fast weight redistributions as a function of the carrier density are a natural consequence of the strong correlations in these materials. The redistribution of intensity in the optical conductivity is connected with the dilution of the spin system by the added holes. For the one-particle spectrum the decrease of weight in the upper Hubbard band away from half filling can be understood by state counting at U=\ensuremath{\infty}, and the effect is strongly enhanced for small doping by both the first- and second-order terms in t/U. Altogether, the one-particle and optical spectra show the importance of the three-site hopping term. Therefore the t-J model does not represent well the spectral properties of the Hubbard model at large U. The local spin order is essential and determines both the k\ensuremath{\rightarrow} dependence of the one-particle spectrum and the intensities in the optical spectrum.
We study the negative-U Hubbard model by exact diagonalization of a 4\ifmmode\times\else\texttimes\fi{}4 cluster with different numbers of particles. We calculate the ground-state energy and wave function as well as the one-particle, charge- and spin-excitation spectra. We show that the BCS approximation gives results in good agreement with the exact ones for the ground-state energy and wave function in the whole range of U. In the one-particle excitation spectrum the BCS dispersion relation gives the position of the most intense peaks of the exact calculation. Lower-energy (quasiparticle) excitations, which are interpreted as strongly dressed Bogoliubov quasiparticles, also appear. The occurrence of pair-breaking and collective excitations in the charge- and spin-excitation spectra and the interplay between one-particle and collective modes are discussed.
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.
We calculate the single-particle spectral function for the Hubbard model within the framework of a projection technique equivalent to the two-pole approximation. We show that the two-pole approximation can be well understood as an average characterization of the upper and the lower Hubbard bands, rather than describing a dispersive quasiparticle. By comparing with numerical spectra of finite Hubbard rings and of a 4\ifmmode\times\else\texttimes\fi{}4 cluster [P. W. Leung et al., Phys. Rev. B 46, 11 779 (1992)], we show that the present approximation is capable of reproducing essential properties of the single-particle spectral function. In particular, the two-pole spectrum is characterized by a direct gap, in agreement with the exact spectrum. We emphasize the role of local antiferromagnetic correlations.
Abstract. In this paper we derive general expressions for few-electron spectral functions of the one-dimensional Hubbard model for values of the excitation energy in the vicinity of the M th upper-Hubbard band lower limit. Here M = 1, 2,... is the rotated-electron double occupation, which vanishes for the ground state and is a good quantum number for all values of the on-site Coulomb repulsion U. Our studies rely on a combination of symmetries of the model with a recent finite-energy holon and spinon description of the quantum problem. We apply our general scheme to the one-electron addition spectral function, dynamical structure factor, and spin singlet Cooper pair addition spectral function. Our results provide physically interesting information about the finite-energy spectral properties of the many-electron one-dimensional quantum liquid. PACS numbers: 71.10.Fd, 64.60.Fr, 11.30.-j, 71.10.PmOne and Two-Electron Spectral Function Expressions... 2 1.
A variational estimate for the spin-wave velocity of the one-band Hubbard model on a square lattice in the spin-density-wave regime is studied. The estimate is given by the ratio of the f-sum rule to the static structure factor of the transverse-spin response function. The known results for the Heisenberg model are used to obtain results for those quantities in the large-U limit of the Hubbard model. The f-sum rule and static structure factor are calculated using the random-phase approximation (RPA). The spin-wave velocity calculated in the RPA, in the spin-density wave ground state, violates the variational bound.