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Physical review. B, Condensed matter

Collective modes in theU=∞ Hubbard model

WENJIE HUANG, J. W. Rasul

Published Feb 15, 1992• 8 citations
Publisher

Abstract

We investigate with the aid of the 1/N expansion the dispersion relation of the collective mode in the infinite-U Hubbard model in the presence of a long-range Coulomb term. We calculate this numerically in three dimensions and for a parabolic dispersion for the full range of fillings. For moderate fillings the results are consistent with Landau theory and weak-coupling approaches, but for fillings close to one-half the nature of the collective mode is completely changed. In this limit it becomes equivalent to a movement of the electron system as a whole, thereby avoiding the U=\ensuremath{\infty} restriction. This mode never decays into single-particle excitations, and the mass scale on which it appears is that of the free-electron system.

DOI
10.1103/physrevb.45.3995

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We investigate with the aid of the 1/N expansion the dispersion relation of the collective mode in the infinite-U Hubbard model in the presence of a long-range Coulomb term. We calculate this numerically in three dimensions and for a parabolic dispersion for the full range of fillings. For moderate fillings the results are consistent with Landau theory and weak-coupling approaches, but for fillings close to one-half the nature of the collective mode is completely changed. In this limit it becomes equivalent to a movement of the electron system as a whole, thereby avoiding the U=\ensuremath{\infty} restriction. This mode never decays into single-particle excitations, and the mass scale on which it appears is that of the free-electron system.
The auxiliary boson method and 1/N expansion are applied to the solution of the Hubbard model in the U = ∞\infty∞ limit, in functional integral form. The zero order term in the expansion gives the mean-field result of non-interacting enhanced mass quasiparticles, and the next order gives Gaussian fluctuations around the mean-field value, representing effective interactions between the quasiparticles. From the density-density correlation function, the collective mode indicates a very different behavior away from and close to half filling. The superconducting instability is examined to 1/$N\sp2$ order from the dominant double collective mode exchange. For a parabolic band, there is a p-wave triplet instability. But for a 2D square lattice, there is no superconducting pairing to this order. The effects of disorder on this system are also investigated. For moderate filling, the specific heat and scattering rate have the same results as predicted in the weakly interacting theories. Close to half-filling, the specific heat and scattering rate show a much sharper temperature and energy dependence.
By use of a diagrammatic method we determine the density-density response functions in the random-phase approximation (RPA) and the collective-excitation spectrum for the extended Hubbard model with on-site attraction and arbitrary electron density, in the superconducting ground state. The energy of the collective modes, given by the poles of these response functions, is found to be a linear function of the wave vector (for small k and short-range intersite interaction) with the velocity interpolating smoothly between the weak- (\ensuremath{\propto}2zt, \ensuremath{\Vert}U\ensuremath{\Vert}\ensuremath{\ll}2zt) and strong-coupling (\ensuremath{\propto}zt2{\mathit{zt}}^{2}zt2/\ensuremath{\Vert}U\ensuremath{\Vert}, \ensuremath{\Vert}U\ensuremath{\Vert}\ensuremath{\gg}2zt) limits. The latter agrees with the results obtained from an effective pseudospin Hamiltonian valid in the strong-coupling limit. The numerical analysis for a two-dimensional (2D) square lattice shows the occurrence of a rotonlike minima near the zone boundary. In the weak-coupling regime we find that apart from a commensurate charge-density-wave (CDW) instability, an increase in the intersite Coulomb repulsion can give rise to a CDW incommensurate with the lattice period, away from half-filling. The resulting phase diagram for the 2D square lattice, including a singlet superconducting ground state, electronic-droplet formation, and CDW's, is determined. Finally, the effects of a long-range Coulomb interaction are analyzed briefly and it is shown that the energy of collective excitations evolves smoothly from the weak- to the strong-coupling limit for a 2D lattice.
We investigate the two-dimensional Hubbard model with the long-range Coulomb interaction. We assume the form of the effective Coulomb interaction, which is treated as a perturbation. For the half-filling case, the self-energy is numerically evaluated up to the second-order terms of the perturbation expansion with the full momentum dependence included. The k -mass is calculated from this self-energy. It is found that the first-order term makes a dominant contribution to the reduction of the k -mass.
We discuss the density-density response functions and the collective excitation spectrum for the extended Hubbard model with on-site attraction and arbitrary electron density, in the superconducting ground state.For the short-range intersite interaction we find the sound wave-like modes with the velocity interpolating smoothly between the weak (v 2Zt, |UD <2Zt) and strong coupling (v > 2Zt) limits.The latter agrees with the results obtained from an effective pseudospin Hamiltonian valid in the strong coupling limit.In the weak coupling regime, we obtain that apart from a commensurate charge density wave instability, an increase in the intersite Coułomb repulsion can give rise to a charge density wave incommensurate with the lattice period, away from half-frlling.The ground state phase diagram for 2D square lattice, including singlet superconducting ground state, electronic droplet formation, and charge density waves is determined.It is also shown that the energy of collective excitations evolves smoothly from weak to strong coupling limit for a 2D lattice, in the case of a long-range Coulomb interaction.Finally, the mode coupling effects in the frequency behavior of the density-density response functions are analyzed.
We define Landau quasiparticles within the Gutzwiller variational theory and derive their dispersion relation for general multiband Hubbard models in the limit of large spatial dimensions D. Thereby we reproduce our previous calculations which were based on a phenomenological effective single-particle Hamiltonian. For the one-band Hubbard model we calculate the first-order corrections in 1/D1/D1/D and find that the corrections to the quasiparticle dispersions are small in three dimensions. They may be largely absorbed in a rescaling of the total bandwidth, unless the system is close to half band filling. Therefore, the Gutzwiller theory in the limit of large dimensions provides quasiparticle bands which are suitable for a comparison with real, three-dimensional Fermi liquids.