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Physical review. B, Condensed matter

Collective excitations in the Penson-Kolb model: A generalized random-phase-approximation study

G. K. Roy, Bibhas Bhattacharyya

Published Jun 15, 1997• 6 citations
Publisher

Abstract

The evolution of the superconducting ground state of the half-filled Penson-Kolb model is examined as a function of the coupling constant using a mean-field approach and the generalized random phase approximation (RPA) in two and three dimensions. On-site singlet pairs hop to compete against single-particle motion in this model, giving the coupling constant a strong momentum dependence. There is a pronounced bandwidth enhancement effect that converges smoothly to a finite value in the strong-coupling (Bose) regime. The low-lying collective excitations evaluated in generalized RPA show a linear dispersion and a gradual crossover from the weak-coupling (BCS) limit to the Bose regime; the mode velocity increases monotonically in sharp contrast to the attractive Hubbard model. Analytical results are derived in the asymptotic limits.

DOI
10.1103/physrevb.55.15506

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The evolution of the superconducting ground state of the half-filled Penson-Kolb model is examined as a function of the coupling constant using a mean-field approach and the generalized random phase approximation (RPA) in two and three dimensions. On-site singlet pairs hop to compete against single-particle motion in this model, giving the coupling constant a strong momentum dependence. There is a pronounced bandwidth enhancement effect that converges smoothly to a finite value in the strong-coupling (Bose) regime. The low-lying collective excitations evaluated in generalized RPA show a linear dispersion and a gradual crossover from the weak-coupling (BCS) limit to the Bose regime; the mode velocity increases monotonically in sharp contrast to the attractive Hubbard model. Analytical results are derived in the asymptotic limits.
We study the evolution of the ground state and the excitation spectrum of the two- and three-dimensional attractive (negative-U) Hubbard model as the system evolves from a Cooper-pair regime for U\ensuremath{\ll}t, to a composite boson regime for U\ensuremath{\gg}t. Our work is motivated by the observation that the high-temperature superconductors, with their short coherence lengths and unusual normal-state properties, may be in an intermediate coupling regime between these two limits. A mean-field analysis of pairing, suitably generalized to account for a shift in the chemical potential, is known to be able to describe the ground-state crossover as a function of U/t. We compute the collective-mode spectrum using a generalized random-phase-approximation analysis within the equations-of-motion formalism. We find a smooth evolution of the Anderson mode for weak coupling into the Bogoliubov sound mode for hard-core bosons. We then include a long-range Coulomb interaction and show that it leads to a plasmon which again evolves smoothly from weak to strong coupling.
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including d=1d=1d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For d=1,d=1,d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum q=0q=0q=0 and q=Qq=Qq=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.
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.
Different kinds of instabilities (charge-density wave, spin-density wave, singlet superconductivity) in the one-dimensional Hubbard model with pair-hopping interaction are investigated using an approximate Bethe-Salpeter equation. The study is performed at any density of electrons and for arbitrary values of the model parameters. In the absence of the on-site interaction, no transition occurs at half-filling for any finite negative pair-hopping parameter, in agreement with recent results; our calculation suggests that the Penson-Kolb model (t,W) with \ensuremath{\Vert}W/t\ensuremath{\Vert}/sinkF{\mathit{k}}_{\mathit{F}}kF​ behaves qualitatively like the Hubbard model (t,U). Phase diagrams of the Penson-Kolb-Hubbard model (t,U,W) at various densities are presented. \textcopyright{} 1996 The American Physical Society.
The ground state of the one-dimensional half-filled Penson-Kolb-Hubbard model (with competing single-particle and pair hoppings in the presence of on-site Coulomb repulsion) has been investigated by a real space renormalization group method. The ground state energy, energy gap, local moment and several correlation functions are studied to obtain the phase diagram. The phase diagram consists of a spin density wave (SDW) state, a superconducting (SC) phase and a quasi-metallic phase dominated by short range SC correlations for positive values of the pair hopping amplitude; when this amplitude is negative, the phase diagram contains a near-metallic commensurate charge density wave and an SC phase of pairs with centre-of-mass momentum q= pi apart from an SDW phase.
The Penson-Kolb-Hubbard model, i.e. the Hubbard model with the pair-hopping interaction J is studied.We focus on the properties of the superconducting state with the Cooper-pair center-of-mass momentum q = Q (η-phase).The transition into the η-phase, which is favorized by the repulsive J (J < 0) is found to occur only above some critical value |J c |, dependent on band filling, on-site interaction U and band structure, and the system never exhibits standard BCS-like features.This is in obvious contrast with the properties of the isotropic s-wave state, stabilized by the attractive J and attractive U , which exhibit at T = 0 a smooth crossover from the BCS-like limit to that of tightly bound pairs with increasing pairing strength.