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arXiv:cond-mat/9310021 · Physical review. B, Condensed matter

Crossover from Cooper pairs to composite bosons: A generalized RPA analysis of collective excitations

Lotfi Belkhir, Mohit Randeria

Published Mar 1, 1994• 64 citations
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Abstract

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.

DOI
10.1103/physrevb.49.6829

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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 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.
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.
The superfluid characteristics of the attractive Hubbard model are analyzed for any coupling |U| and arbitrary electron concentration (0n2) by means of the slave-boson mean-field method and also by the perturbative treatment of the strong-coupling limit. The slave boson method takes into account correlations of electrons and yields a reliable description of the crossover from BCS-type superconductivity to local pair (composite bosons) superconductivity with increasing |U|. The results for the ground state (the free energy, the gap in the excitation spectrum) and the electromagnetic characteristics (the critical magnetic field, the London penetration depth, the coherence length) are compared with those obtained by the Hartree-Fock approximation and by the self-consistent second-order perturbation theory in the weak-coupling limit as well as with those obtained using perturbational approaches in the strong-coupling limit. We show that the slave-boson method, in contrast to the Hartree-Fock approximation, gives credible results for all investigated quantities in the whole interaction range, interpolating smoothly between the BCS and local pair regimes. A comparison of theoretical predictions for our simple model with experimental data for various families of short-coherence-length superconductors suggests that the best agreement can be obtained for intermediate values of the local attraction. \textcopyright{} 1996 The American Physical Society.
The spectrum of collective (pair) excitations in the ground state of a dilute twodimensional (2D) attractive Fermi-gas is studied within the functional integral formalism. The linearized equations for the fluctuations about the non-trivial saddle point are analyzed for all coupling regimes, which are characterized by the ratio ε 0 /ε F , where ε 0 is the two-fermion binding energy and ε F is the Fermi energy. The approximation takes into account propagation of the fluctuations and their interaction with the condensate. In the strong-coupling, or ‘;Bosegas’, regime (ε 0 /ε F > 1) the spectrum is continuous and has the Bogolubov form, but in the weak-coupling limit (ε 0 /ε F «1) there are two types of excitations (different from the two-fermion scattering states): (i) long-wavelength sound-like excitations with the cut-off at momentum q c ≃ l/ξ 0 (where ξ 0 is the Cooper pair size), and (ii) pair excitations with q ≃ (8 m μ) ½ , where μ is the chemical potential and m is the fermion mass. The crossover between weakand strong-coupling behavior of the excitation spectrum is found to occur at the value of the coupling parameter ε 0 /ε F ≃¼. Introduction Since the discovery of high- T c superconductivity there has been growing interest in studying 2D Fermi gases with attractive interaction, especially in the crossover regime, when the pair size ξ 0 is of the order of the interparticle distance. The importance of such a model, which can be regarded as a semiphenomenological model of a 2D superconductor, is highlighted by experimental evidence that the high- T c superconductors, most of which have a layered structure, have a short coherence length, so that k F ξ 0 ≃1, where k F is the Fermi momentum.
Elucidating the impact of strong electronic interactions on the collective excitations of metallic systems has been of longstanding interest, mainly due to the inadequacy of the random phase approximation (RPA) in the strongly correlated regime. Here, we adopt our newly developed radial Kotliar and Ruckenstein slave boson representation to analyze the charge excitation spectrum of a Hubbard model, extended with long range interactions. Working on the face centered cubic lattice, at half filling, and in different coupling regimes ranging from uncorrelated to the metal-to-insulator transition, we compare our results to conventional RPA as a benchmark. We focus on the influence of the local and long range couplings on the particle-hole excitation continuum and the plasmon and upper Hubbard band collective modes. Beyond the weak coupling regime, we find numerous quantitative and even qualitative discrepancies between our method and standard RPA. Our work thus deepens the understanding of charge collective modes in correlated systems and lays the foundations for future studies of a broad series of materials.