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Physical review. B, Condensed matter

Static properties of a particle coupled to a fermionic environment

Per Hedegrd, A. O. Caldeira

Published Jan 1, 1987• 20 citations
Publisher

Abstract

We study the ground-state properties of a particle moving in a tight-binding band and coupled to a Fermi sea of metal electrons. Using an effective Hamiltonian and a variational ground state, we find a band narrowing of ${W}_{\mathrm{e}\mathrm{f}\mathrm{f}/{W}_{0}\ensuremath{\simeq}}$ (${W}_{0}$/${\ensuremath{\varepsilon}}_{F}$${)}^{\ensuremath{\alpha}/1\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$, where \ensuremath{\alpha} measures the coupling strength between particle and environment. A renormalization-group analysis gives the same result for \ensuremath{\alpha}<1 and an effective bandwidth ${W}_{\mathrm{eff}=0}$ for \ensuremath{\alpha}>1, indicating a (zero temperature) localization transition.

DOI
10.1103/physrevb.35.106

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arXiv:0707.0142•Jul 2, 2007•2 citations
We study the ground-state properties of a particle moving in a tight-binding band and coupled to a Fermi sea of metal electrons. Using an effective Hamiltonian and a variational ground state, we find a band narrowing of Weff/W0\ensuremath≃{W}_{\mathrm{e}\mathrm{f}\mathrm{f}/{W}_{0}\ensuremath{\simeq}}Weff/W0​\ensuremath≃​ (W0{W}_{0}W0​/{\ensuremath{\varepsilon}}_{F}$${)}^{\ensuremath{\alpha}/1\mathrm{\ensuremath{-}}\ensuremath{\alpha}}, where \ensuremath{\alpha} measures the coupling strength between particle and environment. A renormalization-group analysis gives the same result for \ensuremath{\alpha}<1 and an effective bandwidth Weff=0{W}_{\mathrm{eff}=0}Weff=0​ for \ensuremath{\alpha}>1, indicating a (zero temperature) localization transition.
The reduced density operator of a particle coupled to a fermionic environment (metal electrons) is written in terms of a Feynman-Vernon influence functional. Two different approaches are used in order to achieve this goal. Firstly, the environmental interacting electron gas is treated in RPA and the coupling is correctly accounted for up to second order in perturbation theory. Then the local and tight-binding models are studied. For the latter a second approach for obtaining the influence functional is used in the narrow-band limit. It is found that the popular representation of the environment in terms of a single set of harmonic oscillators is not feasible unless the particle is confined to move in a local region of radial extension r ≃ k F -1 .
Some ground state properties of a fermion system are investigated. An initial Hamiltonian being a generalization of the lattice Anderson model describes this system as a two-band electron-phonon one where both of electron bands are narrow (one is wider than the other). Two bands are hybridized and electrons from both of them are strongly coupled to the same local phonons. After performing the Lang–Firsov type transformation, a new polaron Hamiltonian is obtained. As a result, there appear three effective polaron-polaron interactions in this Hamiltonian and the bandwidths of both of electron bands are reduced. It is assumed that the reduction of the bandwidth of one of the bands is sufficiently strong to make this band dispersionless. Each of the interactions can lead to superconductivity on its own but only one of them is taken into account at this stage; it is the interaction between electrons from the band with a dispersion. The Bogolyubov’s mean field approach is used in order to find the spectrum of the reduced effective Hamiltonian. The effect of the hybridization on the superconducting gap and the relationship between the critical hybridization and the average number of electrons from the wider band per lattice site are investigated.
The variational scheme of Gutzwiller has been reexamined so as to apply it to the case of degenerate narrow bands. The ground-state wave function for electrons is investigated for an arbitrary density of electrons and an arbitrary strength of interaction. The typical features of a metal, particularly the Fermi surface, are preserved in this approach as opposed to some other methods. The Coulomb repulsion among electrons scatters the electrons from below to above the Fermi surface of uncorrelated bands. Under the assumption that only the intra-atomic Coulomb interaction is important, the probability of electrons being scattered is larger for the paramagnetic state than for the ferromagnetic state. Therefore the ferromagnetic ground state is favored if the density of states is large at the band edges. The degenerate-band-model Hamiltonian is spin dependent, hence the results obtained from it do not have to obey the theorem of Lieb and Mattis.
Correlation and magnetic effects in narrow bands are studied using a new type of self-consistent cluster treatment of Hubbard's Hamiltonian. Here the dynamics consist of electron hopping between lattice sites and Coulomb repulsion between electrons only on the same site. The environment of the cluster is regarded as a particle reservoir, one such for each spin. Hopping between the cluster and reservoirs is described by fermion source terms. It is required that the thermodynamic average of the particle currents of each spin within the cluster equal the corresponding particle exchange between cluster and reservoir. Electron motion in the system is examined for varying ratios of the strength of the hopping to the Coulomb repulsion. This is done as a function of electron density, spin, temperature, and external magnetic field. A Mott transition is found under dynamical conditions close to those predicted by Hubbard when there is one particle per site. For 〈n〉 ≠ 1, no metal insulator transition exists although discontinuous changes in the cluster-reservoir coupling occur. The magnetic susceptibility of the cluster reveals antiferromagnetic behavior for 〈n〉 = 1, ferromagnetic behavior for 〈n〉 > 1, and partial ferromagnetic character for n < 1. This appears to agree with Nagoaka's results.
The Hubbard model in the strong-coupling regime is mainly studied by Kondo-lattice theory or 1/d expansion theory, with d the spatial dimensionality. In two dimensions and higher, the ground state within the Hilbert subspace with no order parameter is a normal Fermi liquid except for n=1 and U/W=+infinity, with n the electron density per unit cell, U the on-site repulsion, and W the bandwidth; the cooperation between the Kondo effect, which favors a local singlet on each unit cell, and a resonating-valence-bond effect, which favors a local singlet on each pair of nearest-neighbor unit cells, stabilizes the Fermi liquid, whose ground state is a singlet as a whole, in the strong-coupling regime. In the whole Hilbert space with no restriction, the normal Fermi liquid is unstable at least against a magnetic or superconducting state. This analysis confirms an early Fermi-liquid theory of high-temperature superconductivity, F. J. Ohkawa, Jpn. J. Appl. Phys. 26, L652 (1987). The ground state for n=1 and U/W=+infinity is a Mott insulator. Actual metal-insulator transitions cannot be explained within the Hubbard model. In order to explain them, the electron-phonon interaction, multi-band or multi-orbital effects, and effects of disorder should be considered beyond the Hubbard model. The crossover between local-moment magnetism and itinerant-electron magnetism corresponds to that between a localized spin and a normal Fermi liquid in the Kondo effect and it is simply a Mott metal-insulator crossover.