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physica status solidi (b)

Spectral Density Method for the Spinless Fermion Model

Dieter Ihle, B. Lorenz

Published Jun 1, 1981• 1 citation
Publisher

Abstract

Abstract A self‐consistent approximation scheme for the spectral densities of the one‐dimensional spinless fermion model is developed, where the one‐particle spectral density fulfils the first four moments. At T = 0 the one‐particle excitation spectrum is calculated in dependence on the Coulomb and band energy parameters. The exact results for the atomic and band limits and the gas—liquid like phase transition are reproduced.

DOI
10.1002/pssb.2221050215

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Abstract A self‐consistent approximation scheme for the spectral densities of the one‐dimensional spinless fermion model is developed, where the one‐particle spectral density fulfils the first four moments. At T = 0 the one‐particle excitation spectrum is calculated in dependence on the Coulomb and band energy parameters. The exact results for the atomic and band limits and the gas—liquid like phase transition are reproduced.
The Holstein model of spinless fermions interacting with dispersionless phonons in one dimension is studied by a Green's function Monte Carlo technique. The ground-state energy, first fermionic excited state, density wave correlations, and mean lattice displacements are calculated for lattices of up to 16 sites, for one fermion per two sites, i.e., a half-filled band. Results are obtained for values of the fermion hopping parameter of t=0.1\ensuremath{\omega}, \ensuremath{\omega}, and 10\ensuremath{\omega}, where \ensuremath{\omega} is the phonon frequency. At a finite fermion-phonon coupling g there is a transition from a metallic phase to an insulating phase in which there is charge-density-wave order. Finite-size scaling is found to hold in the metallic phase and is used to extract the coupling dependence of the Luttinger liquid parameters, ${\mathit{u}}_{\mathrm{\ensuremath{\rho}}}$ and ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$, the velocity of charge excitations, and the correlation exponent, respectively. For free fermions (g=0) and for strong coupling (g2{\mathit{g}}^{2}g2\ensuremath{\gg}t\ensuremath{\omega}) our results agree well with known analytic results. \textcopyright{} 1996 The American Physical Society.
Abstract The frequency‐dependent response of a one‐dimensional fermion system is investigated using Current Density Functional Theory (CDFT) within the local approximation (LDA). DFT‐LDA, and in particular CDFT‐LDA, reproduces very well the dispersion of the collective excitations. Unsurprisingly, however, the approximation fails for details of the dynamic response for large wavevectors. In particular, we introduce CDFT for the one‐dimensional spinless fermion model with nearest‐neighbor interaction, and use CDFT‐LDA plus exact (Bethe ansatz) results for the groundstate energy as function of particle density and boundary phase to determine the linear response. The successes and failures of this approach are discussed in detail (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract Band models with intrasite (Hubbard) and intersite Coulomb interactions are treated in a cluster approximation taking into account the interactions and the transfer within clusters of sites rigorously. Between different clusters the interaction is decoupled in a mean field approximation and the transfer term is treated as a perturbation. The ground‐state energy of the one‐dimensional Hubbard model is calculated up to fourth order in the perturbation and is compared with other approximations and the exact result. In the one‐dimensional spinless fermion model the perturbation is treated without further approximations by solving Dyson's equation for the one‐particle Green's function. The order‐disorder transition in this model is discussed.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
The selfconsistent high-density expansion for thermodynamic quantities of the quantum spin-polarized fermion lattice gas, derived in the preceding paper, is extended to higher orders. The (1/z)2 terms in the free energy are evaluated by summing two-loop diagrams and the corresponding equation for critical temperature in the classical limit is discussed. Ground state phase diagram of the system for arbitrary electron density is derived in the first and second order of the expansion. Properties of the half-filled, one-dimensional spinless fermion model at T = 0, obtained here in the first three orders in 1/z, are compared with the results of exact calculations and of other approximate methods. This provides a test for the unrenormalized high-density expansion method.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.