The low-temperature thermodynamics of the one-dimensional Hubbard model with attractive on-site interaction of the electrons is discussed within the framework of the Bethe ansatz for the wave function. At low T the dominant states consist of spin-paired electrons (Cooper-pair-like) and excitations of such pairs (without breaking up the singlet bound states). A critical field Hc (energy required to depair a Cooper pair) is obtained at T=0 (note that the superconducting Tc =0), which disappears for T≠0. There is no magnetic response at T=0 for H<Hc . The one-electron correlation function falls off exponentially with distance, while the singlet pair–singlet pair correlation decreases with a power law. The specific heat is linear in temperature, which is possibly a consequence of the dimension. The elementary excitations of the system are discussed. The excitation spectrum shows some analogies to that of resonant valence bonds in two dimensions.
The low-temperature thermodynamics of the one-dimensional Hubbard model with attractive on-site interaction of the electrons is discussed within the framework of the Bethe ansatz for the wave function. At low T the dominant states consist of spin-paired electrons (Cooper-pair-like) and excitations of such pairs (without breaking up the singlet bound states). A critical field Hc (energy required to depair a Cooper pair) is obtained at T=0 (note that the superconducting Tc =0), which disappears for T≠0. There is no magnetic response at T=0 for H<Hc . The one-electron correlation function falls off exponentially with distance, while the singlet pair–singlet pair correlation decreases with a power law. The specific heat is linear in temperature, which is possibly a consequence of the dimension. The elementary excitations of the system are discussed. The excitation spectrum shows some analogies to that of resonant valence bonds in two dimensions.
The thermodynamic Bethe-ansatz equations for the one-dimensional Hubbard model with attractive on-site interaction of the electrons are solved numerically at low temperatures. At low temperatures the dominant states consist of spin-paired electrons (Cooper-pair--like) and excitations of such pairs (without breaking up the singlet bound states). A critical field (energy required to depair a Cooper pair) is obtained at T=0 (note that the superconducting Tc=0), which disappears for T\ensuremath{\ne}0. The specific heat is linear in temperature, and the dependence of the coefficient \ensuremath{\gamma} on band filling and magnetic field is discussed. The excitation spectrum shows some analogies with that of resonant valence bonds in two dimensions.
The thermodynamic Bethe-ansatz equations for the one-dimensional Hubbard model are derived for an attractive on-site interaction of the electrons, extending in this way Takahashi's result for the repulsive Hubbard chain. The results are discussed in several special limits, e.g., interaction much larger than the hopping, U\ensuremath{\rightarrow}0 case, and the high- and low-temperature limits. The dominating states at low temperatures consist of spin-paired electrons (resembling Cooper pairs) and bound states of such pairs. The excitation spectrum and its properties are discussed in the context of superconductivity and resonant valence bonds.
One- and two-dimensional attractive Hubbard models in the presence of magnetic field h are studied in a wide range of on-site interaction strength U, temperature T and electron concentration n by means of the generalized self-consistent field (GSCF) approach with renormalized chemical potential. Pairing, the smooth crossover from the BCS regime into the Bose condensation state and the inhomogeneous superconductivity with the non-uniform order parameter are investigated. The excitation spectrum, the momentum distribution function and thermodynamic properties, including the concentration of double occupied sites, the kinetic energy, and the chemical potential, are calculated. The numerical calculations are found in a good agreement with the Bethe-ansatz results and quantum Monte-Carlo calculations in a wide range of system parameters U, n and h. The GSCF theory correctly displays the separation of the energy gap from the order parameter (crossover) as n or |U| decrease. The concentration of double occupied sites and the bound electron pairs n pair are calculated and we found a simple relationship between the order parameter and n pair , valid for the arbitrary U and all n in any dimension. The developed GSCF approach in magnetic field suggests a smooth transition into the inhomogeneous superconducting state.
[[abstract]]one- and two-dimensional attractive Hubbard models in the presence of magnetic field h are studied in a wide range of on-site interaction strength U, temperature T and electron concentration n by means of the generalized self-consistent field (GSCF) approach with renormalized chemical potential. Pairing, the smooth crossover from the BCS regime into the Bose condensation state and the inhomogeneous superconductivity with the non-uniform order parameter are investigated. The excitation spectrum, the momentum distribution function and thermodynamic properties, including the concentration of double occupied sites, the kinetic energy, and the chemical potential, are calculated. The numerical calculations are found in a good agreement with the Bethe-ansatz results and quantum Monte-Carlo calculations in a wide range of system parameters U, n and h. The GSCF theory correctly displays the separation of the energy gap from the order parameter (crossover) as n or \U\ decrease. The concentration of double occupied sites and the bound electron pairs n(pair) are calculated and we found a simple relationship between the order parameter and n(pair), valid for the arbitrary U and ail n in any dimension. The developed GSCF approach in magnetic field suggests a smooth transition into the inhomogeneous superconducting state.
One-dimensional conductors are a long-standing topic of research with direct applications to organic conductors and mesoscopic rings. The discovery of the ceramic high-temperature superconductors has revitalized the interest in low-dimensional charge and spin fluctuations of highly correlated electron systems. Several mechanisms proposed to explain the high-T c superconductors invoke properties of the two-dimensional Hubbard model, but probably also some one-dimensional aspects are relevant. Numerous one-dimensional models for correlated electrons have been studied with various approximate, asymptotically exact and exact methods. These results lead to the concept of Luttinger liquid for interacting electron gases without excitation gaps (metallic systems). Characteristic of Luttinger liquids are the charge and spin separation, marginal Fermi liquid properties, e.g. the absence of quasiparticles in the vicinity of the Fermi surface, nonuniversal power-law singularities in the one-particle spectral function and the related absence of a discontinuity in the momentum distribution at the Fermi level, the power-law decay of correlation functions for long times and large distances, persistent currents in finite rings, etc. Due to the peculiarities of the phase space in one dimension some of the models have sufficient conserved currents to be completely integrable. We review exact results derived within the framework of Bethe's ansatz for integrable one-dimensional models of correlated electrons. The Bethe-ansatz method is presented by explicitly showing the steps leading to the solution of the N-component electron gas interacting via a δ-function potential (repulsive and attractive interaction), which is probably the simplest model of correlated electrons. Emphasis is given to the procedure to extract the groundstate properties, the classification of states, the excitation spectrum, the thermodynamics and finite size effects, such as critical exponents of correlation functions and persistent currents. The method is then applied to numerous other models, e.g. (i) a two-band model involving attractive and repulsive potentials and crystalline fields splitting the bands, (ii) the traditional Hubbard chain with attractive and repulsive U, (iii) the degenerate Hubbard model with repulsive U, which displays a metal–insulator transition at a finite U, (iv) a two-band Hubbard model with repulsive U, (v) the traditional supersymmetric t–J model (vi) a two-band supersymmetric t–J model with band-splitting and (vii) the N-component supersymmetric t–J model. Finally, results for models with long-range interactions, in particular r-2 and sinh -2(r) potentials, are briefly reviewed.
Ground-state properties of the attractive Hubbard model in one dimension are studied by means of both the exact Bethe-ansatz formalism and the self-consistent field (SCF) approach with renormalized chemical potential for general band fillings n and a wide range of coupling strength U/t. The energy, the concentration of double occupied sites, the kinetic energy and the chemical potential of the ground state are in a good numerical agreement with the exact results over a wide range of parameters U/t and n. The concentration of local pairs or double occupied sites in the Bethe-ansatz solution serves as a suitable parameter measuring the electron pairing correlations. The SCF theory provides a simple analytical relationship between the concentration of double occupied sites, the band filling and the BCS order parameter, valid for arbitrary U/t and n. The calculated energy gap, the BCS order parameter, the phase diagram and the compressibility are also discussed. The SCF theory in one dimension distinguishes the order parameter from the excitation gap and suggests a smooth crossover away from half-filling (at $n\ensuremath{\ne}1)$ from the BCS pairing to the Bose condensation regime under the variation of U/t and n.