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Physical review. B, Condensed matter

Thermodynamic Bethe-ansatz equations for the Hubbard chain with an attractive interaction

Kong-Ju-Bock Lee, P. Schlottmann

Published Dec 1, 1988• 29 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.38.11566

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International Journal of Modern Physics B•May 30, 1995•3 citations
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.
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{T}_{c}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 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.
We present the exact solution via Bethe's ansatz to an integrable model consisting of two parabolic bands of electrons of equal mass with a local \ensuremath{\delta}-function exchange interaction. The interaction is attractive or repulsive depending on whether the interacting particles are in a spin-singlet or spin-triplet state. The structure of the Bethe ansatz and the classification of states is similar to the one of the two-channel Kondo problem. The interest in two-band models arises from the possibility that both the 3dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 and 3dz2{\mathit{d}}_{\mathit{z}}^{2}dz2​ orbitals may play a role in high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprates. Some low-temperature properties of this model are discussed as a function of the interaction strength, a magnetic field, and the (crystalline field) splitting between the bands. The attractive interaction leads to Cooper-pair-like bound states, which exist at every temperature. There is no long-range order and no condensation of the Cooper pairs. A threshold magnetic field is required to overcome the binding energy of the Cooper pairs and there is no response to fields smaller than the critical one. The low-temperature specific heat is proportional to T, except at critical points (van Hove singularities of the bands) where C\ensuremath{\propto}T1/2{\mathit{T}}^{1/2}T1/2. The spectrum of elemental excitations is approximately parabolic.
We study mesoscopic pairing in the one-dimensional repulsive Hubbard model and its interplay with the BCS model in the canonical ensemble. The key tool is comparing the Bethe ansatz equations of the two models in the limit of small Coulomb repulsion. For the ordinary Hubbard interaction the BCS Bethe equations with infinite pairing coupling are recovered; a finite pairing is obtained by considering a further density-dependent phase-correlation in the hopping amplitude of the Hubbard model. We find that spin degrees of freedom in the Hubbard ground state are arranged in a state of the BCS type, where the Cooper pairs form a noncondensed liquid on a ``lattice'' of single particle energies provided by the Hubbard charge degrees of freedom; the condensation in the BCS ground state corresponds to Hubbard excitations constituted by a sea of spin singlets.
The ground-state instabilities for a one-dimensional lattice system of electrons with onsite (Hubbard) and bond-site (hopping) interactions are analyzed in a perturbative approach. The zero temperature phase diagram at different band fillings is drawn; an attractive (repulsive) bond-site interaction favors the appearance of a superconductor state at low concentrations of electrons (holes). A comparison with the exact results for the Hubbard model and previous works for particular cases is also discussed.