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Physical review. B, Condensed matter

Low-temperature properties of the Hubbard chain with an attractive interaction

Kong-Ju-Bock Lee, P. Schlottmann

Published Nov 1, 1989• 28 citations
Publisher

Abstract

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 ${T}_{c}$=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.

DOI
10.1103/physrevb.40.9104

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Physical Review Letters•Nov 13, 1989•26 citations
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.
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 exact solution via Bethe's ansatz to a model consisting of two parabolic bands of electrons with local attractive and repulsive interactions is presented. Some low-temperature properties of this model are discussed as a function of the interaction strength, a magnetic field, and the crystal-field splitting between the bands. The attractive interaction leads to Cooper-pair-like bound states and a threshold magnetic field is required to overcome the binding energy. The low-T specific heat is proportional to T, except at critical points where C\ensuremath{\propto}T1/2{\mathit{T}}^{1/2}T1/2.
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.
Unusual properties of the Hubbard chain of arbitrary degeneracy N and excluded-site occupations of more than two electrons are presented: (i) In zero field and in the limit of large N the ground state resembles an interacting Bose gas. The 1/N contributions vanish identically, so that corrections to leading order are 1/N2{\mathrm{N}}^{2}N2. (ii) The susceptibility at T=0 shows logarithmic singularities as H\ensuremath{\rightarrow}0. (iii) The Hubbard model and the Heisenberg chain have the same low-energy spin excitations for all U. (iv) The specific-heat coefficient \ensuremath{\gamma} is singular. These properties hold for arbitrary band filling and (ii)--(iv) hold for all N\ensuremath{\ge}2.