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Physical Review Letters

Integrable two-band model with attractive and repulsive interactions

P. Schlottmann

Published Mar 23, 1992• 37 citations
Publisher

Abstract

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}${\mathit{T}}^{1/2}$.

DOI
10.1103/physrevlett.68.1916

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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.
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.
We consider an integrable model consisting of n parabolic bands of electrons of equal mass with a δ-function exchange interaction. The interaction is attractive or repulsive depending on whether the interacting particles are in a spin-singlet or triplet state. The structure of the Bethe ansatz and the classification of states is similar to that of the n-channel Kondo problem. The attractive interaction leads to Cooper-pair-like boundstates. Some ground-state properties and the excitation spectrum are discussed for the special situation of degenerate bands.
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.
We consider an exactly solvable two-band model of electrons moving in one dimension and interacting with a $\ensuremath{\delta}$-function spin-exchange potential. The relative population of the bands is determined by the band splitting, the magnetic field, and the temperature. The interaction leads to the formation of spin-triplet bound states of the Cooper type (hard-core bosons that do not condensate and exist at all temperatures). In the zero-field ground state the spin-triplet states are effectively free bosons. A threshold band splitting is required to break up a triplet pair. The low-temperature and finite size properties are discussed in the context of anomalous superconducting and antiferromagnetic fluctuations.