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Journal of Applied Physics

Ground-state properties and excitations of an integrable one-dimensional model with δ-function interaction involving several bands

P. Schlottmann

Published May 15, 1994• 1 citation
Publisher

Abstract

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.

DOI
10.1063/1.356852

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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.
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We consider a gas of fermions with parabolic dispersion and N spin components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. The ground-state Bethe ansatz integral equations are solved numerically for both repulsive and attractive interactions to obtain the energy, the chemical potential, and the magnetic susceptibility as a function of the band filling and the interaction strength. For the repulsive interaction the Fermi gas has the properties of a Luttinger liquid. In the attractive case, on the other hand, the fermions in the ground-state form bound states of up to N fermions of different spin components. The spectrum of elemental charge and spin excitations is derived for the repulsive and attractive situations. The spectrum is discussed in the limits of vanishing interaction strength and very strong coupling. For the repulsive interaction the low-lying charge excitations can be characterized by the Fermi momentum and the Fermi velocity. The range of the spin-wave excitations is correlated with the Fermi momentum of the charges. The spin wave velocity is inversely proportional to the magnetic susceptibility. The spin-wave excitations become soft in the infinite-repulsive-coupling limit. In the attractive case in zero field all excitation branches except that of bound states of N fermions have an energy gap. It requires a finite energy to break these bound states and hence there is no response to a field smaller than a critical field. The low-T specific heat is proportional to the temperature.
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 consider an integrable model consisting of two one-dimensional parabolic bands of opposite mass (m=+or- 1/2 , respectively) separated by a gap, 2 Delta . The bands contain spinless fermions, one band corresponds to the conduction band and the other one to the valence band of a semiconductor or semimetal. The holes in the valence band and the spinless electrons in the conduction band are locally attracted via a delta -function potential. This model can be mapped onto the two-component Fermi gas with delta -function interaction, so that the two components label the bands: the chemical potential corresponds to a magnetic field, and Delta to the chemical potential. We use Gaudin and Yang's exact solution of the many-body problem to study the formation of exciton bands. The properties of the ground and metastable states, the excitation spectrum and the thermodynamics of the model are obtained. In the ground or metastable states the particles (electrons in the conduction band and holes in the valence band) are either paired in exciton bound states or unpaired. Their spectrum of elemental excitations is approximately parabolic. At finite T many-particle bound states (string solutions of the Bethe ansatz equations) can be populated; at low T these states are strongly delocalized and can be interpreted as electron-hole droplets. The low-T properties of the model are discussed.
The author considers a gas of fermions with parabolic dispersion and N spin-components (or spin S, N=2S+1) with SU(N) symmetry in one dimension interacting via a delta -function potential. The model is integrable and its solution has been obtained by Sutherland in terms of N nested Bethe ansatze. He analyses the discrete Bethe ansatz equations and classifies their solutions according to the string hypothesis. The thermodynamic Bethe-ansatz equations are derived for arbitrary band-filling for both repulsive and attractive interaction in terms of the thermodynamic energy potentials for the classes of eigenstates of the Hamiltonian. These equations are then discussed in several limiting cases, e.g. S=1/2, the ground state (T to 0), for vanishing interaction strength, for strong coupling, in the high-temperature limit, and the large-N limit.
We calculate the critical exponent of the temperature dependence of the drag current between two parallel quantum wires. The two wires have carriers with equal mass but they may have different electron density. The electrons interact via a $\ensuremath{\delta}$-function spin-exchange potential, which is attractive or repulsive depending on whether the interacting particles are in a spin-singlet or spin-triplet state. This interaction leads to the formation of spin-singlet or spin-triplet bound states of the Cooper type (preformed hard core bosons that do not condensate). Depending on the parameters, the drag current can be parallel or opposite to the driving current. The critical exponent of the drag current is calculated using the Bethe Ansatz solution of the model and conformal field theory.