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Moscow University Physics Bulletin

On the wave functions of an exactly solvable model of interacting fermions with spin 1/2 in an external field

V. I. Inozemtsev, Наталья Германовна Иноземцева, Б. И. Садовников

Published Jun 1, 2008• 0 citations
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Abstract

It is shown that hyperbolic systems of Sutherland spin 1/2 particles in an external field with the Morse potential characterized by parameter τ 2 have a discrete part of their spectrum in the case of a particular constraint on the τ, parameter of two-particle interaction λ, and the number of particles. The ground state is described by a Jastrow wave function. The results known for systems with an interaction inversely proportional to the squared distance between particles are reproduced in the limit τ → ∞.

DOI
10.3103/s0027134908030028

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It is shown that hyperbolic systems of Sutherland spin 1/2 particles in an external field with the Morse potential characterized by parameter τ 2 have a discrete part of their spectrum in the case of a particular constraint on the τ, parameter of two-particle interaction λ, and the number of particles. The ground state is described by a Jastrow wave function. The results known for systems with an interaction inversely proportional to the squared distance between particles are reproduced in the limit τ → ∞.
The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (SU( n ) spins) interacting with an external field with the Morse potential of an arbitrary strength τ 2 . These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form. The known results for the systems with inverse-square interaction within harmonic well are reproduced in the limit τ → ∞ .
Starting from a Calogero-Sutherland model with the hyperbolic interaction confined by an external field with Morse potential, we construct a Heisenberg spin chain with exchange interaction varies as 1/sinh 2 x on a lattice given in terms of the zeros of Laguerre polynomials. Varying the strength of the Morse potential, the Haldane-Shastry and harmonic spin chains are reproduced. The spectrum of the models in this class is found to be that of a classical one-dimensional Ising chain with non-uniform nearest-neighbour coupling in a non-uniform magnetic field which allows us to study the thermodynamics in the limit of infinite chains.
Starting from a Calogero--Sutherland model with hyperbolic interaction confined by an external field with Morse potential we construct a Heisenberg spin chain with exchange interaction ∝1/sinh⁡2x\propto 1/\sinh^2 x∝1/sinh2x on a lattice given in terms of the zeroes of Laguerre polynomials. Varying the strength of the Morse potential the Haldane--Shastry and harmonic spin chains are reproduced. The spectrum of the models in this class is found to be that of a classical one-dimensional Ising chain with nonuniform nearest neighbour coupling in a nonuniform magnetic field which allows to study the thermodynamics in the limit of infinite chains.
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.