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arXiv:hep-th/9210109 · Physical Review Letters

Lattice integrable systems of Haldane-Shastry type

Alexios P. Polychronakos

Published Apr 12, 1993• 193 citations
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Abstract

We present a new lattice integrable system in one dimension of the Haldane-Shastry type. It consists of spins positioned at the static equilibrium positions of particles in a corresponding classical Calogero system and interacting through an exchange term with strength inversely proportional to the square of their distance. We achieve this by viewing the Haldane-Shastry system as a high-interaction limit of the Sutherland system of particles with internal degrees of freedom and identifying the same limit in a corresponding Calogero system. The commuting integrals of motion of this system are found using the exchange operator formalism.

DOI
10.1103/physrevlett.70.2329

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We present a new lattice integrable system in one dimension of the Haldane-Shastry type. It consists of spins positioned at the static equilibrium positions of particles in a corresponding classical Calogero system and interacting through an exchange term with strength inversely proportional to the square of their distance. We achieve this by viewing the Haldane-Shastry system as a high-interaction limit of the Sutherland system of particles with internal degrees of freedom and identifying the same limit in a corresponding Calogero system. The commuting integrals of motion of this system are found using the exchange operator formalism.
In this paper, we investigate a family of one-dimensional multicomponent quantum many-body systems. The interaction is an exchange interaction based on the familiar family of integrable systems which includes the inverse square potential. We show these systems to be integrable, and exploit this integrability to completely determine the spectrum including degeneracy, and thus the thermodynamics. The perioidic inverse square case is worked out explicitly. Next, we show that in the limit of strong interaction of the ``spin'' degrees of freedom decouple. Taking this limit for our example, we obtain a complete solution to a lattice system introduced recently by Shastry and by Haldane; our solution reproduces the numerical results. Finally, we emphasize the simple explanation for the high multiplicities found in this model.
The su(1|1) symmetric version of the Haldane-Shastry spin chain is diagonalized by means of a linear transformation. The same transformation applied to the original su(2) model yields simple expressions for the Hamiltonian and the generators of the Yangian symmetry of the model in terms of spin wave operators. Several years ago Haldane and Shastry independently discovered a soluble spin chain with exchange interaction falling off as the inverse square of the distance between the spins [1, 2]. This model can be thought of as the static limit [3] of the Sutherland model with internal degrees of freedom [4, 5, 6]. The idea of putting internal degrees of freedom into the Calogero-Sutherland models has proved to be very fruitful since then. A rich family of related models has been found. These long range interacting systems are under active investigation now. A common algebraic structure underlying their integrability is still waiting for its discovery. In this letter we discuss a linear transformation, that turns the su(m|n) supersymmetric extension of the original Haldane-Shastry Hamiltonian into a more compact form,
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.
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.
The spectrum of a one-dimensional chain of SU(n) spins positioned at the static equilibrium positions of the particles in the corresponding classical Calogero system with an exchange interaction inversely proportional to the square of their distance is studied. As in the translationally invariant Haldane-Shastry model the spectrum is found to exhibit a very simple structure containing highly degenerate 'super-multiplets'. The algebra underlying this structure is identified and several sets of raising and lowering operators are given explicitly. On the basis of this algebra and numerical studies the authors give the complete spectrum and thermodynamics of the SU(2) system.