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arXiv:cond-mat/9212028 · Physical Review Letters

Solution of some integrable one-dimensional quantum systems

Bill Sutherland, B. Sriram Shastry

Published Jul 5, 1993• 115 citations
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Abstract

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.

DOI
10.1103/physrevlett.71.5

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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.
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 present the exact solution (in the thermodynamic limit) to a one-dimensional, two-component, quantum many-body system in which like particles interact with a pair potential s(s+1)/sinh2{\mathrm{sinh}}^{2}sinh2(r), while unlike particles interact with a pair potential -s(s+1)/cosh2{\mathrm{cosh}}^{2}cosh2(r). We first give a proof of integrability, then derive the coupled equations determining the complete spectrum. All singularities occur in the ground state when there are equal numbers of the two components; we give explicit results for the ground state and low-lying states in this case.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.
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.