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Physical review. B, Condensed matter

Squeezed strings and Yangian symmetry of the Heisenberg chain with long-range interaction

Zou Ha, F. D. M. Haldane

Published May 15, 1993• 47 citations
Publisher

Abstract

The exact and complete set of highest-weight eigenstates of the finite-size SU(n) Heisenberg chain with inverse-square exchange (ISE) are constructed using the occupation-number representation of the ``strings,'' which can be considered as the unbroken resonating-valence bonds. The string description, which is believed to be exact only in the thermodynamic limit for the Bethe ansatz solvable models, becomes exact for any size ISE model due to the squeezing effect of the strings in the ISE limit. This feature provides a basis for the realization of the Yangian symmetry in the ISE model.

DOI
10.1103/physrevb.47.12459

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The exact and complete set of highest-weight eigenstates of the finite-size SU(n) Heisenberg chain with inverse-square exchange (ISE) are constructed using the occupation-number representation of the ``strings,'' which can be considered as the unbroken resonating-valence bonds. The string description, which is believed to be exact only in the thermodynamic limit for the Bethe ansatz solvable models, becomes exact for any size ISE model due to the squeezing effect of the strings in the ISE limit. This feature provides a basis for the realization of the Yangian symmetry in the ISE model.
The SU(n) quantum chains with inverse-square exchange exhibit a novel form of Yangian symmetry compatible with periodic boundary conditions, allowing states to be countable. We characterize the ``supermultiplets'' of the spectrum in terms of generalized ``occupation numbers.'' We embed the model in the k=1 SU(n) Kac-Moody algebra and obtain a new classification of the states of conformal field theory, adapted to particlelike elementary excitations obeying fractional statistics.
In this thesis we theoretically investigate the close relation between spin chains and Yangians. First, we investigate the Haldane-Shastry model, which symmetry algebra is a representation of the Yangian. We show this in full detail for the spin-1/2 model. We also give an argument why a spin-1 chain with Yangian symmetry cannot exist. \nAfter that, we turn to integrable spin chains which are constructed using the quantum inverse scattering method. Using this method, an integrable spin chain can be constructed from each so called R-matrix, which is a solution of the Yang-Baxter equation. Our main result is the construction of an R-matrix that is invariant under the adjoint representation of SU(n), which is found using an intertwining operator for the corresponding Yangian representation. The integrable\nspin chain that corresponds to this R-matrix is non-Hermitian, making its physical interpretation unclear. These results have been published recently as a preprint under arXiv:1606.02516. \nFinally, integrable spin chains with defects are studied. We present the Hamiltonian for the Heisenberg model and the Babujian-Takhtajan model with defects, such that these models keep there integrability while staying Lie algebra invariant.
Using the algebraic Bethe ansatz, we consider the correlation functions of the integrable higher spin chains. We apply a method recently developed for the spin ½ Heisenberg chain, based on the solution of the quantum inverse problem. We construct a representation for the correlation functions on a finite chain for arbitrary spin. Then we show how the string solutions of the Bethe equations can be considered in the framework of this approach in the thermodynamic limit. Finally, a multiple integral representation for the spin 1 zero-temperature correlation functions is obtained in the thermodynamic limit.
The solution of the Heisenberg model ( XXX -spin chain) is reviewed to define main ingredients of the quantum inverse scattering method. The spectrum and eigenfunctions are constructed by the algebraic Bethe Ansatz. A dynamical symmetry algebra is identified with a quantum group —the sl (2) Yangian. The Yang-Baxter equation and its GL ( n )-invariant solutions ( R -matrices) are discussed. Integral equations describing the Heisenberg model in the limit of an infinite chain are given. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.