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Journal of the Physical Society of Japan

Infinite Symmetry of the Spin Systems with Inverse Square Interactions

Kazuhiro Hikami, Miki Wadati

Published Dec 15, 1993• 65 citations
Publisher

Abstract

One-dimensional quantum particle system with S U (ν) spins interacting through inverse square interactions is studied. We reveal algebraic structures of the system: hidden symmetry is the \(U(\nu )\cong SU(\nu )\otimes U(1)\) current algebra. This is consistent with the fact that the ground state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, which is the w 1+∞ algebra. With this W algebra we can clarify simultaneously the structures of the Calogero type (1/ x 2 -interactions) and Sutherland type (1/sin 2 x -interactions). The Yangian symmetry is briefly discussed.

DOI
10.1143/jpsj.62.4203

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One-dimensional quantum particle system with S U (ν) spins interacting through inverse square interactions is studied. We reveal algebraic structures of the system: hidden symmetry is the (U(\nu )\cong SU(\nu )\otimes U(1)) current algebra. This is consistent with the fact that the ground state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, which is the w 1+∞ algebra. With this W algebra we can clarify simultaneously the structures of the Calogero type (1/ x 2 -interactions) and Sutherland type (1/sin 2 x -interactions). The Yangian symmetry is briefly discussed.
Quantum integrable particle systems with inverse square interactions in one-dimensional space are investigated by use of the quantum inverse scattering method. For the case of particles with $\mathrm{SU}(\ensuremath{\nu})$ spins, a hidden symmetry is the $\mathrm{SU}(\ensuremath{\nu})\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\simeq}\mathrm{U}(\ensuremath{\nu})$ current algebra. The system has a higher symmetry, the ${W}_{1+\ensuremath{\infty}}$ algebra. This WWW algebra structure unifies the Calogero type (1x2\frac{1}{{x}^{2}}x21​ interactions) and the Sutherland type (1sin2x\frac{1}{{sin}^{2}x}sin2x1​ interactions). The energy spectra of the systems are obtained in a simple manner based on the fermion representation of the algebra.
Recent developments in the theory of quantum integrable particle systems in one-dimension with inverse square interactions are reviewed. First the Yangian symmetry is introduced and the energy spectra of the related spin models are discussed. The character of the su(n)1 WZNW theory is shown to be closely related with the Rogers–Szegö polynomial. Second, the infinite dimensional representation for solutions of the Yang–Baxter equation and the reflection equation is given. Based on the representation, the Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Finally, some results on short-range interacting systems are mentioned.
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,
One-dimensional quantum systems show fascinating properties beyond the scope of the mean-field approximation. However, the complicated mathematics involved is a high barrier to non-specialists. Written for graduate students and researchers new to the field, this book is a self-contained account of how to derive the exotic quasi-particle picture from the exact solution of models with inverse-square interparticle interactions. The book provides readers with an intuitive understanding of exact dynamical properties in terms of exotic quasi-particles which are neither bosons nor fermions. Powerful concepts, such as the Yangian symmetry in the Sutherland model and its lattice versions, are explained. A self-contained account of non-symmetric and symmetric Jack polynomials is also given. Derivations of dynamics are made easier, and are more concise than in the original papers, so readers can learn the physics of one-dimensional quantum systems through the simplest model.
A few years ago, Matsuo and Cherednik proved that from some solutions of the Knizhnik-Zamolodchikov (KZ) equations, which first appeared in conformal field theory, one can obtain wave functions for the Calogero integrable system. In the present communication, it is shown that from some solutions of generalised KZ equations, one can construct wave functions, characterized by any given permutational symmetry, for some Calogero-Sutherland-Moser integrable models with exchange terms. Such models include the spin generalizations of the original Calogero and Sutherland ones, as well as that with δ-function interaction.