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Physical Review Letters

Integrable Systems with Long-Range Interactions, W∞ Algebra, and Energy Spectrum

Kazuhiro Hikami, Miki Wadati

Published Aug 29, 1994• 39 citations
Publisher

Abstract

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 $W$ algebra structure unifies the Calogero type ($\frac{1}{{x}^{2}}$ interactions) and the Sutherland type ($\frac{1}{{sin}^{2}x}$ interactions). The energy spectra of the systems are obtained in a simple manner based on the fermion representation of the algebra.

DOI
10.1103/physrevlett.73.1191

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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 SU(\ensuremathν)\mathrm{SU}(\ensuremath{\nu})SU(\ensuremathν) spins, a hidden symmetry is the SU(\ensuremathν)\ensuremath⨂U(1)\ensuremath≃U(\ensuremathν)\mathrm{SU}(\ensuremath{\nu})\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\simeq}\mathrm{U}(\ensuremath{\nu})SU(\ensuremathν)\ensuremath⨂U(1)\ensuremath≃U(\ensuremathν) current algebra. The system has a higher symmetry, the W1+\ensuremath∞{W}_{1+\ensuremath{\infty}}W1+\ensuremath∞​ 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.
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(ν)≅SU(ν)⊗U(1)U(\nu )\cong SU(\nu )\otimes U(1)U(ν)≅SU(ν)⊗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.
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.
We investigate the ground state properties of a family of N-body systems in one dimension, trapped in a polynomial potential and having long-range two-body interaction in addition to the inverse square potential studied in the Calogero-Sutherland model (CSM). We show that for such a Hamiltonian, the ground state energy is similar to that of free fermions in a harmonic well with a displacement that depends on the number of particles and depth of the well. We obtain the ground state wave function and using random matrix results, study the particle density and pair correlation function (PCF). We observe that the particles are arranged in bands. Due to the presence of long-range interaction, the PCF shows a departure from the CSM.
There is developed a current algebra representation scheme for reconstructing algebraically factorized quantum Hamiltonian and symmetry operators in the Fock type space and its application to quantum Hamiltonian and symmetry operators in case of quantum integrable spatially many- and one-dimensional dynamical systems. As examples, we have studied in detail the factorized structure of Hamiltonian operators, describing such quantum integrable spatially many- and one-dimensional models as generalized oscillatory, Calogero-Sutherland, Coulomb type and nonlinear Schrödinger dynamical systems of spinless bose-particles.
In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, G2G_2G2​ rational, or I6I_6I6​) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index kkk. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure g(k)g^{(k)}g(k) with various kkk, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra g(k)g^{(k)}g(k) for a certain kkk. In all presented cases the algebra of integrals is a 4-generated (H,I1,I2,I12≡[I1,I2])(H, I_1, I_2, I_{12}\equiv[I_1, I_2])(H,I1​,I2​,I12​≡[I1​,I2​]) infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.