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Generalized Calogero-Sutherland systems from many-matrix models

Polychronakos, A P

Published Jan 1, 1999• 12 citations
OpenAlex

Abstract

We construct generalizations of the Calogero-Sutherland-Moser system by appropriately reducing a model involving many unitary matrices. The resulting systems consist of particles on the circle with internal degrees of freedom, coupled through modifications of the inverse-square potential. The coupling involves SU(M) non-invariant (anti)ferromagnetic interactions of the internal degrees of freedom. The systems are shown to be integrable and the spectrum and wavefunctions of the quantum version are derived.

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Multi-component Sutherland model

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The Sutherland model has a number of variants. One of them is the U( K ) Sutherland model [71, 85, 86, 132]. This model describes N particles moving along a circle of perimeter L , and each particle possesses an internal degree of freedom with K possible values. This corresponds to spin with K = 2, and more generally a color. In the U( K ) Sutherland model, all particles obey common statistics: bosonic or fermionic. We can generalize the model further. The U( K B , K F ) Sutherland model [177] consists of bosons having K B possible colors and fermions having K F colors. The multi-component Sutherland model has a degeneracy in energy levels which is described in terms of a Yangian. The Yangian is an algebra related to quantum groups [43, 44]. The Yangian is nicely realized by variants of Jack polynomials which are modified so as to conform to the internal symmetry. Elementary excitations in the multi-component Sutherland model are described in a few alternative ways: interacting bosonic or fermionic particles, or non-interacting particles obeying generalized exclusion statistics. Furthermore, the lattice models such as the Haldane–Shastry models [77, 161] and 1/ r 2 supersymmetric t – J model [119] are obtained in the strong coupling limit of U(2) and U(2, 1) Sutherland models, respectively. The Sutherland models in the continuum space are much more tractable mathematically than the corresponding lattice models. Hence, the mapping to lattice models turns out to be useful to derive the explicit results on thermodynamics and dynamics in lattice models. In the present chapter, we extend our treatment for the single-component Sutherland model in order to include the internal degrees of freedom. We shall discuss the energy spectrum, thermodynamics, and dynamical correlation functions.

Cambridge University Press eBooks•Aug 6, 2009•0 citations
We construct generalizations of the Calogero-Sutherland-Moser system by appropriately reducing a model involving many unitary matrices. The resulting systems consist of particles on the circle with internal degrees of freedom, coupled through modifications of the inverse-square potential. The coupling involves SU(M) non-invariant (anti)ferromagnetic interactions of the internal degrees of freedom. The systems are shown to be integrable and the spectrum and wavefunctions of the quantum version are derived.
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.
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.
Relations between free motion on the GL + ( n , ℝ) group manifold and the dynamics of an n -particle system with spin degrees of freedom on a line interacting with a pairwise 1/sinh 2 x “potential” (Euler-Calogero-Sutherland model) are discussed within a Hamiltonian reduction. Two kinds of reductions of the degrees of freedom are considered: that which is due to continuous invariance and that which is due to discrete symmetry. It is shown that, upon projecting onto the corresponding invariant manifolds, the resulting Hamiltonian system represents the Euler-Calogero-Sutherland model in both cases.
We review recent results on new physical models constructed as PT-symmetrical deformations or extensions of different types of integrable models. We present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero-Moser-Sutherland type and nonlinear integrable field equations of Korteweg-de Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero-Moser-Sutherland models, we provide three alternative deformations: a complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real-valued field equations of nonlinear integrable systems; and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of Korteweg-de Vries type are studied with regard to different kinds of PT-symmetrical scenarios. A reduction to simple complex quantum mechanical models currently under discussion is presented.
The Sutherland model has a number of variants. One of them is the U( K ) Sutherland model [71, 85, 86, 132]. This model describes N particles moving along a circle of perimeter L , and each particle possesses an internal degree of freedom with K possible values. This corresponds to spin with K = 2, and more generally a color. In the U( K ) Sutherland model, all particles obey common statistics: bosonic or fermionic. We can generalize the model further. The U( K B , K F ) Sutherland model [177] consists of bosons having K B possible colors and fermions having K F colors. The multi-component Sutherland model has a degeneracy in energy levels which is described in terms of a Yangian. The Yangian is an algebra related to quantum groups [43, 44]. The Yangian is nicely realized by variants of Jack polynomials which are modified so as to conform to the internal symmetry. Elementary excitations in the multi-component Sutherland model are described in a few alternative ways: interacting bosonic or fermionic particles, or non-interacting particles obeying generalized exclusion statistics. Furthermore, the lattice models such as the Haldane–Shastry models [77, 161] and 1/ r 2 supersymmetric t – J model [119] are obtained in the strong coupling limit of U(2) and U(2, 1) Sutherland models, respectively. The Sutherland models in the continuum space are much more tractable mathematically than the corresponding lattice models. Hence, the mapping to lattice models turns out to be useful to derive the explicit results on thermodynamics and dynamics in lattice models. In the present chapter, we extend our treatment for the single-component Sutherland model in order to include the internal degrees of freedom. We shall discuss the energy spectrum, thermodynamics, and dynamical correlation functions.