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Journal of the Physical Society of Japan

Integrability of the Quantum Calogero-Moser Model

Hideaki Ujino, Kazuhiro Hikami, Miki Wadati

Published Oct 15, 1992• 50 citations
Publisher

Abstract

The integrability of the quantum Calogero-Moser model is proved through the quantum inverse scattering method. By use of the Lax operator L , a new formula for the conserved operators is found. The formula suggests that Tr L n , which are conserved in the classical case, are not necessarily conserved operators. The Hénon-type conserved operators are also presented.

DOI
10.1143/jpsj.61.3425

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The integrability of the quantum Calogero-Moser model is proved through the quantum inverse scattering method. By use of the Lax operator L , a new formula for the conserved operators is found. The formula suggests that Tr L n , which are conserved in the classical case, are not necessarily conserved operators. The Hénon-type conserved operators are also presented.
For the quantum Calogero-Moser model, we construct a set of conserved operators and another set of operators, named boost operators, from its Lax operator. We prove that each conserved operator satisfies both the Lax equation and a remarkable relation named additional relation. From these knowledge, we show that the conserved operators are involutive. Moreover, the conserved operators and the boost operators constitute the U(1)-current algebra. All the proofs are simplified a great deal due to the Lax equations and additional relations.
The integrability of the one-dimensional (1D) fermion chain model is investigated in the framework of the Quantum Inverse Scattering Method (QISM). We introduce a new R-operator for the fermion chain model, which is expressed in terms of the fermion operators. The R-operator satisfies a new type of the Yang-Baxter relation with fermionic L-operator. We derive the fermionic Sutherland equation from the relation, which is equivalent to the fermionic Lax equation. It also provides a mathematical foundation of the boost operator approach for the fermion model. In fact, we obtain some higher conserved quantities of the fermion model using the boost operator.
A generalization of the quantum inverse scattering method is proposed replacing the quantum group RLLRLLRLL commutation relations of Lax operators by reflection equation type RLRLRLRLRLRL commutation relations. Under some natural assumptions the most general algebra of this type allowing to construct the neccessary integrals of motion is found. It serves to describe Lax operators with completely non-ultralocal commutation relations. An example of this new formalism is an integrable model on monodromies of flat connections on a Riemann surface which is related to the XXZ quantum spin chain.
We investigate the exact integrability of the one-dimensional (1D) Bariev model in the framework of the quantum inverse scattering method (QISM). Using the Jordan - Wigner transformation, the 1D Bariev model can be regarded as a coupled spin model. We construct the higher conserved currents which commute with the Hamiltonian. The explicit form of the conserved currents helps us to infer the L-operator of the 1D Bariev model. From the L-operator, we construct a transfer matrix which is a generating function of the conserved currents. We also find the corresponding R-matrix which satisfies the Yang - Baxter relation. Thus the exact integrability of the 1D Bariev model is established. The R-matrix does not have the `difference property' for the spectral parameter, as in the case of the 1D Hubbard model. We also provide the Lax representation and the fermionic formulation of the Yang - Baxter relation.