An explicit demonstration of integrability for the linear Heisenberg chain of <i>N</i> = 9 nodes

Topolewicz, Stanisław; Lulek, T.

doi:10.1002/pssb.200674624

Cited by 3 publications

(1 citation statement)

References 5 publications

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“…We use the algebraic Bethe ansatz approach of Faddeev and Takhtajan [9,10] to derive explicitly the set of N − 1 mutually commuting operators (involving the Heisenberg Hamiltonian for the XXX model) from the corresponding monodromy matrix, point out some their general properties and special cases, and compare on some examples the resulting eigenstates with the exact Bethe ansatz solutions, classified by rigged string configurations, as defined by Kerov, Kirillov and Reshetikhin [7]. Some preliminary results on these operators have been reported in our previous papers [11,12].…”

Section: Introductionmentioning

confidence: 99%

A complete set of commuting operators for the Bethe ansatz

Lulek¹,

Topolewicz²

2010

J. Phys. A: Math. Theor.

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We have derived an explicit formula for a complete set of commuting operators for the XXX model of the Heisenberg magnetic ring, using an algebraic Bethe ansatz approach of Faddeev and Takhtajan. Each operator turns out to be the sum of increasing l-cycles in the symmetric group acting on the set of nodes of the ring. It is demonstrated that the resulting algebra of operators encloses the total spin S, the Hamiltonian and the quasimomentum. We point out that it is a maximal Abelian subalgebra in the algebra of the symmetric group, associated with the basis of exact Bethe ansatz eigenstates, the latter classified by rigged string configurations of Kerrov, Kirillov and Reshetikhin. This algebra is also conjugated to the Jucys-Murphy algebra, responsible for the Young orthogonal basis of standard tableaux along the Schur-Weyl duality.

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Section: Introductionmentioning

confidence: 99%

A complete set of commuting operators for the Bethe ansatz

Lulek¹,

Topolewicz²

2010

J. Phys. A: Math. Theor.

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Bethe ansatz, quantum computers, and unitary geometry

Lulek

2008

J. Phys.: Conf. Ser.

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Schwinger Geometry, Bethe Ansatz, and a Magnonic Qudit

Jakubczyk

Topolewicz

Wal

et al. 2009

Open Syst. Inf. Dyn.

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Schwinger approach of unitary geometry for a finite-dimensional Hilbert space is interpreted in terms of a magnonic qudit — a hypothetic elementary unit of memory of a quantum computer. The space is interpreted within the Heisenberg model for a magnetic ring, its calculational basis as the classical configuration space for a single spin deviation, treated as a Bethe pseudoparticle, and the dual basis corresponds to quasimomenta, so that the classical phase space spans the quantum algebra of observables. Effects of the Schur-Weyl duality and Bethe ansatz exact eigenstates of the Heisenberg Hamiltonian for the XXX model on properties of the magnonic qudit are presented.

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An explicit demonstration of integrability for the linear Heisenberg chain of N = 9 nodes

Cited by 3 publications

References 5 publications

A complete set of commuting operators for the Bethe ansatz

A complete set of commuting operators for the Bethe ansatz

Bethe ansatz, quantum computers, and unitary geometry

Schwinger Geometry, Bethe Ansatz, and a Magnonic Qudit

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