Algebraic bethe ansatz for the XYZ Gaudin model

Sklyanin, E. K.; Takebe, Takashi

doi:10.1016/0375-9601(96)00448-3

Cited by 70 publications

(100 citation statements)

References 12 publications

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“…It seems rather plausible in view of the formal similarities of the rational, trigonometric and elliptic models that some version of Sklyanin's functional Bethe ansatz should also be feasible in the latter case as has already been achieved for the XY Z Gaudin magnet in [10].…”

Section: Resultsmentioning

confidence: 97%

“…The monodromy matrix T (λ, {λ i }) (generalized to the inhomogeneous chain [9], [10]) is given as the ordered product of Lax operators…”

Section: Xy Z Model and Its Relation To Icetype Modelsmentioning

confidence: 99%

“…The monodromy matrix related to this modified Yang-baxter equation is 10) where 0 denotes the horizontal auxiliary space (with the asssociated spectral parameter λ 0 ), the positive integers 1, . .…”

Section: Xy Z Model and Its Relation To Icetype Modelsmentioning

confidence: 99%

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The Drinfel'd twisted XYZ model

Poghossian¹

2000

Proceedings of Non-Perturbative Quantum Effects 2000 — PoS(tmr2000)

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Section: Resultsmentioning

confidence: 97%

“…The monodromy matrix T (λ, {λ i }) (generalized to the inhomogeneous chain [9], [10]) is given as the ordered product of Lax operators…”

Section: Xy Z Model and Its Relation To Icetype Modelsmentioning

confidence: 99%

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The Drinfel'd twisted XYZ model

Poghossian¹

2000

Proceedings of Non-Perturbative Quantum Effects 2000 — PoS(tmr2000)

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“…Following [36,37], the elliptic Gaudin operators (5.4) are obtained by expanding the double-row transfer matrix (2.15) at the point u = z j around w = 0:…”

Section: Results For the Associated Gaudin Modelmentioning

confidence: 99%

Multiple reference states and complete spectrum of the Belavin model with open boundaries

Yang

Zhang

2008

Nuclear Physics B

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The multiple reference state structure of the Z n Belavin model with non-diagonal boundary terms is discovered. It is found that there exist n reference states, each of them yields a set of eigenvalues and Bethe Ansatz equations of the transfer matrix. These n sets of eigenvalues together constitute the complete spectrum of the model. In the quasi-classic limit, they give the complete spectrum of the corresponding Gaudin model.

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“…Following a review [DPS04] it has become commonplace to refer to these models as Richardson-Gaudin models. The elliptic case (see, e.g., [ST96,ED15]) is more challenging, since it breaks u(1) symmetry leading to non-conservation of particle number. In this thesis we will focus on the rational and trigonometric constructions, leaving the elliptic case for future work.…”

Section: Introductionmentioning

confidence: 99%

Richardson-Gaudin models from the Boundary Quantum Inverse Scattering Method

Lukyanenko¹

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Richardson-Gaudin models are a class of quantum integrable models connected to many physical systems, including pairing Hamiltonians from the theory of superconductivity. They can be obtained in the quasi-classical limit of the Quantum Inverse Scattering Method, which is based on an R-matrix and a Lax operator satisfying the Yang-Baxter equation. They can also be obtained from the Boundary Quantum Inverse Scattering Method, which relies on solutions of the reflection equations known as K-matrices. In this thesis we study these latter models systematically, explore the connections between them and investigate the interpretation of the "boundary".First of all, we consider Richardson-Gaudin models obtained from the spin-1/2 su(2) Boundary Quantum Inverse Scattering Method with diagonal K-matrices. We prove that the trigonometric boundary construction is equivalent to its rational limit, through a change of variables, rescaling, and a basis transformation. Moreover, we prove that the twisted-periodic and boundary constructions are equivalent in the trigonometric case, but not in the rational limit. Thus, including the "boundary" does not lead to a new model in this case.Next, we investigate Richardson-Gaudin models obtained from the spin-1/2 su(2) Boundary Quantum Inverse Scattering Method with non-diagonal K-matrices. Here the situation is different. The conserved operators in the boundary construction are no longer equivalent to the ones in the twisted-periodic construction. Also, the rational and the trigonometric boundary constructions are not equivalent. In the rational case this allows us to construct a generalisation of the p+ip pairing Hamiltonian with external interaction terms. In the trigonometric case the expressions for the conserved operators involve several free parameters, which can be adjusted to construct a variety of Hamiltonians. This result offers opportunities for future investigations.Finally, we study the case of the q-deformed bosonic Lax operator. This case is much more challenging than the case of the spin Lax operator. It is not straightforward to define the rational and quasi-classical limits of the bosonic Lax operator. Even after making modifications to the Lax operator for these limits to be well defined, it turns out that the limits do not commute. We state some open questions for future work.

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Algebraic bethe ansatz for the XYZ Gaudin model

Cited by 70 publications

References 12 publications

The Drinfel'd twisted XYZ model

The Drinfel'd twisted XYZ model

Multiple reference states and complete spectrum of the Belavin model with open boundaries

Richardson-Gaudin models from the Boundary Quantum Inverse Scattering Method

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