Eigenvectors of open<i>XXZ</i>and ASEP models for a class of non-diagonal boundary conditions

Crampé, Nicolas; Ragoucy, É.; Simon, Damien

doi:10.1088/1742-5468/2010/11/p11038

Cited by 57 publications

(79 citation statements)

References 44 publications

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“…While the "faulty" pseudovacuum |Ω remains an eigenstate of the S z (u) operator defined in (18), it no longer is an eigenstate of the resulting S 2 (u). This lack of a proper vacuum reference state is also typical of integrable models without U(1) symmetry and consequently a wide variety of approaches have been built in order to address this specific issue: from the diagonalisation of the XXZ open spin chain from the representation theory of the q-Onsager algebra [12], to the generalisation of the coordinate Bethe ansatz used in the XXZ chain and the antisymmetric simple exclusion process (ASEP) models in [13] as well as the functional Bethe ansatz used to treat various boundaries problems in XXX and XXZ chains [14,15,16].…”

Section: Quantum Inverse Scattering Methodsmentioning

confidence: 99%

Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields

Faribault

Tschirhart

2017

SciPost Phys.

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In this work we demonstrate a simple way to implement the quantum inverse scattering method to find eigenstates of spin-1/2 XXX Gaudin magnets in an arbitrarily oriented magnetic field. The procedure differs vastly from the most natural approach which would be to simply orient the spin quantisation axis in the same direction as the magnetic field through an appropriate rotation.Instead, we define a modified realisation of the rational Gaudin algebra and use the quantum inverse scattering method which allows us, within a slightly modified implementation, to build an algebraic Bethe ansatz using the same unrotated reference state (pseudovacuum) for any external field. This common framework allows us to easily write determinant expressions for certain scalar products which would be highly non-trivial in the rotated system approach.

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Section: Quantum Inverse Scattering Methodsmentioning

confidence: 99%

Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields

Faribault

Tschirhart

2017

SciPost Phys.

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“…Now remarking that: c hn (n, α, β, γ)q hn = c hn (n, α, β/q, γq), (A. 45) as the effect of q hn is to bring k n to k n + 1 in the state Ω n,α,β,γ |, this, for the gauge choice (A.28), being equivalent to the above redefinitions of the gauge parameters. So that we get:…”

Section: A1 Gauge Transformed Yang-baxter Generatorsmentioning

confidence: 99%

Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II

2018

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This article is a direct continuation of [1] where we begun the study of the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. There we addressed this problem for the case where one of the K-matrices describing the boundary conditions is triangular. In the present article we consider the most general integrable boundary conditions, namely the most general boundary K-matrices satisfying the reflection equation. The spectral analysis is developed by implementing the method of Separation of Variables (SoV). We first design a suitable gauge transformation that enable us to put into correspondence the spectral problem for the most general boundary conditions with another one having one boundary K-matrix in a triangular form. In these settings the SoV resolution can be obtained along an extension of the method described in [1]. The transfer matrix spectrum is then completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions and equivalently as the set of solutions to an analogue of Baxter's T-Q functional equation. We further describe scalar product properties of the separate states including eigenstates of the transfer matrix.

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“…How to apply the Bethe ansatz in these cases is still an open question. However, let us mention that new methods appeared recently in order to solve similar problems where there is no conserved charge due to the boundary conditions (generalization of the CBA [17], Onsager approach [18], separation of variables [19], inhomogeneous Bethe equation [20], modified algebraic Bethe ansatz [21]). A generalization of these methods may be possible to deal with the eigenvalue problem (3.13).…”

Section: Coordinate Bethe Ansatzmentioning

confidence: 99%

3-state Hamiltonians associated to solvable 33-vertex models

Crampé

Frappat

Ragoucy

et al. 2016

Journal of Mathematical Physics

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Using the nested coordinate Bethe ansatz, we study 3-state Hamiltonians with 33 non-vanishing entries, or 33-vertex models, where only one global charge with degenerate eigenvalues exists and each site possesses three internal degrees of freedom. In the context of Markovian processes, they correspond to diffusing particles with two possible internal states which may be exchanged during the diffusion (transmutation). The first step of the nested coordinate Bethe ansatz is performed providing the eigenvalues in terms of rapidities. We give the constraints ensuring the consistency of the computations. These rapidities also satisfy Bethe equations involving 4 × 4 R-matrices, solutions of the YangBaxter equation which implies new constraints on the models. We solve them allowing us to list all the solvable 33-vertex models.

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Eigenvectors of openXXZand ASEP models for a class of non-diagonal boundary conditions

Cited by 57 publications

References 44 publications

Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields

Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields

Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II

3-state Hamiltonians associated to solvable 33-vertex models

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