Eigenvalue-based determinants for scalar products and form factors in Richardson–Gaudin integrable models coupled to a bosonic mode

Claeys, Pieter W.; Baerdemacker, Stijn De; Raemdonck, Mario Van; Neck, Dimitri Van

doi:10.1088/1751-8113/48/42/425201

Cited by 15 publications

(17 citation statements)

References 51 publications

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“…A common approach, known as the eigenvaluebased method, maps the Richardson-Gaudin equations (3) to an equivalent set of equations for the variables 13,14,25,52,[93][94][95][96][97][98][99]…”

Section: Numericsmentioning

confidence: 99%

Variational method for integrability-breaking Richardson-Gaudin models

et al. 2017

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We present a variational method for approximating the ground state of spin models close to (Richardson-Gaudin) integrability. This is done by variationally optimizing eigenstates of integrable Richardson-Gaudin models, where the toolbox of integrability allows for an efficient evaluation and minimization of the energy functional. The method is shown to return exact results for integrable models and improve substantially on perturbation theory for models close to integrability. For large integrability-breaking interactions, it is shown how (avoided) level crossings necessitate the use of excited states of integrable Hamiltonians in order to accurately describe the ground states of general non-integrable models.

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Section: Numericsmentioning

confidence: 99%

Variational method for integrability-breaking Richardson-Gaudin models

et al. 2017

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“…These equations are also known as the substituted or quadratic Bethe equations. Originally obtained by Babelon and Talalaev [19], these were later extended in a series of articles towards all known Richardson-Gaudin models [21,22,26,34,36]. They arose as a way of circumventing the singular behaviour of the regular Richardson-Gaudin equations, since they do not exhibit the singularities associated with the poles in the regular Bethe equations [37][38][39].…”

Section: Eigenvalue-based Frameworkmentioning

confidence: 99%

“…In the second approach, the so-called eigenvalue-based approach, eigenstates are characterized by solving for a set of variables determining the eigenvalues of the conserved charges for each eigenstate. The correlation coefficients can then also be calculated as DWPFs depending only on these new variables, keeping the rapidities implicit [19][20][21][22]. In this work, the connection between these approaches is made explicit, highlighting the Cauchy structure of all involved matrices, which follows from the rational functions defining these models.…”

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confidence: 99%

Inner products in integrable Richardson-Gaudin models

2017

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We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula.Furthermore, this framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.

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“…One should also point out that a completely distinct approach using a pseudo-deformation of the algebra has also recently been used in [14] to demonstrate the results we previously published in [13] concerning spin-boson realizationss used to define Tavis-Cummings-like integrable hamiltonians.…”

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confidence: 90%

Determinant representation of the domain-wall boundary condition partition function of a Richardson–Gaudin model containing one arbitrary spin

Faribault¹,

Tschirhart²,

Müller³

2016

J. Phys. A: Math. Theor.

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In this work we present a determinant expression for the domain-wall boundary condition partition function of rational (XXX) Richardson-Gaudin models which, in addition to N 1 -spins 1 2, contains one arbitrarily large spin S. The proposed determinant representation is written in terms of a set of variables which, from previous work, are known to define eigenstates of the quantum integrable models belonging to this class as solutions to quadratic Bethe equations. Such a determinant can be useful numerically since systems of quadratic equations are much simpler to solve than the usual highly nonlinear Bethe equations. It can therefore offer significant gains in stability and computation speed.

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Eigenvalue-based determinants for scalar products and form factors in Richardson–Gaudin integrable models coupled to a bosonic mode

Cited by 15 publications

References 51 publications

Variational method for integrability-breaking Richardson-Gaudin models

Variational method for integrability-breaking Richardson-Gaudin models

Inner products in integrable Richardson-Gaudin models

Determinant representation of the domain-wall boundary condition partition function of a Richardson–Gaudin model containing one arbitrary spin

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