Quadratic operator relations and Bethe equations for spin-1/2 Richardson–Gaudin models

Dimo, Claude; Faribault, Alexandre

doi:10.1088/1751-8121/aaccb4

Cited by 12 publications

(11 citation statements)

References 52 publications

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“…still commute with one another therefore defining an integrable model allowing us to retain the major simplifications that integrability has to offer. Our numerically study of the model's eigenstates makes use of recent work [28,29,33] which has shown explicitly that the set of eigenvalues (r 1 , r 2 . .…”

Section: Richardson-gaudin Modelsmentioning

confidence: 99%

Read-Green points and level crossings in XXZ central spin models and px+ipy topological superconductors

2019

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In this work, we study the full set of eigenstates of a px + ipy topological superconductor coupled to a particle bath which can be described in terms of an integrable Hamiltonian of the Richardson-Gaudin class. The results derived in this work also characterise the behaviour of an anisotropic XXZ central spin model in a external magnetic field since both types of Hamiltonian are know to share the exact same conserved quantities making them formally equivalent.We show how by ramping the coupling strength (or equivalently the magnetic field acting in the z-direction on the central spin), each individual eigenstate undergoes a sequence of gain/loss of excitations when crossing the specific values known as Read-Green points. These features are shown to be completely predictable, for every one of the 2 N eigenstates, using only two integers obtainable easily from the zero-coupling configuration which defines the eigenstate in question.These results provide a complete map of the particle-number sectors (superconductor) or magnetisation sectors (central spin) involved in the large number of level-crossings which occur in these systems at the Read-Green points. It further allows us to define quenching protocols which could create states with remarkably large excitation-number fluctuations.

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Section: Richardson-gaudin Modelsmentioning

confidence: 99%

Read-Green points and level crossings in XXZ central spin models and px+ipy topological superconductors

2019

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“…(7) was given in Ref. 11. Hence at least in the case when all the j p = 1/2, all the higher powers of h p can be written as multi-linear combinations of h p s. Here we derive similar expressions for higher values of j p .…”

Section: Introductionmentioning

confidence: 60%

Symmetries of Hamiltonians describing systems with arbitrary spins

Cervia

Patwardhan

Balantekin

2019

Int. J. Mod. Phys. E

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“…For all of the models parametrised by (2), it was shown [28] that integrability is sufficient to establish that the resulting N conserved charges obey, at the operator level, the following quadratic relations:…”

Section: The Modelsmentioning

confidence: 99%

“…While similar constructions are possible for higher spins s > 1/2 (see [20] for example), they must contain additional local terms which takes them outside of the class of RG models studied here which is assumed to only contain S x i S x j , S y i S y j and S z i S z j coupling terms between spins i and j. It was recently demonstrated that, for these spin-1/2 systems in an external field, the conserved charges systematically obey quadratic relations so that the eigenvalues characterising their eigenstates can always be found as the solutions to a small set of quadratic equations [28]. Expectation values of local spin operators, in any of these eigenstates, can then also be computed easily by making use of the Hellmann-Feynman theorem giving access to some of the physics of these systems [29].…”

Section: Introductionmentioning

confidence: 99%

"Bethe-Ansatz-free" eigenstates of spin-1/2 Richardson-Gaudin integrable models

Faribault

Dimo

2018

Preprint

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In this work we construct the eigenstates of the most general spin-1/2 Richardson-Gaudin model integrable in an external magnetic field. This includes the possibility for fully anisotropic XYZ coupling such that the S x i S x j , S y i S y j and S z i S z j terms all have distinct coupling strengths. While insuring that integrability is maintained in the presence of an external field excludes the elliptic XYZ model which is only integrable at zero field, this work still covers a wide class of fully anisotropic (XYZ) models associated with non skew-symmetric r-matrices.The eigenstates, as constructed here, do not require any usable Bethe ansatz and therefore: no proper pseudo-vacuum, Bethe roots, or generalised spin raising (Gaudin) operators have to be defined. Indeed, the eigenstates are generically built only through the conserved charges which define the model of interest and the specification of the set of eigenvalues defining the particular eigenstate. Since these eigenvalues are, in general, solutions to a simple set of quadratic equations, the proposed approach is simpler to implement than any Bethe ansatz and, moreover, it remains completely identical independently of the symmetries of the model. Indeed, the construction removes any distinction between XYZ models and XXZ/XXX models and, generically, that between models with or without U(1) so that any difficulties associated with the use of a Bethe ansatz in any of these cases are avoided.

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Quadratic operator relations and Bethe equations for spin-1/2 Richardson–Gaudin models

Cited by 12 publications

References 52 publications

Read-Green points and level crossings in XXZ central spin models and px+ipy topological superconductors

Read-Green points and level crossings in XXZ central spin models and px+ipy topological superconductors

Symmetries of Hamiltonians describing systems with arbitrary spins

"Bethe-Ansatz-free" eigenstates of spin-1/2 Richardson-Gaudin integrable models

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