Chern–Simons theory and BCS superconductivity

Asorey, M.; Falceto, Fernando; Sierra, Germȧn

doi:10.1016/s0550-3213(01)00614-9

Cited by 51 publications

(65 citation statements)

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“…The number of these parameters E a k is: (19), (21), and (23) extend the results presented in [8,9] for the rational case, to the trigonometric and hyperbolic ones.…”

Section: Richardson-gaudin Models From a Generalized Gaudin Algebrasupporting

confidence: 68%

“…In practice this method consists of using the already known solutions of the general-Lie-algebra Yang-Baxter Equations (obtained by using the Quantum Inverse Scattering method) to obtain the corresponding solution of the RG models by taking the classical limit of the respective Bethe equations. From a Conformal Field Theory context, Asorey, Falceto and Sierra [8] obtained the integrals of motion and respective eigenvalues of the rational RG models for any simple Lie algebra as a limiting case of the Chern-Simons theory.…”

Section: Introductionmentioning

confidence: 99%

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SU(3) Richardson–Gaudin models: three-level systems

Errea¹

2007

J. Phys. A: Math. Theor.

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Abstract. We present the exact solution of the Richardson-Gaudin models associated with the SU(3) Lie algebra. The derivation is based on a Gaudin algebra valid for any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For the rational case additional cubic integrals of motion are obtained, whose number is added to that of the quadratic ones to match, as required from the integrability condition, the number of quantum degrees of freedom of the model. We discuss different SU(3) physical representations and elucidate the meaning of the parameters entering in the formalism. By considering a bosonic mapping limit of one of the SU(3) copies, we derive new integrable models for three level systems interacting with two bosons. These models include a generalized Tavis-Cummings model for three level atoms interacting with two modes of the quantized electric field.PACS numbers: 02.30. Ik, 03.65.Fd, 31.15.Hz, Submitted to: J. Phys. A: Math. Gen. IntroductionThe Richardson-Gaudin (RG) exactly solvable models [1,2] can be traced back to the exact solution of the BCS Hamiltonian given by Richardson in the early 1960's [3] and to the integrable spin model developed by Gaudin in the seventies [4]. They are diagonalizable by Bethe Ansatz techniques and are the so-called classical limit of two dimensional vertex models. We refer the reader to [5], where the connection between the RG models and the inhomogeneous XXX vertex model with twisted boundary conditions is established in detail for the case of the rank-1 SU(2) algebra. Previously, in [6] it was shown that the solution of the Gaudin models [those without linear term in the integrals of motion, cf.(19) below] associated with more general Lie algebras, can be used to get a solution of the corresponding Classical Yang-Baxter Equations (CYBE). This connection has been largely exploited to diagonalize the RG models (see [2,7] for some examples). In practice this method consists of using the already known solutions of the general-Lie-algebra Yang-Baxter Equations (obtained by using the Quantum Inverse Scattering method) to obtain the corresponding solution of the RG models by taking the classical limit of the respective Bethe equations. From a Conformal Field Theory context, Asorey, Falceto and Sierra [8] obtained the integrals SU(3) Richardson-Gaudin models 2 of motion and respective eigenvalues of the rational RG models for any simple Lie algebra as a limiting case of the Chern-Simons theory.Here we follow an alternative and more direct approach to diagonalize the RG models. Though essentially equivalent, it differs in practice. This approach, where no reference to the YBE is necessary, is similar to the presented by Ushveridze [9] for the rational case. The method is based on the introduction of an infinite dimensional algebra (the Gaudin algebra) associated with the Lie-algebra of a simple group. By taking a Casimir-like operator in the Gaudin algebra, one gets the transfer matrix of the YBE approach and, consequently, a set of independent quadratic ...

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“…The number of these parameters E a k is: (19), (21), and (23) extend the results presented in [8,9] for the rational case, to the trigonometric and hyperbolic ones.…”

Section: Richardson-gaudin Models From a Generalized Gaudin Algebrasupporting

confidence: 68%

Section: Introductionmentioning

confidence: 99%

SU(3) Richardson–Gaudin models: three-level systems

Errea¹

2007

J. Phys. A: Math. Theor.

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“…£ [2] ÒÑÍÂÊÂÐÑ, ÚÕÑ ÍÑÓÐË ¢ÇÕÇ ÔÑÑÕÄÇÕÔÕÄÖáÕ ÄÑÊÃÖÉAEÈÐÐÞÏ ÍÖÒÇÓÑÄÔÍËÏ ÒÂÓÂÏ. ³ ÕÑÚÍË ÊÓÇÐËâ ÍÑÐ×ÑÓÏÐÑÌ ÕÇÑÓËË ÒÑÎâ ÍÖÒÇÓÑÄÔÍËÇ ÒÂÓÞ ÔÑÑÕÄÇÕÔÕÄÖáÕ ÑÒÇÓÂÕÑÓÂÏ àÍÓÂ-ÐËÓÑÄÂÐËâ [34,35].…”

Section: ²çðñóïåóöòòñäñì ùëíî ä åóâ×çðçunclassified

Limit cycles in renormalization group dynamics

Bulycheva¹,

Gorskii²

2014

Usp. Fiz. Nauk

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“…The main result of [16] is to show that, for a generic simple simply connected, compact Lie group G, the Richardson equations, the CRS conserved quantities and their eigenvalues arise from the KZB connection and their associated horizontal sections. This is done in a limit where the torus degenerates into the cylinder and then into the complex plane.…”

Section: Bcs and Chern-simons Theory(2001)mentioning

confidence: 99%