Symmetries of Spin Calogero Models

Caudrelier, Vincent; Crampé, Nicolas

doi:10.3842/sigma.2008.090

Cited by 3 publications

(4 citation statements)

References 38 publications

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“…(4.22) and (4.11) also suggests that, except for the above mentioned special cases where either m 1 " n 1 " 0 or m 2 " n 2 " 0, the symmetry algebra of the Hamiltonian H pm 1 ,m 2 |n 1 ,n 2 q N would be a proper subalgebra of the Y pglpm|nqq Yangian algebra. Even though the symmetry algebras of some spin Calogero models associated with the BC N root system have been studied earlier [55,56], as far as we know the symmetry algebras of the BC N type of PF spin chains like (2.9) have not received much attention. So it might be interesting as a future study to find out the symmetry algebras of these BC N type of spin chains and explore how the representations of such symmetry algebras are connected with the q-deformed and the restricted super Schur polynomials.…”

Section: Discussionmentioning

confidence: 99%

Appearance of branched motifs in the spectra of BC type Polychronakos spin chains

Basu-Mallick

Sinha

2020

Nuclear Physics B

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As is well known, energy levels appearing in the highly degenerate spectra of the A N´1 type of Haldane-Shastry and Polychronakos spin chains can be classified through the motifs, which are characterized by some sequences of the binary digits like '0' and '1'. In a similar way, at present we classify all energy levels appearing in the spectra of the BC N type of Polychronakos spin chains with Hamiltonians containing supersymmetric analogue of polarized spin reversal operators. To this end, we show that the BC N type of multivariate super Rogers-Szegö (SRS) polynomials, which at a certain limit reduce to the partition functions of the later type of Polychronakos spin chains, satisfy some recursion relation involving a q-deformation of the elementary supersymmetric polynomials. Subsequently, we use a Jacobi-Trudi like formula to define the corresponding q-deformed super Schur polynomials and derive a novel expression for the BC N type of multivariate SRS polynomials as suitable linear combinations of the q-deformed super Schur polynomials. Such an expression for SRS polynomials leads to a complete classification of all energy levels appearing in the spectra of the BC N type of Polychronakos spin chains through the 'branched' motifs, which are characterized by some sequences of integers of the form pδ 1 , δ 2 , ..., δ N´1 |lq, where δ i P t0, 1u and l P t0, 1, ..., Nu. Finally, we derive an extended boson-fermion duality relation among the restricted super Schur polynomials and show that the partition functions of the BC N type of Polychronakos spin chains also exhibit similar type of duality relation.

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Section: Discussionmentioning

confidence: 99%

Appearance of branched motifs in the spectra of BC type Polychronakos spin chains

Basu-Mallick

Sinha

2020

Nuclear Physics B

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“…The choice of the folding constants is dictated by the sym- [23,24]. The method presented in this letter can be easily applied to any integrable spin chains.…”

Section: Discussionmentioning

confidence: 99%

How to fold a spin chain: Integrable boundaries of the Heisenberg XXX and Inozemtsev hyperbolic models

Gomez¹,

MacKay²,

Regelskis³

2017

Physics Letters A

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We present a general method of folding an integrable spin chain, defined on a line, to obtain an integrable open spin chain, defined on a half-line. We illustrate our method through two fundamental models with sl 2 Lie algebra symmetry: the Heisenberg XXX and the Inozemtsev hyperbolic spin chains. We obtain new long-range boundary Hamiltonians and demonstrate that they exhibit Yangian symmetries, thus ensuring integrability of the models we obtain. The method presented provides a "bottom-up" approach for constructing integrable boundaries and can be applied to any spin chain model.

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“…The search for integrable systems associated to the action of more complicated finite groups, and in particular of the cyclic group Z N , has attracted a lot of attention recently (see e.g. [8,9,10,11,12,13,14,15,16]). In this paper, we introduce a new equation which generalizes the classical reflection equation (2) as well as algebraic structures related to Z N (cyclotomic) models.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Classical N-reflection equation and Gaudin models

Caudrelier¹,

Crampé²

2018

Lett Math Phys

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We introduce the notion of N -reflection equation which provides a generalization of the usual classical reflection equation describing integrable boundary conditions. The latter is recovered as a special example of the N = 2 case. The basic theory is established and illustrated with several examples of solutions of the N -reflection equation associated to the rational and trigonometric r-matrices. A central result is the construction of a Poisson algebra associated to a non skew-symmetric r-matrix whose form is specified by a solution of the N -reflection equation. Generating functions of quantities in involution can be identified within this Poisson algebra. As an application, we construct new classical Gaudin-type Hamiltonians, particular cases of which are Gaudin Hamiltonians of BC L -type.

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Symmetries of Spin Calogero Models

Cited by 3 publications

References 38 publications

Appearance of branched motifs in the spectra of BC type Polychronakos spin chains

Appearance of branched motifs in the spectra of BC type Polychronakos spin chains

How to fold a spin chain: Integrable boundaries of the Heisenberg XXX and Inozemtsev hyperbolic models

Classical N-reflection equation and Gaudin models

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