The nested Bethe ansatz for ‘all’ closed spin chains

Belliard, S.; Ragoucy, É.

doi:10.1088/1751-8113/41/29/295202

Cited by 84 publications

(127 citation statements)

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“…In the context of the Quantum Inverse Scattering Method (QISM) [6][7][8][9], an algebraic version of this method (NABA) was developed in [10][11][12]. One more approach based on the qKZ equation and Jackson integrals was proposed in [13][14][15] and generalized to the superalgebras in [16]. We should also mention a method to construct Bethe vectors via certain projection of Drinfel'd currents, that was developed in a series of works [17][18][19][20][21].…”

Section: Jhep06(2018)018mentioning

confidence: 99%

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On Bethe vectors in $$ \mathfrak{g}{\mathfrak{l}}_3 $$-invariant integrable models

Liashyk

Slavnov

2018

J. High Energ. Phys.

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We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl 3 -invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.

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Section: Jhep06(2018)018mentioning

confidence: 99%

“…Different methods to construct Bethe vectors were developed in [12,[15][16][17]. Several equivalent explicit representations were found in [25].…”

Section: Jhep06(2018)018mentioning

confidence: 99%

On Bethe vectors in $$ \mathfrak{g}{\mathfrak{l}}_3 $$-invariant integrable models

Liashyk

Slavnov

2018

J. High Energ. Phys.

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“…This R matrix differs from the one of section 2, since it intertwines between the fundamental representation of the corresponding quantum group and the representation V from which the spin chain is built. One constructs the monodromy M and transfer matrix T as in (18), with the fundamental representation placed in the auxiliary space and diagonalizes the transfer matrix T using the Nested Bethe Ansatz [23]. One can of course not obtain the Hamiltonian directly in this way, since the T thus constructed does not reduce to the shift operator for any value of the spectral parameter, i.e.…”

Section: Bethe Equations For Gl(n|m) Spin Chainsmentioning

confidence: 99%

“…The Bethe equations we will investigate can be found in [23][24][25][26][27] and in a compact notation…”

Section: Bethe Equations For Gl(n|m) Spin Chainsmentioning

confidence: 99%

Dynamic Lattice Supersymmetry in $$\mathfrak {gl}\left( {n}|{m}\right) $$ gl n | m Spin Chains

Meidinger

Mitev

2014

J Stat Phys

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Supersymmetry operators that change a spin chain's length have appeared in numerous contexts, ranging from the AdS/CFT correspondence to statistical mechanics models. In this article, we present, via an analysis of the Bethe equations, all homogeneous, rational and trigonometric, integrable gl(n|m) spin chains for which length-changing supersymmetry can be present. Furthermore, we write down the supercharges explicitly for the simplest new models, namely the sl(n|1) spin chains with the (n − 1)-fold antisymmetric tensor product of the fundamental representation at each site and check their compatibility with integrability.

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“…Sometimes we will use the more detailed notationT In general, they are non-local operators involving spins at all sites of the chain. As is shown in [50] (see also [51][52][53]), the eigenvalues of the transfer matrix and the Hamiltonians are of the form…”

Section: Jhep01(2014)070mentioning

confidence: 89%

Spectrum of quantum transfer matrices via classical many-body systems

Gorsky

Zabrodin

Zotov

2014

J. High Energ. Phys.

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In this paper we clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous gl n -invariant XXX spin chain on N sites with twisted boundary conditions can be found in terms of velocities of particles in the rational N -body Ruijsenaars-Schneider model. The possible values of the velocities are to be found from intersection points of two Lagrangian submanifolds in the phase space of the classical model. One of them is the Lagrangian hyperplane corresponding to fixed coordinates of all N particles and the other one is an N -dimensional Lagrangian submanifold obtained by fixing levels of N classical Hamiltonians in involution. The latter are determined by eigenvalues of the twist matrix. To support this picture, we give a direct proof that the eigenvalues of the Lax matrix for the classical Ruijsenaars-Schneider model, where velocities of particles are substituted by eigenvalues of the spin chain Hamiltonians, calculated through the Bethe equations, coincide with eigenvalues of the twist matrix, with certain multiplicities. We also prove a similar statement for the gl n Gaudin model with N marked points (on the quantum side) and the Calogero-Moser system with N particles (on the classical side). The realization of the results obtained in terms of branes and supersymmetric gauge theories is also discussed.

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The nested Bethe ansatz for ‘all’ closed spin chains

Cited by 84 publications

References 50 publications

On Bethe vectors in $$ \mathfrak{g}{\mathfrak{l}}_3 $$-invariant integrable models

On Bethe vectors in $$ \mathfrak{g}{\mathfrak{l}}_3 $$-invariant integrable models

Dynamic Lattice Supersymmetry in $$\mathfrak {gl}\left( {n}|{m}\right) $$ gl n | m Spin Chains

Spectrum of quantum transfer matrices via classical many-body systems

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