New compact construction of eigenstates for supersymmetric spin chains

Gromov, Nikolay; Levkovich-Maslyuk, Fedor

doi:10.1007/jhep09(2018)085

Cited by 17 publications

(19 citation statements)

References 88 publications

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“…This conjecture has been then proven in the ABA framework in[70]. These observations and conjectures have also been extended to the super-symmetric case in[78].…”

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confidence: 66%

On scalar products in higher rank quantum separation of variables

2020

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Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models , we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states, that are transfer matrix eigenstates or some simple generalization of them. In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector \langle\underline{\mathbf{h}}\rangle⟨𝐡̲⟩, there could be more than one non-vanishing overlap \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle⟨𝐡̲|𝐤̲⟩ with the vectors |{\underline{\mathbf{k}}}\rangle|𝐤̲⟩ of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the \mathcal{Y}(gl_3)𝒴(gl3) lattice model with NN sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality (by standard we mean \langle{\underline{\mathbf{h}}}|{\underline{\mathbf{k}}}\rangle \propto \delta_{\underline{\mathbf{h}}, \underline{\mathbf{k}}}⟨𝐡̲|𝐤̲⟩∝δ𝐡̲,𝐤̲). In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the \mathcal{Y}(gl_2)𝒴(gl2) rank one case.

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“…This conjecture has been then proven in the ABA framework in[70]. These observations and conjectures have also been extended to the super-symmetric case in[78].…”

mentioning

confidence: 66%

On scalar products in higher rank quantum separation of variables

2020

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“…Indeed, suppersymmetry can "linearise" representation theory of gl(N ) by reducing it to study of a chain of gl(1|1) subalgebra imbeddings [40], and one can expect that the same is true for representations of Yangian as is hinted in [32,41]. The paper [19] contains the first promising results on the properties of B in the supersymmetric case, and further study is needed.…”

Section: Conclusion and Final Remarksmentioning

confidence: 99%

Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame

Ryan

Volin

2019

Journal of Mathematical Physics

119

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We propose a basis for rational gl(N ) spin chains in an arbitrary rectangular representation (S A ) that factorises the Bethe vectors into products of Slater determinants in Baxter Q-functions. This basis is constructed by repeated action of fused transfer matrices on a suitable reference state. We prove that it diagonalises the so-called Boperator, hence the operatorial roots of the latter are the separated variables. The spectrum of the separated variables is also explicitly computed and it turns out to be labelled by Gelfand-Tsetlin patterns. Our approach utilises a special choice of the spin chain twist which substantially simplifies derivations.

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“…These results were later proven and shown to hold beyond the fundamental representation, first for slp3q in [38] and then for slpN q in [39] where the spectrum of separated variables x in more general cases was also obtained 6 . The eigenstates construction (1.5) was extended to the supersymmetric case in [47]. However, the factorisation property (1.7) does not guarantee the existence of a simple formula for the scalar product.…”

Section: )mentioning

confidence: 99%

Separation of variables and scalar products at any rank

Cavaglià

Gromov

Levkovich-Maslyuk

2019

J. High Energ. Phys.

Self Cite

110

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Separation of variables (SoV) is a special property of integrable models which ensures that the wavefunction has a very simple factorised form in a specially designed basis. Even though the factorisation of the wavefunction was recently established for higher rank models by two of the authors and G. Sizov, the measure for the scalar product was not known beyond the case of rank one symmetry. In this paper we show how this measure can be found, bypassing an explicit SoV construction. A key new observation is that the measure for spin chains in a highest-weight infinite dimensional representation of slpN q couples Qfunctions at different nesting levels in a non-symmetric fashion. We also managed to express a large number of form factors as ratios of determinants in our new approach. We expect our method to be applicable in a much wider setup including the problem of computing correlators in integrable CFTs such as the fishnet theory, N " 4 SYM and the ABJM model.

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New compact construction of eigenstates for supersymmetric spin chains

Cited by 17 publications

References 88 publications

On scalar products in higher rank quantum separation of variables

On scalar products in higher rank quantum separation of variables

Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame

Separation of variables and scalar products at any rank

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