Scalar products of Bethe vectors in the 8-vertex model

Slavnov, N. A.; Zabrodin, A.; Zotov, A.

doi:10.1007/jhep06(2020)123

Cited by 14 publications

(14 citation statements)

References 66 publications

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“…In particular, the obtention of a generalization of the Slavnov's formula [30] for the scalar products of Bethe states, and more generally of a similar determinant representation for the matrix elements of local operators, as in [17], may be a very difficult problem if the combinatorial structure of the Bethe states is too involved. This is for instance the case in the XYZ model, for which first results about scalar products within ABA were obtained only very recently in [73] (but for which the obtention of a compact formula for matrix elements of local operators in finite volume remains an open problem 3 ), using the fact that the scalar products of on-shell/off-shell Bethe vectors can be characterized as solutions to a system of linear equations, as initially proposed in [76]. This is also the case for models based on higher rank algebras, see for instance the works [77][78][79][80][81][82][83][84][85].…”

Section: Introductionmentioning

confidence: 99%

Correlation functions by separation of variables: the XXX spin chain

2021

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We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.

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Section: Introductionmentioning

confidence: 99%

Correlation functions by separation of variables: the XXX spin chain

2021

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“…The problematics that comes up in these models, is that the magnon-form picture is not evident here, hence the issues with Bethe Ansätze. For example, appropriately adopted inhomogeneous nested CBA already fails at level 2, but it is important to note that not all possibilities have been tried, including generalised ABA [85] or QSC. On the other hand, analysis on short spin chains or small number of excitations indicates the highly nontrivial eigenspectrum.…”

Section: R Matrices Are Found To Bementioning

confidence: 99%

Automorphic Symmetries, String integrable structures and Deformations

Pribytok¹

2022

Preprint

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We address the novel structures arising in quantum and string integrable theories, as well as construct methods to obtain them and provide further analysis. Specifically, we implement the automorphic symmetries on periodic lattice systems for obtaining integrable hierarchies, whose commutativity along with integrable transformations induces a generating structure of integrable classes. This prescription is first applied to 2-dim and 4-dim setups, where we find the new sl 2 sector, su(2) ⊕ su(2) with superconductive modes, Generalised Hubbard type classes and more. The corresponding 2-and 4-dim R matrices are resolved through perturbation theory, that allows to recover an exact result. We then construct a boost recursion that allows to address the systems, whose R-/S-matrices exhibit arbitrary spectral dependence, that also is an apparent property of the scattering operators in AdS integrability. It is then possible to implement the last for Hamiltonian Ansätze in D = 2, 3, 4, which leads to new models in all dimensions. We also provide a method based on a coupled differential system that allows to resolve for R matrices exactly. Importantly, one can isolate a special class of models of non-difference form in 2-dim case (6vB/8vB), which provides a new structure consistently arising in AdS 3 and AdS 2 string backgrounds. We prove that these classes can be represented as deformations of the AdS {2,3} models. We also work out that the latter satisfy free fermion constraint, braiding unitarity, crossing and exhibit deformed algebraic structure that shares certain properties with AdS 3 × S 3 × M 4 and AdS 2 × S 2 × T 6 models. The embedding and mappings of known AdS {2,3} models to 6vB/8vB deformations are demonstrated, along with a discussion on the associated candidates of sigma models.

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“…It is worth mentioning that the aforementioned results concern essentially the simplest integrable models solvable by ABA, namely the XXZ Heisenberg spin chain or the quantum non-linear Schrödinger model with periodic boundary conditions (see also [41][42][43][44][45] for some results about correlation functions in more complicated models within ABA, and [46][47][48][49][50] for some generalizations of the scalar products and/or form factor determinant representations to more complicated models). For quantum integrable models not solvable within ABA but within the quantum version of the Separation of Variables (SoV) approach [51][52][53][54], the problem of computing correlation functions still remains widely open.…”

Section: Introductionmentioning

confidence: 99%

On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model

Pei¹,

Terras²

2020

Preprint

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We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by Separation of Variables (SoV). We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations which are similar, in their form, to their counterparts in the periodic case as obtained in the Bethe ansatz framework. This is quite remarquable since the analytic description of the spectrum, and in particular the periodicity properties of the Q-function solving the Baxter TQ-equation, is rather different in the antiperiodic case with respect to the periodic one.

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Scalar products of Bethe vectors in the 8-vertex model

Cited by 14 publications

References 66 publications

Correlation functions by separation of variables: the XXX spin chain

Correlation functions by separation of variables: the XXX spin chain

Automorphic Symmetries, String integrable structures and Deformations

On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model

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