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arXiv:2102.13570 · Symmetry Integrability and Geometry Methods and Applications

ompleteness of SoV Representation for SL(2,R) Spin Chains

Sergey E. Derkachov, Karol Kajetan Kozlowski, Univ Lyon, France, Alexander N. Manashov, Universität Hamburg, Germany

Published Jun 25, 2021• 4 citations
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Abstract

This work develops a new method, based on the use of Gustafson's integrals and on the evaluation of singular integrals, allowing one to establish the unitarity of the separation of variables transform for infinite-dimensional representations of rank one quantum integrable models. We examine in detail the case of the SL(2,R) spin chains.

DOI
10.3842/sigma.2021.063

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This work develops a new method, based on the use of Gustafson's integrals and on the evaluation of singular integrals, allowing one to establish the unitarity of the separation of variables transform for infinite-dimensional representations of rank one quantum integrable models. We examine in detail the case of the SL(2,R) spin chains.
We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state.
Abstract 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 𝔰𝔩(N) couples Q-functions 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 \mathcal{N} N N = 4 SYM and the ABJM model.
Abstract Separation of variables (SoVs) is a powerful method expected to be applicable for a wide range of quantum integrable systems, from models in condensed matter physics to gauge and string theories. Yet its full implementation for many higher rank examples, such as SU(N) spin chains with N > 2, has remained elusive for a long time. In this pedagogical review we discuss the major progress achieved recently in understanding SoV for models of this type. In particular, for rational SU(N) spin chains we describe different constructions of the SoV basis, novel compact forms for spin chain eigenstates, the functional SoV approach, and explicit computation of the SoV measure. We also discuss key first applications of these results, namely the new compact determinant representations for many observables such as scalar products and correlators.
This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of gl(n)\mathfrak{gl}(n)gl(n) in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for gl(n)\mathfrak{gl}(n)gl(n) spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.
Abstract Integrable sl \mathfrak{sl} sl sl (N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin’s SoV B operator. As a result, we are able to derive the B operator for sl \mathfrak{sl} sl sl (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.