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arXiv:1805.03927 · Journal of High Energy Physics

New compact construction of eigenstates for supersymmetric spin chains

Nikolay Gromov, Fedor Levkovich-Maslyuk

Published Sep 1, 2018• 22 citations
arXivHTMLPDF

Abstract

Abstract The problem of separation of variables (SoV) in supersymmetric spin chains is closely related to the calculation of correlation functions in $$ \mathcal{N}=4 $$ N = 4 SYM theory which is integrable in the planar limit. To address this question we find a compact formula for the spin chain eigenstates, which does not have any sums over auxiliary roots one usually gets in the widely adopted nested Bethe ansatz. Our construction only involves one application of a simple B g (u k ) operator to the reference state for each of the magnons, in complete analogy with the $$ \mathfrak{s}\mathfrak{u}(2) $$ s u 2 algebraic Bethe ansatz. This generalizes our SoV based construction for $$ \mathfrak{s}\mathfrak{u}(n) $$ s u n to the supersymmetric $$ \mathfrak{s}\mathfrak{u}\left(1\Big|2\right) $$ s u 1 | 2 case.

DOI
10.1007/jhep09(2018)085

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•Dec 11, 2014•0 citations
Abstract The problem of separation of variables (SoV) in supersymmetric spin chains is closely related to the calculation of correlation functions in N=4\mathcal{N}=4N=4 N = 4 SYM theory which is integrable in the planar limit. To address this question we find a compact formula for the spin chain eigenstates, which does not have any sums over auxiliary roots one usually gets in the widely adopted nested Bethe ansatz. Our construction only involves one application of a simple B g (u k ) operator to the reference state for each of the magnons, in complete analogy with the su(2)\mathfrak{s}\mathfrak{u}(2)su(2) s u 2 algebraic Bethe ansatz. This generalizes our SoV based construction for su(n)\mathfrak{s}\mathfrak{u}(n)su(n) s u n to the supersymmetric su(1∣2)\mathfrak{s}\mathfrak{u}\left(1\Big|2\right)su(1​2) s u 1 | 2 case.
We conjecture a new way to construct eigenstates of integrable XXX quantum spin chains with SU(N) symmetry. The states are built by repeatedly acting on the vacuum with a single operator B good(u) evaluated at the Bethe roots. Our proposal serves as a compact alternative to the usual nested algebraic Bethe ansatz. Furthermore, the roots of this operator give the separated variables of the model, explicitly generalizing Sklyanin’s approach to the SU(N) case. We present many tests of the conjecture and prove it in several special cases. We focus on rational spin chains with fundamental representation at each site, but expect many of the results to be valid more generally.
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 In this paper we take further steps towards developing the separation of variables program for integrable spin chains with gl(N) \mathfrak{gl}(N) gl(N) gl N symmetry. By finding, for the first time, the matrix elements of the SoV measure explicitly we were able to compute correlation functions and wave function overlaps in a simple determinant form. In particular, we show how an overlap between on-shell and off-shell algebraic Bethe states can be written as a determinant. Another result, particularly useful for AdS/CFT applications, is an overlap between two Bethe states with different twists, which also takes a determinant form in our approach. Our results also extend our previous works in collaboration with A. Cavaglia and D. Volin to general values of the spin, including the SoV construction in the higher-rank non-compact case for the first time.
Abstract We consider the XXX open spin-1/2 chain with the most general non-diagonal boundary terms, that we solve by means of the quantum separation of variables (SoV) approach. We compute the scalar products of separate states, a class of states which notably contains all the eigenstates of the model. As usual for models solved by SoV, these scalar products can be expressed as some determinants with a non-trivial dependance in terms of the inhomogeneity parameters that have to be introduced for the method to be applicable. We show that these determinants can be transformed into alternative ones in which the homogeneous limit can easily be taken. These new representations can be considered as generalizations of the well-known determinant representation for the scalar products of the Bethe states of the periodic chain. In the particular case where a constraint is applied on the boundary parameters, such that the transfer matrix spectrum and eigenstates can be characterized in terms of polynomial solutions of a usual T - Q equation, the scalar product that we compute here corresponds to the scalar product between two off-shell Bethe-type states. If in addition one of the states is an eigenstate, the determinant representation can be simplified, hence leading in this boundary case to direct analogues of algebraic Bethe ansatz determinant representations of the scalar products for the periodic chain.
In this thesis we give accounts on the solution of the open XXZ and XYZ quantum spin-1/2 chains with the most generic integrable boundary terms. By using the the Separation of Variables method (SoV), due to Sklyanin, we are able, in the inhomogeneous case, to build the complete set of eigenstates and the associated eigenvalues. The characterization of these quantities is made through a maximal system of N quadratic equations, where N is the size of the chain. Different methods, like the Algebraic Bethe ansatz (ABA) or other generalized Bethe ansatz techniques, have been used, in the past, in order to tackle these problems. None of them resulted effective in the reproduction of the full set of eigenstates and eigenvalues in the case of most general boundary conditions. A Vandermonde determinant formula for the scalar products of SoV states is obtained as well. The scalar product formula represents a first step towards the calculation of form factors and correlation functions.