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Journal of Mathematical Physics

Topics in quantum integrable systems

Kazuhiro Hikami, Miki Wadati

Published Jul 24, 2003• 14 citations
Publisher

Abstract

Recent developments in the theory of quantum integrable particle systems in one-dimension with inverse square interactions are reviewed. First the Yangian symmetry is introduced and the energy spectra of the related spin models are discussed. The character of the su(n)1 WZNW theory is shown to be closely related with the Rogers–Szegö polynomial. Second, the infinite dimensional representation for solutions of the Yang–Baxter equation and the reflection equation is given. Based on the representation, the Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Finally, some results on short-range interacting systems are mentioned.

DOI
10.1063/1.1588743

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•Jan 1, 2016•0 citations
Recent developments in the theory of quantum integrable particle systems in one-dimension with inverse square interactions are reviewed. First the Yangian symmetry is introduced and the energy spectra of the related spin models are discussed. The character of the su(n)1 WZNW theory is shown to be closely related with the Rogers–Szegö polynomial. Second, the infinite dimensional representation for solutions of the Yang–Baxter equation and the reflection equation is given. Based on the representation, the Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Finally, some results on short-range interacting systems are mentioned.
Development in the theory of solvable (integrable) models is reviewed. It covers from basic knowledge on completely integrable systems to recent work by the authors. First, soliton theory is briefly summarized. Through the inverse scattering method and its quantum extension, a central concept, commuting transfer matrices, and a key relation, the Yang-Baxter relation, are introduced. Second, it is shown that there exists at least ∞ × ∞ number of solvable models in two-dimensional statistical mechanics. Third, quantum spin chains corresponding to solvable statistical mechanical models are discussed. In particular, finite temperature extension of Baxter's formula is given. Fourth, a new approach is presented to the classification problem of knots and links. It is shown that link polynomial, topological invariant for knots and links, can be associated with any solvable model in statistical mechanics. In the presentation, universality of the soliton picture in field theory, spin systems and statistical mechanics is observed.
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.
Inspired by the integrable structures appearing in weakly coupled planar N=4 super Yang-Mills theory, we study Q-operators and Yangian invariants of rational integrable spin chains. We review the quantum inverse scattering method along with the Yang-Baxter equation which is the key relation in this systematic approach to study integrable models. Our main interest concerns rational integrable spin chains and lattice models. We recall the relation among them and how they can be solved using Bethe ansatz methods incorporating so-called Q-functions. In order to remind the reader how the Yangian emerges in this context, an overview of its so-called RTT-realization is provided. The main part is based on the author's original publications. Firstly, we construct Q-operators whose eigenvalues yield the Q-functions for rational homogeneous spin chains. The Q-operators are introduced as traces over certain monodromies of R-operators. Our construction allows us to derive the hierarchy of commuting Q-operators and the functional relations among them. We study how the nearest-neighbor Hamiltonian and in principle also higher local charges can be extracted from the Q-operators directly. Secondly, we formulate the Yangian invariance condition, also studied in relation to scattering amplitudes of N=4 super Yang-Mills theory, in the RTT-realization. We find that Yangian invariants can be interpreted as special eigenvectors of certain inhomogeneous spin chains. This allows us to apply the algebraic Bethe ansatz and derive the corresponding Bethe equations that are relevant to construct the invariants. We examine the connection between the Yangian invariant spin chain eigenstates whose components can be understood as partition functions of certain 2d lattice models and tree-level scattering amplitudes of the four-dimensional gauge theory. Finally, we conclude and discuss some future directions.
The solution of the Heisenberg model ( XXX -spin chain) is reviewed to define main ingredients of the quantum inverse scattering method. The spectrum and eigenfunctions are constructed by the algebraic Bethe Ansatz. A dynamical symmetry algebra is identified with a quantum group —the sl (2) Yangian. The Yang-Baxter equation and its GL ( n )-invariant solutions ( R -matrices) are discussed. Integral equations describing the Heisenberg model in the limit of an infinite chain are given. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this thesis we theoretically investigate the close relation between spin chains and Yangians. First, we investigate the Haldane-Shastry model, which symmetry algebra is a representation of the Yangian. We show this in full detail for the spin-1/2 model. We also give an argument why a spin-1 chain with Yangian symmetry cannot exist. \nAfter that, we turn to integrable spin chains which are constructed using the quantum inverse scattering method. Using this method, an integrable spin chain can be constructed from each so called R-matrix, which is a solution of the Yang-Baxter equation. Our main result is the construction of an R-matrix that is invariant under the adjoint representation of SU(n), which is found using an intertwining operator for the corresponding Yangian representation. The integrable\nspin chain that corresponds to this R-matrix is non-Hermitian, making its physical interpretation unclear. These results have been published recently as a preprint under arXiv:1606.02516. \nFinally, integrable spin chains with defects are studied. We present the Hamiltonian for the Heisenberg model and the Babujian-Takhtajan model with defects, such that these models keep there integrability while staying Lie algebra invariant.