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arXiv:hep-th/9810220 · Physical Review Letters

Algebraic Approach in Unifying Quantum Integrable Models

Anjan Kundu

Published May 17, 1999• 27 citations
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Abstract

A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given $R$ matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known discrete and field models a new class of inhomogeneous and impurity models is obtained.

DOI
10.1103/physrevlett.82.3936

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A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given RRR matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known discrete and field models a new class of inhomogeneous and impurity models is obtained.
Exploiting the quantum integrability condition we construct an ancestor model associated with a new underlying quadratic algebra. This ancestor model represents an exactly integrable quantum lattice inhomogeneous anisotropic model and at its various realizations and limits can generate a wide range of integrable models. They cover quantum lattice as well as field models associated with the quantum RRR-matrix of trigonometric type or at the undeformed qto1q \\to 1qto1 limit similar models belonging to the rational class. The classical limit likewise yields the corresponding classical discrete and field models. Thus along with the generation of known integrable models in a unifying way a new class of inhomogeneous models including variable mass sine-Gordon model, inhomogeneous Toda chain, impure spin chains etc. are constructed.
An operator defoemed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its q → 1 limit seem to be the most general Yang-Baxter algebra underlying quantum integrable systems. Three different directions of application of this algebra in integrable systems depending on different sets of values of deforming operators are identified. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, while the associated Lax operators generate and classify them in an unified way. Variable values construct a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. PACS numbers 02.30.Ik, 02.20.Uw, 05.50+q, 03.65.Fd 1
The aim of this review is to present the list of by now a significant collection of quantum integrable models, ultralocal as well as nonultralocal, in a systematic way stressing on their underlying unifying algebraic structures. We restrict to quantum and statistical models belonging to trigonometric and rational classes with (2 x 2)- Lax operators. The ultralocal models can be classified successfully through their associated quantum algebra and are governed by the Yang-Baxter equation, while the nonultralocal models, the theory of which is still in the developmental stage, allow systematization through the braided Yang-Baxter equation. Along with the known integrable models some possible directions for investigation in this field and generation of such new models are suggested.
The aim of this review is to present the list of by now a significant collection of quantum integrable models, ultralocal as well as nonultralocal, in a systematic way stressing on their underlying unifying algebraic structures.We restrict to quantum and statistical models belonging to trigonometric and rational classes with 2 × 2 Lax operators.The ultralocal models can be classified successfully through their associated quantum algebra and are governed by the Yang-Baxter equation, while the nonultralocal models, the theory of which is still in the developmental stage, allow systematization through the braided Yang-Baxter equation.Along with the known integrable models some possible directions for investigation in this field and generation of such new models are suggested.
An algebraic construction more general and intimately connected with that of Faddeev1, along with its application for generating different classes of quantum integrable models are summarised to complement the recent results of ref. 1 ( L.D. Faddeev, Int. J. Mod. Phys. A10, 1845 (1995)). 1 A significant portion of an excellent recent review by Faddeev 1 is devoted to an important result showing deep relation between the Yang–Baxter equation and the quantum group. We are delighted to note that a scheme for systematic construction of the Lax operators through Yang–Baxterisation of the Faddeev–Reshetikhin–Takhtajan (FRT) algebra persued by us over the last few years 2,3 has an intimate connection with this approach. In fact our formulation gives a more general framework allowing construction of a wide range of Lax operators of lattice models and recovers the corresponding result of ref 1 related to Uqsl(2) as a particular case. Another convincing demonstration of the usefulness of our approach is that, the well known Lax operator of the lattice Liouville model 4 as well as the recently discovered 5 nontrivial spectral parameter dependent variant of it can be constructed directly and easily from our general form,