One-boundary Temperley–Lieb algebras in the XXZ and loop models (original) (raw)

Baxter’s relations and spectra of quantum integrable models

Duke Mathematical Journal, 2015

Generalized Baxter's relations on the transfer-matrices (also known as Baxter's T Q relations) are constructed and proved for an arbitrary untwisted quantum affine algebra. Moreover, we interpret them as relations in the Grothendieck ring of the category O introduced by Jimbo and the second author in [HJ] involving infinite-dimensional representations constructed in [HJ], which we call here "prefundamental". We define the transfer-matrices associated to the prefundamental representations and prove that their eigenvalues on any finite-dimensional representation are polynomials up to a universal factor. These polynomials are the analogues of the celebrated Baxter polynomials. Combining these two results, we express the spectra of the transfer-matrices in the general quantum integrable systems associated to an arbitrary untwisted quantum affine algebra in terms of our generalized Baxter polynomials. This proves a conjecture of Reshetikhin and the first author formulated in 1998 [FR1]. We also obtain generalized Bethe Ansatz equations for all untwisted quantum affine algebras. Contents 1. Introduction 2. Quantum loop algebra and Borel algebras 3. Representations of Borel algebras 4. Baxter's relations in category O 13 5. Transfer-matrices and polynomiality 18 6. A grading of positive prefundamental representations 28 7. End of the proof of Theorem 5.9 34 References 39

Integrability and quantum symmetries

Nuclear Physics B - Proceedings Supplements, 1991

1. Quantum Groups are expected to be the appropriate mathematical framework to describe both the symmetry properties of rational conformal field theories and integrable models [4][5]. The first common feature between integrable models and conformal field theories is of course the appearance of finite dimensional representations of the braid group. For conformal field theories the braid group is represented on the space of conforreal blocks, which is finite dimensional for rational theories . Similarly in the context of integrable models the representation of the braid group is defined by the Yang-Baxter operators of IRF-models in t~e limit of the spectral parameter equal to infinity. In particular the closest relation between both representations of the braid group is obtained for the case of Prigonometric solutions of IRF-models defined on a graph [6] when the graph is builded up using as incidence matrix the one obtained from the fusion algebra by Aij = N~ for k-fixed [7 I.

Bethe Ansatz Solutions for Temperley–Lieb Quantum Spin Chains

International Journal of Modern Physics A, 2000

We solve the spectrum of quantum spin chains based on representations of the Temperley–Lieb algebra associated with the quantum groups [Formula: see text] for Xn=A1, Bn, Cn and Dn. The tool is a modified version of the coordinate Bethe ansatz through a suitable choice of the Bethe states which give to all models the same status relative to their diagonalization. All these models have equivalent spectra up to degeneracies and the spectra of the lower-dimensional representations are contained in the higher-dimensional ones. Periodic boundary conditions, free boundary conditions and closed nonlocal boundary conditions are considered. Periodic boundary conditions, unlike free boundary conditions, break quantum group invariance. For closed nonlocal cases the models are quantum group invariant as well as periodic in a certain sense.

Exact S matrices for integrable quantum spin chains

Arxiv preprint hep-th/9501109, 1995

We begin with a review of the antiferromagnetic spin 1/2 Heisenberg chain. In particular, we show that the model has particle-like excitations with spin 1/2, and we compute the exact bulk S matrix. We then review our recent work which generalizes these results. We first consider an integrable alternating spin 1/2 -spin 1 chain. In addition to having excitations with spin 1/2, this model also has excitations with spin 0. We compute the bulk S matrix, which has some unusual features. We then consider the open antiferromagnetic spin 1/2 Heisenberg chain with boundary magnetic fields. We give a direct calculation of the boundary S matrix. (Talk presented at the conference on Statistical Mechanics and Quantum Field Theory at USC, 16 -21 May 1994)

Quantum integrable systems related to lie algebras

Physics Reports, 1983

Introduction 316 14. Systems with v(q) = I~I 367 1. Examples. Systems with one degree of freedom 318 15. Systems with v(q) = 8(q). Bethe Ansatz 369 2. General description 327 16*. Miscellanea 372 3*~Abstract quantum systems, related to root systems 331 16.1. Factorization of the ground-state wave function 372 4. The proof of complete integrability of the systems 335 16.2. Green's functions on symmetric spaces 376 5~.Complete integrability in the abstract case 337 Appendices 6*. Wave functions 340 A. Groups generated by reflections and their root systems 378 7*~Systems of type I (v(q) = q 2) 344 B. Symmetric spaces 38'7 8*. Systems of type II (v(q) = sinh2 q) 350 C. Laplace operators and spherical functions 394 9* Systems of type III (v(q) = sin2 q) 352 D. Connection between Hamiltonians and Laplace operators 31 0*. Systems of type IV (v(q) =~(q)) 354 E. Proof of Propositions of section 5 401 11*. Systems of type V (v(q) = q2+ w2q2) 355 References 402 12*. Systems of type VI (v(q)=expq) 363 13*. Systems of type VI' (Generalized periodic Toda lattices) 366

Remarks towards the spectrum of the Heisenberg spin chain type models

Physics of Particles and Nuclei, 2015

The integrable close and open chain models can be formulated in terms of generators of the Hecke algebras. In this review paper, we describe in detail the Bethe ansatz for the XXX and the XXZ integrable close chain models. We find the Bethe vectors for two-component and inhomogeneous models. We also find the Bethe vectors for the fermionic realization of the integrable XXX and XXZ close chain models by means of the algebraic and coordinate Bethe ansatz. Special modification of the XXZ closed spin chain model ("small polaron model") is consedered. Finally, we discuss some questions relating to the general open Hecke chain models.

Quantum integrability and Kac-Moody algebras

Physics Letters B, 1988

We introduce a quantum "linear" problem, associated to the integrability of certain current algebras. In the conformal invariant limit we interpret the quantum fields of the quantum "linear" problem as Weyl fields of a field theory generated by a Kac-Moody algebra.

Exact solutions forA-DTemperley - Lieb models

Journal of Physics A: Mathematical and General, 1996

We solve for the spectrum of quantum spin chains based on representations of the Temperley-Lieb algebra associated with the quantum groups U q (X n) for X n = A 1 , B n , C n and D n. We employ a generalization of the coordinate Bethe-Ansatz developed previously for the deformed biquadratic spin one chain. As expected, all these models have equivalent spectra, i.e. they differ only in the degeneracy of their eigenvalues. This is true for finite length and open boundary conditions. For periodic boundary conditions the spectra of the lower dimensional representations are containded entirely in the higher dimensional ones. The Bethe states are highest weight states of the quantum group, except for some states with energy zero.

Rectangular Yang–Baxter Algebras and Alternating A-Type Integrable Vertex Models

International Journal of Geometric Methods in Modern Physics, 2005

Given a couple of Yang–Baxter operators 𝖱[k] and 𝖱[l] corresponding to integrable anisotropic vertex models of Ak-1 and Al-1 type, respectively, we construct and study a class of related lattice models whose monodromy matrices alternate between the mentioned operators. In order to do that, we use a natural generalization of the idea of coproduct in a bialgebra, that appears in the scenario of non-commutative algebraic geometry, related to the notion of internal homomorphisms of quantum spaces. We build up all eigenstates and eigenvalues of the transfer matrix by means of algebraic Bethe ansatz technics, where not only one vector, but a pseudovacuum subspace is needed for the process of diagonalization.