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International Journal of Geometric Methods in Modern Physics

RECTANGULAR YANG–BAXTER ALGEBRAS AND ALTERNATING A-TYPE INTEGRABLE VERTEX MODELS

Sergio Grillo, H. Montani

Published Dec 1, 2005• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0219887805000946

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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.
Given a family of monodromy matrices T0,T1,…,TK−1 corresponding to integrable anisotropic vertex models of Anμ−1 type, μ=0,1,…,K−1, we built up a related mixed vertex model by means of gluing the lattices on which they are defined, in such a way that integrability property is preserved. The gluing process is implemented through one-dimensional representations of rectangular quantum matrix algebras A(Rnμ−1:Rnμ), namely, the gluing matrices ζμ. Algebraic Bethe ansatz is applied on a pseudovacuum space with a selected basis and, for each element of this basis, it yields a set of nested Bethe ansatz equations matching up to the ones corresponding to an Am−1 quasiperiodic model, with m equal to minμ∈ZK{rank ζμ}.
A crash course on the Yang–Baxter equation and its applications in the statistical mechanics of lattice vertex models is presented. Using the simple example of a one-dimensional lattice gas, the basic terminology and standard mathematical procedures of statistical mechanics are illustrated. The coproduct notation is introduced via a discussion of the two-dimensional dimer model, and the transfer matrix formulation is elaborated further. The algebraic Bethe Ansatz (ABA) is introduced in the context of the ice model, leading to the Yang–Baxter equations. The motivation for introducing the ABA is developed, emphasizing the similarity to the quantum oscillator problem, in features such as the generation of “excited states”, and the determination of the eigenvalue spectrum. Finally, these methods from the theory of lattice models are applied to the case of a quantum many-body problem, the Heisenberg chain. From the condition that the monodromy matrices satisfy the Yang–Baxter equation, a complete set of mutually commuting operators is derived, and the Heisenberg chain problem is completely solved.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y
This is a review on infinite non-abelian symmetries in two-dimensional field theories. We show how any integrable QFT enjoys the existence of infinitely many {\bf conserved} charges. These charges {\bf do not commute} between them and satisfy a Yang--Baxter algebra. They are generated by quantum monodromy operators and provide a representation of q−q-q−deformed affine Lie algebras $U_q({\hat\G})$. We review the work by de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators in classically scale invariant theories where the classical monodromy matrix is conserved. Then, the recent generalization to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue is given (This provides a representation of SU^(2)qS{\hat U}(2)_qSU^(2)q​). It is then reported on the recent work by Destri and de Vega, where both commuting and non-commuting integrals of motion are systematically obtained by Bethe Ansatz in the light-cone approach. The eigenvalues of the six--vertex alternating transfer matrix τ(ł)τ(ł)τ(ł) are explicitly computed on a generic physical state through algebraic Bethe ansatz. In the thermodynamic limit τ(ł)τ(ł)τ(ł) turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix.[Based on a Lecture at the Clausthal Symposium].
Abstract Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.
The Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and their many applications are the subject of this reivew. I start by the solvable lattice statistical models constructed from YBZF algebras. All two-dimensional integrable vertex models follow in this way and are solvable via Bethe Ansatz (BA) and their generalizations. The six-vertex model solution and its q(2q−1) vertex generalization including its nested BA construction are exposed. YBZF algebras and their associated physical models are classified in terms of simple Lie algebras. It is shown how these lattice models yield both solvable massive quantum field theories (QFT) and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is described as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YBZF algebras. A discussion on braid and quantum groups concludes this review.
A generalization of the Yang-Baxter algebra is found in quantizing the monodromy matrix of two (m)KdV equations discretized on a space lattice. This braided Yang-Baxter equation still ensures that the transfer matrix generates operators in involution which form the Cartan sub-algebra of the braided quantum group. Representations diagonalizing these operators are described through relying on an easy generalization of Algebraic Bethe Ansatz techniques. The conjecture that this monodromy matrix algebra leads, in the cylinder continuum limit, to a Perturbed Minimal Conformal Field Theory description is analysed and supported. b ∗