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arXiv:0712.3301 · Journal of Physics A Mathematical and Theoretical

Baxterization ofGLq(2) and its application to the Liouville model and some other models on a lattice

Andrei Bytsko

Published Apr 29, 2008• 1 citation
arXivHTMLPDF

Abstract

We develop the Baxterization approach to (an extension of) the quantum group GL q (2). We introduce two matrices which play the role of spectral parameter-dependent L -matrices and observe that they are naturally related to two different comultiplications. Using these comultiplication structures, we find the related fundamental R-operators in terms of powers of coproducts and also give their equivalent forms in terms of quantum dilogarithms. The corresponding quantum local Hamiltonians are given in terms of logarithms of positive operators. An analogous construction is developed for the q -oscillator and Weyl algebras using the fact that their algebraic and coalgebraic structures can be obtained as reductions of those for the quantum group. As an application, the lattice Liouville model, the q -DST model, the Volterra model, a lattice regularization of the free field and the relativistic Toda model are considered.

DOI
10.1088/1751-8113/41/19/194003

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We develop the Baxterization approach to (an extension of) the quantum group GL q (2). We introduce two matrices which play the role of spectral parameter-dependent L -matrices and observe that they are naturally related to two different comultiplications. Using these comultiplication structures, we find the related fundamental R-operators in terms of powers of coproducts and also give their equivalent forms in terms of quantum dilogarithms. The corresponding quantum local Hamiltonians are given in terms of logarithms of positive operators. An analogous construction is developed for the q -oscillator and Weyl algebras using the fact that their algebraic and coalgebraic structures can be obtained as reductions of those for the quantum group. As an application, the lattice Liouville model, the q -DST model, the Volterra model, a lattice regularization of the free field and the relativistic Toda model are considered.
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,
We define and study certain integrable lattice models with non-compact quantum group symmetry (the modular double of ) including an integrable lattice regularization of the sinh-Gordon model and a non-compact version of the XXZ model. Their fundamental R -matrices are constructed in terms of the non-compact quantum dilogarithm. Our choice of the quantum group representations naturally ensures self-adjointness of the Hamiltonian and the higher integrals of motion. These models are studied with the help of the separation of variables method. We show that the spectral problem for the integrals of motion can be reformulated as the problem to determine a subset among the solutions to certain finite difference equations (Baxter equation and quantum Wronskian equation) which is characterized by suitable analytic and asymptotic properties. A key technical tool is the so-called -operator, for which we give an explicit construction. Our results allow us to establish some connections to related results and conjectures on the sinh-Gordon theory in continuous spacetime. Our approach also sheds some light on the relations between massive and massless models (in particular, the sinh-Gordon and Liouville theories) from the point of view of their integrable structures.
We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate bases in which their spectral problem is separated, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called “non-fundamental” models we construct two different types of SoV bases. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin’s approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic Y(gl_{2}) Y ( g l 2 ) Yang-Baxter algebra. Our SoV approach also leads to the construction of a Q Q -operator in terms of the fused transfer matrices. Finally, we show that the Q Q -operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases.
Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Map-induced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch–Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang–Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2⊗j of SU(2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a two-parameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spin-chains are outlined.
<!-- *** Custom HTML *** --> The Yang–Baxter algebras (YBA) are introduced and formulated in a general way stressing graphical methods. Their various physical applications are then exposed: lattice statistical models, integrable field theories and factorizable S-matrices. The Bethe Ansatz (BA) and its generalizations provide the explicit solutions of all these models using the appropiate YBA. The six-vertex model solution is exposed. YB algebras and their associated physical models are classified in terms of simple Lie algebras. It is exposed how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is exposed 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 YB algebras. To conclude braid and quantum groups are derived from the YBA in the limit of infinite spectral parameter.
We have constructed series of the spectral parameter dependent solutions to the Yang–Baxter equations defined on the tensor product of reducible representations with a symmetry of quantum (super)algebra. These series are produced as descendant solutions from the s l q ( 2 ) -invariant Hecke type R r r ( u ) -matrices. The analogues of the matrices of Hecke type with the symmetry of the quantum super-algebra o s p q ( 1 | 2 ) are obtained precisely. For the homogeneous solutions R r 2 − 1 r 2 − 1 there are constructed Hamiltonian operators of the corresponding one-dimensional quantum integrable models, which describe rather intricate interactions between different kind of spin states. Centralizer operators defined on the products of the composite states are discussed. The inhomogeneous series of the operators R r R ( u ) , extended Lax operators of Hecke type, also are suggested.