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Communications in Theoretical Physics

New R(x) -Matrix, L(x) -Operator and Quantum Algebra in Spin Model

Mo‐Lin Ge, Xu‐Feng Liu, Kang Xue

Published Sep 1, 1991• 0 citations
Publisher

Abstract

New solution of Yang-Bax ter equation for spin-1 ⊗ spin- tensorial space is derived based on the direct calculation and extended Yang-Baxterization prescription. It contains the standard solution as a special case. By employing Faddeev-Reshetikhin-Takhtajan approach we give the correspondent quantum algebraic structure. The Yang-Baxterization of the L -operator is also made explicitly.

DOI
10.1088/0253-6102/16/2/173

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New solution of Yang-Bax ter equation for spin-1 ⊗ spin- tensorial space is derived based on the direct calculation and extended Yang-Baxterization prescription. It contains the standard solution as a special case. By employing Faddeev-Reshetikhin-Takhtajan approach we give the correspondent quantum algebraic structure. The Yang-Baxterization of the L -operator is also made explicitly.
The Yang-Baxterization of L operator is performed for spin-1 model, namely, for given braid group representation (BGR) the x -dependent L ( x ) operator can be determined where x is the spectral parameter. It is pointed out that in order to establish such an x -dependence besides the usual L ± operators in accordance with the Faddeev-Reshetikhin-Takhtajan (FRT) approach an additional operator L 0 should be introduced. The explicit expression of L 0 is given. This Yang-Baxterization prescription preserves the FRT and the YBE is satisfied by L ( x ) and R ( x ).
In this paper we review the theory of the Yang–Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). In the simplest case of the spin s=1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators.
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,
Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced ℓ 1 -norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigner's D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to 3 parafermion model based on SU (3) principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity rather than Galilean rule ( u + v ). Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for 3 parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces .
Abstract Starting from a quantum dilogarithm over a Pontryagin self-dual LCA group A , we construct an operator solution of the Yang–Baxter equation generalizing the solution of the Faddeev–Volkov model. Based on a specific choice of a subgroup and by using the Weil transformation, we also give a new non-operator interpretation of the Yang–Baxter relation. That allows us to construct a lattice QFT-model of IRF-type with gauge invariance under independent B -translations of local ‘spin’ variables.