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Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups

Vladimir V. Bazhanov, С. М. Сергеев

Published Sep 23, 2005• 59 citations
OpenAlex

Abstract

The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra $U_q(\\hat{sl}(n))$, where the rank $n$ coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the $sl(n)$-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2+1 dimensions.

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The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra Uq(hatsl(n))U_q(\\hat{sl}(n))Uq​(hatsl(n)), where the rank nnn coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the sl(n)sl(n)sl(n)-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2+1 dimensions.
The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra Uq(sl^(n))U_q(\hat{sl}(n))Uq​(sl^(n)), where the rank nnn coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the sl(n)sl(n)sl(n)-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2+1 dimensions.
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.
An overview of the mathematical structure of the three-dimensional (3D) Ising model is given, from the viewpoints of topologic, algebraic and geometric aspects. By analyzing the relations among transfer matrices of the 3D Ising model, Reidemeister moves in the knot theory, Yang-Baxter and tetrahedron equations, the following facts are illustrated for the 3D Ising model: 1) The complexified quaternion basis constructed for the 3D Ising model represents naturally the rotation in a (3 + 1) - dimensional space-time, as a relativistic quantum statistical mechanics model, which is consistent with the 4-fold integrand of the partition function by taking the time average. 2) A unitary transformation with a matrix being a spin representation in 2^(nlo)-space corresponds to a rotation in 2nlo-space, which serves to smooth all the crossings in the transfer matrices and contributes as the non-trivial topologic part of the partition function of the 3D Ising model. 3) A tetrahedron relation would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model, and its existence is guaranteed also by the Jordan algebra and the Jordan-von Neumann-Wigner procedures. 4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases Φx, Φy, and Φz. The relation with quantum field and gauge theories, physical significance of weight factors are discussed in details. The conjectured exact solution is compared with numerical results, and singularities at/near infinite temperature are inspected. The analyticity in β = 1/(kB T) of both the hard-core and Ising models has been proved for β > 0, not for β = 0. Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.
The formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide quantum group representations even when qqq is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact SU(2)qSU(2)_qSU(2)q​ invariance on the lattice. The algebra of the Yang-Baxter (YB) and SU(2)qSU(2)_qSU(2)q​ generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the SU(2)qSU(2)_qSU(2)q​ generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of SU(2)qSU(2)_qSU(2)q​. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS(ppp) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of SU(2)qSU(2)_qSU(2)q​, at q=eiπ/pq=e^{iπ/p}q=eiπ/p, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J
We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang–Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.