杨-巴克斯特方程在量子纠缠及新型量子模型中的应用

Yang-Baxter equation (YBE) is the parametrization of braid group that describes a large class of quantum integrability systems. During the 60's and 90's in the 20th centrary it almost dominated the research in Mathematical Physics. Espacially, in the statistical models and quantum chain models it plays very important role and leads to the Quantum Groups. However, since 2006 there appears a new family of solutions of YBE that is different from all of the known solutions, it is related to the quantum entangling states and Majorana fermions. The proposed project is going to solve some of the challenge problem in this respect, including: the establishment of the connection between the new solutions and Kitaev models, discribtion of 3-body entangling states based on YBE, finding the topological quantum field theory (TQFT) and the Hamiltonian associated with Birman-Wenzl algebra and the commection with minimization of L_1-norm in quantum mechanics. With the solutions of the above issues a more general TQFT based on the Wigner rotation functions may be proposed.

杨-Baxter方程（YBE）是辫子群的参数化，它描述一大类量子可积系统。从上世纪60年代末到90年代末，它成为数学物理研究的主流之一，在研究统计模型与量子链模型等物理领域起了很重要的作用，并导致量子群的出现。但自2006年以后，出现了YBE 新解系，它完全不同于上述传统的解系，而是与量子纠缠相联系，并用Majorana费米子描述。本项目在解决与T-L代数相关的问题基础上力求解决这方面引起的的新课题，包括：建立新解系与Kitaev模型间的联系，给出三体纠缠的杨-Baxter描述，尤其是解决与Birman-Wenzl代数相关的哈密顿量与拓扑量子场论的难题，找到它们与量子力学中L_1-模极值问题的联系。在解决这些问题基础上，可以初步形成一个以Wigner转动函数为一般表示的与多体量子纠缠相关的拓扑量子场论描述。

建立杨振宁-巴克斯特方程(YBE)的新型解，它与量子信息密切相关，揭示了与一系列物理模型的联系。如YBE与连续纠缠态，Berry相因子，Kitaev模型，多体分立对称性(Z_2,Z_3)等的联系，并给出三体散射S-矩阵具体形式，指出量子信息与L_1-norm极值的联系。