The Applications Of The Yang-Baxter Equation In Topological Physics

Since C. N. Yang in 1967 and R. J. Baxter in 1972 separately established quantum Yang- Baxter Equation (For short, QYBE). Since then the investigations on quantum integrable mo- dels have been greatly promoted. Worthy of mention especially is that the Yangian and quantum algebra was established by V.G. Drinfeld in 1985 that offer a cogent mathematic method for the studies about the symmetry of quantum integrable models in physics. After several decad- es, very recently, QYBE, Temperley-Lieb algebra and topological basis have been introduced to the field of quantum information and quantum computation. In this thesis, the topological basis has been connected with the XXX model and the physical properties of the topological basis in the XXX model have been discussed. While we have presented a method of costructing the generators of the Temperley-Lieb algebra, and have studied the relations between the topological parameter and entanglement, the entanglement properties and geometric phase in a Yang-Baxter system. This discussions lead to some interesting results, which shed light on understanding the physical implications of geometric phase and quantum entanglement, it also inspires us how QYBE can be applied in more fields of physics. The dissertation consists of six chapters, and the main contents are given in Chapters 2 through 6.In Chapter 1, the background of our study and the importance of the investigation are introduced, the general situation of quantification of quantum information theory, entanglement, geometric phase, as well as QYBE. The method of measuring the quantum entnglement and the derivation of the geometric phase are described in detail.In Chapter 2, we connect the topological basis states with physics models (i.e., XXX model) successfully. In other words, we construct some Hamiltonians from the Temperley-Lieb algebra (TLA) generators, and the topological basis states are the eigenstaes of the Hamiltonians. Then we can investigate the physical properties of the topological basis in the corresponding physics models. When the topological parameter d=2, the particular physical properties of the topological basis in the four-qubit Heisenberg XXX spin chain: the energy single state and the spin single states of the system all fall on the topological basis states. When the topological parameter d= 2 , we present a set of complete orthonormal maximally entangled four-qubit basis, which can be used for teleporting an arbitrary two-qubit state in our subsequent work. And the particular physical properties of the topological basis in the corresponding four-qubit spin chain: it is interesting that whether the system is the anti-ferromagnetic case or the ferromagnetic case, the ground states are all double degenerate, and they all fall on the topological basis states.In Chapter 3, a method of constructing the matrices U which satisfy the Temperley-Lieb algebra with the single loop has been presented. Specifically, we obtain a matrix U with , which is seen to be very important examples for applications to quantum comptuting, particularly the topological model, and has sufficient value in quantum information. Via Yang-Baxterization approach, we obtain a unitary -matrix, a solution of the Yang-Baxter Equation. This Yang-Baxter matrix is universal for quantum computing.In Chapter 4, some relations between the topological parameter d and concurrences of the projective entangled states have been presented. It is shown that for the case with d = n, all the projective entangled states of two n-dimensional quantum systems are the maximally entangled states (i.e. C = 1). And for another case with , C both approach 0 when for n = 2 and 3. Then we study the thermal entanglement and the entanglement sudden death (ESD) for a kind of Yang-Baxter Hamiltonian. It is found that the parameter d not only influences the critical temperature , but also can influence the maximum entanglement value at which the system can arrive at. And we also find that the parameter d has a great influence on the ESD.In Chapter 5, entanglement and Berry Phase in a Yang-Baxter system are investigated. A M-matrix which satisfies the Hecke algebraic relations is presented. Via the Yang-Baxterization approach, we obtain a unitary solution of Yang-Baxter Equation. It is shown that any pure two-qutrit entangled states can be generated via the universal -matrix assisted by local unitary transformations. A Hamiltonian is constructed from the -matrix, and Berry phase of the Yang-Baxter system is investigated. Specifically, for , the Hamiltonian can be represented based on three sets of SU(2) operators, and three oscillator Hamiltonians can be obtained. Under this framework, the Berry phase can be interpreted.In Chapter 6, the thermal entanglement in the two-qubit systems constructed from the Yang-Baxter -matrixsome are studied. By the concept of concurrence, we explore the thermal entanglement of some new Hamiltonians, which are constructed by the unitary Yang-Baxter matrices . We find that the interaction of z-component of two neighboring spins g has great influences on the thermal entanglement. And it not only influences the critical temperature , but also can influence the critical magnetic field .Finally, the conclusions and discussions are presented.