Analyse Of Quantum Phase Transition Of XXZ Model

Content: In this paper, the quantum phase transition of the XXZ model of Heisenberg closespin-1/2 chains in one dimension is studied, the nearest neighbor interaction of Fermition systemis only considered in this model. In half-filling situation, all kinds of [N, k] are discussed. Nis the total number of sites, k is the number of electrons at site spin up, the electrons at rest(N-k)site spin down. Take for [5, 2], [6, 2], [7, 2], [11, 5], [12, 6], [13, 6] examples, the changes ofthe ground energy, encitation spectrum and the ground entanglement along with the change oft parameter are showed. The behavior of quantum phase transition base on them are analyed.There are four kinds of the XXZ model Hamiltonian operators because of the ferromagnetismand antiferromagnetism, they are writen simply as operators with ++, +-, -+, --. Completebasis vectors are produced with the knowledge of the permutation group. Energy matrix areconstructed, which the Hamiltonian operator of the XXZ model Hamiltonian operator functioncomplete basis vectors. Eigenvalue and eigenvector are obtained by diagonalizing the matrix.Firstly,in the ++, +-, -+, -- situations, the eigenvalue and eigenvector of the matrix afterdefiniting [N, k] and t parameter are calculated. For example, the change of energy along witht parameter are observed in the situation that N is 5(0≤k≤5), the energy is the lowest whenN is 2. In general, there is the lowest energy of the system When k is N/2 while N is fixed. Baseon these, the ground energy, encitation spectrum and the ground entanglement along with thechange of t parameter are studied further. Then the situations are discussed respectively, (ⅰ)In afixed N and k(0≤k≤N/2) situation, the ground energy and the ground entanglement of [N, k]change along with the change of t parameter. (ⅱ)In a fixed N and t situation, the ground energyand the ground entanglement of the system change along with the change of k. (ⅲ) In a fixedk and t situation, the ground energy and the ground entanglement of the system change alongwith the change of N. Then in the ++, +-, -+, -- four situations are discussed When N is odd,even, and the biggest number, when N is odd and even, encitation spectrum from the groundto the tenth energy of [N, k](0≤k≤N/2), encitation energy, and the ground entanglement of the system change along with the change of t are showed. then the corresponding laws of thequantum phase transition of the XXZ model are talked. In the end, the character of groundentanglement of [N, N/2] are discussed when N is even.