Quantum Phase Transition Of One-dimensional Ising Model In A Transverse Field With Dzyaloshinkii-Moriva Interaction

Quantum phase transitions will happen when the ground-state properties of many-body quantum system occur dramatic changes resulted by a small variation in an ex-ternal field. Traditionally, we research quantum phase transitions rely on the studies of order parameter and correlation functions, in this paper, we approach to quantum pahse transition of one-dimensional Ising model in a transverse field with Dzyaloshinkii-Moriya interaction through the studies of fidelity and quantum Fisher information pro-posed by the quantum information theory.In chapter 1, We introduce the research background and history of basic concepts related to this paper, such as phase transition and quantum phase transition, Ising model in transverse field, fidelity, Dzyaloshinkii-Moriya interaction and so on. Chapter 2 first reviews expressions of fidelity with multi-parameters in the density-matrix repre-sentation. Then we introduce the definition of the quantum Fisher information based on classic Fisher information, and formulate expressions of quantum Fisher information with multi-parameters in the density-matrix representation. Contrasting these two ex-pressions, we obtain the relationship between fidelity and quantum Fisher information.In Chapter 3, we calculate exactly solution of one-dimensional Ising model in a transverse field with Dzyaloshinkii-Moriya interaction. One can perform Jordan-Wigner transformation that maps spin Pauli operators onto fermionic creation and annihila-tion operator, And diagonalize the Hamiltonian by performing Fourier transform and Bogolubov transformation. After that, we can obtain the ground-state wave function and energy. We discuss four different cases because of the system sizes and parity.Chapter 4 addresses fidelity and quantum Fisher information of one-dimensional Ising model in a transverse field with Dzyaloshinkii-Moriya interaction changes over external field according to the results of previous two chapters. we give the variation curve of fidelity and quantum Fisher information changes over external field, and study the position of critical point changes over the value of Dzyaloshinkii-Moriya interaction.The last Chapter denotes the conclusion and prospect.