Study On Topological Phase Transition In The Spin Systems

The quantum many-body systems are of great importance in the study of condensed matter physics and quantum information.And many fantastic phenomenons are emergent from these interaction many-body systems such as topological phase transition.The spin systems play very fundamental and important roles in studying topological phase tran-sition in many-body systems.For example,the topological transition firstly found by physicists is the Berezinskii-Kosterlitz-Thouless phase transition in the XY spin system.In thesis,we aim at the Berry phase and topological transition of the one-dimensional Ising model and XXZ model.The layout of this thesis is organized as follows:In chapter-1,we point out the meaning and background of our study.We devote to a general introduction to the Berry phase(BP)and quantum topological phase transitions in condensed matter physics.We describe the important foreground of the topological phase applied in the quantum information and quantum computation.In chapter-2,we introduce some concepts and methods in quantum field theory and topological phase transition such as Wick's theorem and topological order parameters.Particularly,we present a geometric approach to understand topological phase transitions,which is based on the topological characterization of the fiber bundles.We write some definitions of topologically characteristic quantities,such as Berry phase,Chern number and topological entanglement entropy and entanglement spectrum.Finally,we introduce some numerical methods in quantum many-body systems.In chapter-3,based on the residue theorem and degenerate perturbation theory,we derive a simple and general formula for Berry phase calculation in a two-level system for which the Hamiltonian is a real symmetric matrix.We verify the correctness of our formula on the spinless Su-Schrieffer-Heeger(SSH)model.Then the Berry phase of one-dimensional quantum anomalous Hall insulator(1DQAHI)is calculated analytically by applying our method,the result is--?/2-?/4 sgn(B)[sgn(?-4B)+sgn(?)].Finally,illuminated by this idea,we investigate the Chern number in the two-dimensional case,and find a very simple way to determine the parameter range of the non-trivial Chern number in the topological phase diagram.Further,we draw the topological phase diagram of Chern number.In chapter-4,for the one-dimensional Ising chain with spin-1/2 and exchange couple J in a steady transverse field(TF),the quantum transition theory has been introduced in the paramagnetic case where J < 0.And an analytical theory has well been developed in terms of some topological order parameters such as Berry phase.There seems to exist a topological phase transition for the one-dimensional Ising chain in a longitudinal field(LF)with the reduced field strength ?.But the theoretic characterization has not been well founded.This chapter tries to aim at this problem.With the Jordan-Wigner transformation,we give the four-fermion interaction form of the Hamiltonian in the onedimensional Ising chain with a LF.Further by the method of Wick's theorem and the mean-field theory,the four-fermion interaction is well dealt with.We solve the ground state energy and the ground wave function in the momentum space.We discuss the BP and suggest that there exist nonzero BPs when ? = 0 in the paramagnetic case where J < 0and when-1 < ? < 1,in the diamagnetic case where J > 0.In chapter-5,we introduce the matrix-product-state(MPS)method.Further,we determine the geometric phase for the one-dimensional XXZ Heisenberg chain with spin-1/2,the exchange couple J and the spin anisotropy parameter ? in a longitudinal field(LF)with the reduced field strength h.Using the Jordan-Wigner transformation and the meanfield theory based on the Wick's theorem,a semi-analytical theory has been developed in terms of order parameters which satisfy the self-consistent equations.The values of the order parameters are numerically computed using the MPS method.The validity of the mean-filed theory could be checked through the comparison between the self-consistent solutions and the numerical results.Finally,we draw the the topological phase diagrams of Berry phase in the case J < 0 and the case J > 0.