Quantum Simulation Of Topological Semi-metallic Under Two Dimensional Spin Orbit Coupling

Topological state is a new kind of quantum state,which has topological properties.It can not be described by the theory of symmetry breaking,but it can be described by topological invariants.Since the quantum Hall effect was discovered in 1980 s,the relevant theories and experiments have been widely concerned and studied.In particular,the Nobel prize in physics was awarded to the three physicists in 2016,who proposed topological phase transitions and topological phases in condensed matter.Thus,one pays much more attentions to topological physics.Some progresses have been made in topological states,such as topological insulators,topological semimetals,topological superconductors and so on.Cold atom is an important branch of atomic physics.Due to three features of cold atoms: the slow speed,low temperature,and the lower collision probability than hot atoms,it has a more stable energy level structure and better controllability.When the cold atoms are trapped in the periodic network trap formed by the mutual interference of the laser,the optical lattice which is similar to the crystal structure can be realized.As a pure and controllable experimental platform,optical lattice has gradually become one of the most convenient tools to simulate physical systems.It is widely used in many subjects,such as magnetism,dynamics and quantum theory,and has important applications in many fields.For the topological state,it can be effectively simulated and explored by optical lattices.Compared to the real material,the simple controllability and operability of the optical lattice provides convenience for the realization of the experiment.In this paper,we study the properties of topological semi-metallic and other topological phases in optical lattices.By the tight binding approximation,the Fourier transform and the diagonalization,we study energy spectrum of system in momentum space,and calculate the topological invariants to determine the corresponding phases.By means of the dimensionreduction method,we study the properties of the edge states and verify the properties of the corresponding phases.In addition,the effects of different strengths on the energy spectrum and edge states are analyzed by changing the strength of the two dimensional spin orbit coupling.The different strengths of spin orbit coupling do not open the energy gap of the system,which shows that the system preserves the semi-metallic properties,and the existence of the edge states and the topological invariants show that the system has nontrivial topological characteristics.