Quantum Simulation Study On Topological Systems

In recent years,the topology related problcems has been well studied.On one hand,the study on topological problems helps to get a further understanding on sorne traditional physical problems like the quantum Hall effect.(On the other hand,it also indicts some novel material like topological insulator,topological superconductor.Majorana Fermion,and Weyl Fermion.Moreover,these novel topological materials have their own special topological properties.which not only help us to understand the beauty of the nature but also has great potential in practical applications.For example,the applications of these material in quantum calculation and low dissipation transport have been well discussed.Due to these special advantage of the topological material,people in the literature pay a lot attention in the discovery and investigation on it.Among them,the quantum simulation method is one of the valuable methods on this topic.The quantum simula-tion method means simulating a designed Hamiltonian by an artificial quantum system.Through this method,the Hamiltonian simulated can be well designed,which means that the parameter in the Hamiltonian can be tuned and the related properties can be well studied.In this thesis,we meanly study the property of Majorana Fermion system and offer a quantum simulation scheme to simulate the Weyl Fermion system.In detail,we first give a simple review on the history of the topological system.and then focus on the introduction of the TKNN number,Majorana Fermion,Weyl Fermion.Then,we give a further intruduction on the quantum simulation method and its applications.Here,we focus on the optical system,which includes photonic crystal,and optical cavity,etc..Moreover,we also show the possibility of using the artificial dimension to reduce the complexity of the system.After the brief introduction,we show our main contribution to the literature.1.For the Majorana Fermion,its.definition is clear,however,its existence in the real solid material is still questionable.This is because that it is difficult to obtain the conclusive experimental evidence.For example.even though the conductance peak around the zero bias voltage in the transport detection is one of the property of the Majorana Fermion.it can also be induced by many other mechanisms,e.g.,the state duo to the impurity in the boundary and Kondo effect.This means that the conductance peak is not enough to identify the Majorana Fermion.On the other hand,the experiment using other special properties of the Majorana Fermion.liking the 4?-periodicity of the Josephson effect,is difficult to be realized.To deal with this problem,we propose a new scheme to detect the special property of the Majorana Fermion via the transport detection.Our model is constituted by two topological supercoiducting wires conncected by a ring.A magnetic flux is added through the center of the ring which changes the superconducting phases of the two wires.Due to the quantization of the magnetic flux,the phase difference between the two ires can be only zero or ?.When it is ? two Majorana mode appear around the ring without interacting with each other.Nevertheless,when the phase difference is zero,these Majorana modes disappear.This difference in the two cases leads to very difference transport properties,which can be used to identih them from each other and hence can be used as the evidence of the existence of the Majorana Fermion.2.We also discuss the possibility of realizing the Weyl Fermion through the quan-tum simulation method.Althouegh,the Weyl Fermion has been observed in the solid material,the band structure in these material is complex and its special boundary prop-erty,i.e.,Fermi are,is degenerate with the bulk band.Hence,it is difficult to direct investigate the property of the boundany states.Therefore,it is valuable to simulate it in a simple model with a simple model Hamiltonian.Moreover,as a topological phe-nomenon in three-dimensional,the simulation of the Weyl Fermion needs lots of grids in the simulation.Hence,the artificial dimension is needed in the simulation.We pro-posed a degenerate optical array system using the optical angular momentum of the light to achieve this purpose.In this system,the related band structure can be detected via the optical transmission detection.From the results,we show that the coupling between the boundary states is highly related to the number of sites in the constrained direction.When the number is odd there are no coupling,nevertheless,when it is even,the cou-pling is momentum and spin related,which leads to momentum and spin related transport properties.3.At last,we discuss the development of the topology in the system with interactions.By now,the existing theories on this topic all have some problems or can not be used in all cases.This means that the description of the topology in the interacting system is a valuable problem still needs to be investigated.On the other hand,although there are still no satisfactory theories,people has already begun to propose experimental schemes on it.Here,we how the dependence of the energy specture on the boundary condition in the fractional Hall system,and Hence,prove that it is possibility to directly detect the topological number of the interacting system via experimental method.