| The preparation of graphene has opened a new era in the research of two-dimensional materials.The charge carriers in graphene are massless Dirac fermions,which can be described by two-dimensional Dirac’s equation.Thus,graphene has become a new platform to investigate relativistic quantum mechanics such that it is possible to investigate novel relativistic quantum phenomena in condensed matter formalism.With the deepening of research,more two-dimensional materials have been made successfully,such as transition metal dichalcogenides,silicene and phosphorene.Due to the dimension limit,the carrier migration and heat diffusion in two-dimensional materials are located in the plane.Therefore,two-dimensional materials exhibit a plethora of extraordinary physical properties,which contribute to various future applications in reality.Twodimensional materials have attracted wide attention and become experimental and theoretical research hotspots in recent years.The investigation of phase transitions remains a key role in condensed matter physics.In Landau phase transition theory,phase transitions appear with spontaneous symmetry breaking,and are characterized by local order parameters.However,there is no spontaneous symmetry breaking in topological phase transitions,and non-trivial topological states are distinguished by topological invariants.Integer quantum Hall effect is the first topological states discovered in experiment.The quantum spin Hall effect,predicted to exist in graphene,were finally verified in Hg Te/Cd Te quantum well in 2007.After that,the topological states including topological insulators have developed fast.And due to the quantum fluctuations,there are more physical phenomena in low dimensional systems,such as fractional quantum Hall effect and non-Fermi liquid.In addition,the development of ultra-cold atomic optical lattice also provides a pure and controllable research platform for two-dimensional systems.The dissertation focuses on two-dimensional systems and investigate the topological and magnetic phase transitions induced by the interplay among various parameters in two specific two-dimensional systems.The first work of this dissertation is to study the competitive effects between spinorbit coupling and Coulomb repulsion on the two-dimensional puckered honeycomb lattice.There are many peculiar phenomena caused by strongly correlated effects in condensed matter physics,such as high-temperature superconductors,heavy fermions,giant magnetoresistance effects and quantum spin liquids.Because the correlation between electrons is strong enough,perturbation theory is no longer applicable.Thus the strongly correlated problems deserve to be investigated.Thanks to the development of computing technology,physicists have developed a variety of numerical methods to tackle them,such as the exact diagonalization method,the density matrix renormalization group method,and the quantum Monte Carlo method.Combining the cellular dynamical mean-field theory with the continuous-time quantum Monte Carlo method,we numerically solve this complex problem.We obtain the phase diagrams as a function of spin-orbit coupling,Coulomb interaction and temperature.We find that the topological insulator induced by the intrinsic spin-orbit coupling is robust against the weak Rashba spin-orbit coupling and interaction.What’s more,the continuous metalantiferromagnetic Mott insulator transition induced by interaction is predicted to transform to a first-order metal-ferromagnetic insulator transition under the effect of Rashba spin-orbit coupling.In addition,we study the effect of the SU(2)gauge field on the bilayer honeycomb lattice thoroughly.We deduce the tight-binding Hamiltonian with SU(2)gauge potential by Peierls substitution at first.We discover a topological Lifshitz transition induced by the non-Abelian SU(2)gauge field,by analysing the topologies of the Fermi surfaces.Then the corresponding phase boundary is obtained by solving the secular equation of the Hamiltonian.Finally,the effects of the gauge field on the localized edge states of the system are also investigated. |
