Investigations On Static And Periodically Driven Weyl Semimetals

In the last decade,topological materials have attracted great interest due to the theoretical proposals and experimental discoveries of topological insulators.The emer-gence of Weyl semimetals(WSMs)has also extended topological phases from gapped insulators to gapless semimetals.WSMs have a lot of novel features,such as surface Fermi arcs,chiral anomaly,negative magneto-resistance and chiral magnetic effects.Since the first proposal of WSMs,how to look for natural WSM materials or realize WSM phases in certain systems under certain external conditions has been a very hot topic.The first part of this thesis puts forward two proposals of realizing WSMs,one by introducing a Zeeman field or a circularly-polarized light field in(strained)?-Sn,the other by introducing a periodically kicked term in the extended Harper model.More-over,it remains an interesting and worthy question how to characterize WSMs' unique band structures and corresponding topology.Therefore,in the second part,entangle-ment spectrum and wave-packet dynamics in optical lattices are used to depict WSMs.The thesis is organized as follows:In chapter one,we first give a brief review of topological materials,from quan-tum hall effects and topological insulators to topological semimetals.We then provide the fundamentals of WSMs,including the definition of Weyl fermions,the topological protection of WSMs,Fermi arcs,chiral anomalies and negative magneto-resistance in WSMs.An brief introduction to other semimetals,such as type-? Weyl semimetals,Dirac semimetals and nodal-line semimetals is given at last.Chapter two discusses how to realize WSM phases in(strained)?-Sn by intro-ducing an external Zeeman field or a circularly-polarized light field.We first derive the k·p Luttinger Hamiltonian based on the crystal structure and corresponding symmetries of ?-Sn.Then we study the cases with strain,a Zeeman field,and a light field,respec-tively,where strain-induced Dirac points,Zeeman-field-induced and light-field-induced(double)Weyl points are found,respectively.Moreover,due to the broken C3,111 sym-metry when the strain term is introduced,the system shows different responses to fields in the[001]and[100]direction.It is worth emphasizing that "ideal" Weyl points can be generated from strained ?-Sn by a Zeeman field or a light filed in the[100]direction.In chapter three,we first briefly discuss how to realize WSM phases both in static and periodically driven extended Harper model,and then give a detailed investigation on the effect of introducing nearest-neighbor p-wave superconducting paring terms in the former model.It is found that,by tuning the strength of pairing potential,the system can switch between gapped phases or gapless phases with point nodes or line nodes.After calculating corresponding topological invariants and through bulk-edge correspondence,we find surface Majorana flat bands and Dirac arcs connecting them for certain system parameters.Therefore,we realize two kinds of topological zero(?)modes,the Kitaev-like Majorana modes and Su-Schrieffer-Heeger-like Dirac modes,within the same system.The fourth chapter investigates the entanglement spectrum of WSMs.We begin by introducing the concept of entanglement spectrum and the method for its calcula-tion.Then through dimensional reduction,which treats a three-dimensional WSM as a collection of two dimensional insulators,we calculate the entanglement spectrum as well as the "trace index" for two lattice models.It is found that the correspondence between the entanglement surface state and the physical surface state persists for the gapless phase;and the "trace inde",remains equivalent to the Chern number of the 2D insulator layers.The fifth chapter studies the wave-packet dynamics of Weyl quasiparticles in op-tical lattices.We first introduce an optical lattice model,which exhibits WSM phases.we then place a Gaussian wave packet on the Weyl point and let it expand freely.We analyze Weyl quasiparticles' dynamics by analytically and numerically calculating its density file and position of the center of mass(PCM).The PCM is proved to be related to the chirality and Chern number of each Weyl point.We conclude and give some future prospects in chapter six.