Topological Defects In Topological Semimetals

New topological materiel in low dimensions is at the cutting edge of the international research of condensed matter physics.It not only is of great significance theoretically,but also has wide potential applications in practice.The emphasis of this Thesis is placed on topological semi-metal.Chapter 1 is the Introduction,which presents a brief review on various quantum Hall effects and several new topological materials.It is stressed that the topological properties of the materials induce the extraordinary band structures.Chapter 2 serves as a preliminary in the physical aspects,providing the concepts of the Landau energy levels and the filling factor ν in the quantum Hall effect.We highlight the essence of νas a a topological invariant,as well as the induced properties such as band structures and edge state.Chapter 3 serves as a preliminary of mathematics,differential geometry and topology in particular,by providing a brief introduction to the method of gauge potential decomposition and topological currents.Chapter 4 is the core part of the Thesis containing the innovative contents.Focusing on various topological defects in topological semi-metals,we present their differential descriptions and the discussions on their solutions in different cases,with associate physical interpretations.The innovative points of this Thesis are as follows.1.For Weyl semi-metals(as classified when a conduction band and a valence band touch each other at isolated single nodes): It is discovered from the differential structures of the defects that the Weyl nodes are the so-called monopoles,a type of three-dimensional defects,in the momentum space.At these singularities the Chern number is ill-defined,accompanied with topological phase transitions.In a Weyl semi-metal there also exist merons,a type of two-dimensional defects,whose topological charges account for the Chern number of the semi-metal.2.For nodal-line semi-metals(as classified when a conduction band and a valence band touch each other at nodal lines): It is discovered from the differential structures of the defects that the system carries four-dimensional point defects.At these singularities the Hopf invariant is ill-defined,accompanied with topological phase transitions.In a nodal-line semi-metal there also exist three-dimensional point defect,we also call the merons,whose topological charges account for the Hopf invariant of the nodal-line semi-metal.3.A U(1)Chern-Simons invariant is introduced to characterise the topological properties of nodal semi-metals: This theory gives the topological origin of nodal line,i.e.,the line defects in three-dimensional Brillouin zone.Nodal line protected by various symmetry can characterised by this invariant.Furthermore,several topological invariants describing the topological properties of nodal semi-metals can be obtained from this theory.