Duan's Topological Current Theory, Duan-Ge's U(1) Gauge Potential Decomposition Theory And Their Applications In Superconductivity Theory And Quantum Hall Effect

The study of topological defects attracts many physicist in high energy physics and condensed matter physics.In condensed matter physics the best known topological defects are the Abrikosov vortices which were put forward by Abrikosov in 1957.Subsequently,the type-â…¡superconductors was found which display the different magnetic properties from Meissner effect.The same kind of topological vortex were found in the superfluid and BEC.However, there exists more topological defects in the condensed matter physics with advent the spinor superconductors and BEC.In this dissertation,we have used the Duan's topological current theory and Duan-Ge's decomposition theory of gauge potential proposed by Prof.YiShi Duan to study the topological properties of superconductivity theory and quantum Hall effect.Compared with other methods,our theory can describe the inner topological structure of the systems and obtain the non-trivial vorticity.Firstly,we give an introduction of the Duan's topological current theory and Duan-Ge's decomposition theory of gauge potential.Secondly,by making use of the Duan-Ge's U(1)gauge potential decomposition and the Duan's topological current theory,we investigate the topological inner structure of the Bogomol'nyi equations and deduce a modified decoupled Bogomol'nyi equation with a nontrivial topological term,which is ignored in conventional model.We find that the nontrivial topological term states the position and topological charges of N-vortex which arises from the zero points of the complex scalar field.Furthermore,we establish a relationship between Ginzburg-Landau free energy and the winding number,we find that GinzburgLandau free energy is a topological invariant.Thirdly,by making use of the Duan-Ge's U(1)gauge potential decom- position and the Duan's topological current theory,we explore the topological properties of Chern-Silnons-Ginzburg-Landau theory.We obtain the equation about the particle density,magnetic flux(vortices),and the external magnetic field.It will play an important role in understanding how quasiparticles formed at self-dual point.Finally,using the Duan's topological current theory,we study the topological property of Mermin-Ho vortex in the triplet superconductors,we establish a relationship between local vorticity and the winding number.Our conclusion complements the present Ginzburg-Landau theory in the topological defects,and provides the theoretical foundation for the topological defects studies of experiments and numerical value methods.