Φ-Mapping Topological Current Theory And Its Applications In The Frontier Of Physics

In this dissertation, based on theφ-mapping topological current theory proposed byProf. Yi-shi Duan, we study the ferromagnetic spin-triplet superconductor, the two-gapsuperconductor, the two-scalar Abelian Chern-Simons model, the brane world cosmicstring model, and the steady-state bifurcations of the vector field.Firstly, using theφ-mapping topological current theory, we argue that ferromagneticspin-triplet superconductors allow formation of unstable magnetic monopoles. We define atopological current of the magnetic monopole, and show that the topological current is exactlythe particle current of the magnetic monopole. The nonvanishing of topological currentindicates the existence of the magnetic monopole. The conserved topological chargeof the topological current is corresponding magnetic charge of the magnetic monopole.To make the energy finite in an infinite volume ferromagnetic spin-triplet superconductor,the magnetic monopoles can exist only in the form of the monopole-antimonopole pairs.In such a pair, the monopole and antimonopole will be connected by a Dirac string, or adoubly-quantized vortex, which belongs to the trivial topological class of the first homotopygroup of SO(3). Besides, we show that the limit points and the bifurcation pointsof theφ-mapping will serve as the interaction points of these magnetic monopoles.Secondly, we derive the exact modified London equation for the two-gap superconductor,compare it with its single-gap counterpart. We find the cores of vortices in thetwo-gap superconductor are soft (or continuous). In particular, we discuss the topologicalstructure of the finite energy vortices (Abrikosov-like vortices), and find that theycan be viewed as the incarnation of the baby skyrmion stretched in the third direction.Besides, we point out that the knot soliton in the two-gap superconductor is the twistedAbrikosov-like vortex with its two periodic ends connected smoothly. The relation betweenthe magnetic monopoles and the Abrikosov-like vortices is also discussed briefly.Thirdly, we introduce the Abelian Chern-Simons model with two complex scalar fields,and usingφ-mapping topological current theory, we investigate the self-dual vortices in this model. We find a nontrivial equation with a topological term for each scalar field.which is missing in many references. Besides, we also find a equation relates two scalarfields by their topological terms. We calculate the angular momentum of the system, andfind that it is exactly the generalization of the angular momentum of single complex scalarvortices. We also calculate the magnetic flux of the system for different boundary conditions.Furthermore, we briefly discussed the evolution processes of the vortices, and foundthat because of the present of the vortex molecule, the detailed evolution processes of thevortices in the present model is more complicated than the vortex evolution processes inthe corresponding single scalar field model.Fourthly, making use ofφ-mapping topological current theory, the structure of thevortex has been obtained from the AH model, which provides an effective description tothe brane world cosmic string system. The advantage of this description of the vortexconfigurations is the physical quantities concerning the topology of the system can beexpressed in an analytical way, and the relationships of these quantities can be provedstrictly. Therefore, for the purpose of studying the topological property of the cosmicstrings, this description is highly significant. Besides, combining the result of decompositionof U(1) gauge potential, we verify two conclusions showed in the previous literaturein an alternative way.Finally, using theφ-mapping topological current theory, we introduce a topologicalcurrent to describe the topological feature of the vector field. In this description, theequilibria of the vector field can be viewed as the particles, which carry the windingnumbers as their topological charges and move in the phase space as the parameter varies.Based on this description, we qualitatively discussed various steady-state bifurcations ofthe vector field, and found a general method to determine the numbers and the directionsof the bifurcation curves at the limit points and the bifurcation points.