Field Theory of Quantum Hall Effects, Composite Bosons, Vortices and Skyrmions
A field theory of quantum Hall effects is constructed based on the \CB picture. It is tightly related with the microscopic wave-function theory. The characteristic feature is that the field operator describes solely the physical degrees of freedom representing the deviation from the Laughlin state. It presents a powerful tool to analyze excited states within the \LLL. It is shown that all excitations are nonlocal topological solitons in the spinless quantum Hall system. On the other hand, in the presence of the spin degree of freedom it is shown that a quantum coherence develops spontaneously, where excitations include a Goldstone mode besides nonlocal topological solitons. Solitons are vortices and Skyrmions carrying the U(1) and SU(2) topological charges, respectively. Their classical configurations are derived from their microscopic wave functions. The Skyrmion appears merely as a low-energy excitation within the \LLL and not as a solution of the effective nonlinear sigma model. We use it as a consistency check of the Skyrmion theory that the Skyrmion is reduced to the vortex in the vanishing limit of the Skyrmion size. We evaluate the activation energy of one Skyrmion and compare it with experimental data.