Quantum electrodynamics with chern-simons term in 2+1 dimensions and anyon

Abstract

The Chern-simons (CS) term is the particular object in 2+1 dimensional field theory. We investigate on what the kinematical and the dynamical roles of the Abelian CS term are and what consequences are obtained in various systems. The CS term makes the matter field obey the fractional statistics by acquiring the fractional spin and become anyon. This is the kinematical role of the CS term and is justified in the symmetric phase. In the broken-symmetric phase, however, the matter field does no longer acquire the fractional spin except for the zero momentum mode. The investigation of the statistics-changing transition between symmetric and broken phase leads us to the negative result, namely that such transition does not appear. Based on the result in the symmetric phase, the composite operators showing the anyonicity are constructed from the CS-matter theory in the presence and the absence of the Maxwell kinetic term. The dynamical role of the CS term is that it leads to the short-range magnetic interaction between charged particles. Due to this interaction, there appears the possibility that two equally charged fermions can form a bound state pair. Concerning with the pairs, the low energy effective action describing them is derived. It is shown that the fermion pairs behave like doubly charged spin 1 bosons and, when they condense, the gauge field acquires the longitudinal mass. The approximate symmetry structure in the effective action is discussed. As the final study, a model related to the high-temperature superconductivity is considered.