Research On One-dimensional Anyons

The theory of anyons and fractional statistcs, which has been the theoretical foundation of the Quantum Hall Effect(QHE), is an important part of the theories of the modern condensed matter physics. And QHE has been a hotspot for both theoretical and experimental researches since the early of 1980s. Today, the theory of anyons has been widely used to describe the low-dimensional quantum systems with fractional statistics. It is becoming of increasing importance in condensed matter physics. In recent years, it is also attracting more and more attentions of the researchers on quantum computation.For many of the past years, because of the difficulties in experimental techniques and the lack of connections with the real physical world, the studies on one-dimensional(lD) anyons was far behind of ones on two-dimensional(2D) anyons. However, with the increasing improvement of experimental techniques, the signature of fractional statistics in two-dimensional systems has recently been observed in experiments. Furthermore, confining two-dimensional systems into one-dimensional is not a difficult problem now. So, the theoretical researches on anyons are connected with experiments more closely, and the theoretical studies on 1D anyons are payed much more attentions now.In this dissertation, the main idea of is to present the study result on 1D anyons. As the main body of this dissertation, the first three chapters are devoted to introduce some basic theory of anyons and fractional statistics, which are necessary for understanding the 1D anyons problem better. In the fourth chapter, we introduce two kinds of 1D anyons that have been payed much attentions in recent years, and some works done by the author himself are presented.For Kundu 1D anyon model, which has been studied more frequently, we suggest a modification, that is making the interacting strength depended on the anyonic parameter. This modification permits a theoretical discussion for a limiting case, whose solution was left out in Kundu's original work, through analyzing its asymptotic behavior. Moreover, the most significant point of this modification perhaps is its feasibility to be realized in experiments, which may be of interest to the experimental physicists.the second contribution of us is to introduce a general theory of constructing 1D anyon models. By constructing a form of statistical interaction, and using the path integral formalism, we recover the known exchange symmetry of 1D anyons. The statistical transmutation between anyons, bosons, and fermions are presented in quantum mechanics formulation. The known 1D anyons, i.e. Kundu's anyons and Girardeau's anyons can both be brought into this general theory, furthermore the definition of 1D anyons is extended.