Farey series, hierarchy structure, and scaling theory of the fractional quantum Hall effect

In this paper, we present a unique procedure, with physical justification, for the hierarchical construction of quantum Hall states (QHS). We have connected the hierarchy structure of the filling fractions of QHS constructed using our rule with the Farey-number hierarchy structure. This connection reveals that our hierarchy structure is self-similar. Based on this hierarchy structure, we propose a renormalization-group (RG) flow diagram for the scaling behavior of transitions between Hall plateaus. We discuss and compare our results with Jain's hierarchy theory and other previous scaling theories of the quantum Hall effect. The difference can be examined by future experiments.