Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces †

The FQH states are shown to have q̃g fold ground state degeneracy on a Riemann surface of genus g, where q̃ is the ground state degeneracy in a torus topology. The ground state degeneracies are directly related to the statistics of the quasi-particles given by θ = p̃π q̃ . The ground state degeneracy is shown to be invariant against weak but otherwise arbitrary perturbations. Therefore the ground state degeneracy provides a new quantum number, in addition to the Hall conductance, characterizing different phases of the FQH systems. The phases with different ground state degeneracies are considered to have different topological orders. For a finite system of size L, the ground state degeneracy is lifted. The energy splitting is shown to be at most of order e− L ξ . We also show that the G-L theory of the FQH states (in the low energy limit) is a dual theory of the U(1) Chern-Simons topological theory. † Published in Phys. Rev. B41, 9377 (1990) * After Dec. 1, 1989, School of Natural Science, Institute for Advanced study, Princeton, NJ 08540.