Effects of weak disorders on quantum Hall critical points

Abstract

We study the consequences of the random mass, random scalar potential, and random vector potential on the line of fixed points between integer and/or fractional quantum Hall states and an insulator. This line of fixed points was first identified in a clean Dirac fermion system with both Chern-Simon coupling and Coulomb interaction [Phys. Rev. Lett. 80, 5409 (1998)]. By performing a renormalization-group analysis in $1/N$ $(N$ is the number of species of Dirac fermions) and the variances of three disorders ${\ensuremath{\Delta}}_{M},{\ensuremath{\Delta}}_{V},{\ensuremath{\Delta}}_{A},$ we find that ${\ensuremath{\Delta}}_{M}$ is irrelevant along this line, and both ${\ensuremath{\Delta}}_{A}$ and ${\ensuremath{\Delta}}_{V}$ are marginal. With the presence of all three disorders, the pure fixed line is unstable. Setting Chern-Simon interaction to zero, we find one nontrivial line of fixed points in the $({\ensuremath{\Delta}}_{A},w)$ plane with dynamic exponent $z=1$ and continuously changing \ensuremath{\nu}; it is stable against small $({\ensuremath{\Delta}}_{M},{\ensuremath{\Delta}}_{V})$ in a small range of the line $1<w<1.31,$ therefore it may be relevant to integer quantum Hall transition. Setting ${\ensuremath{\Delta}}_{M}=0,$ we find a fixed plane with $z=1,$ the part of this plane with $\ensuremath{\nu}>1$ is stable against small ${\ensuremath{\Delta}}_{M},$ therefore it may be relevant to fractional quantum Hall transition. Although we do not find a generic fixed point with all the couplings nonvanishing, we prove that the theory is renormalizable to the order ${(1/N)}^{2},(1/N)\ensuremath{\Delta},{\ensuremath{\Delta}}^{2},$ and we explore the interesting processes which describe the interferences between the Chern-Simon interaction, the Coulomb interaction, and the three kinds of disorders.