Critical wave functions, conformal invariance, and theories for quantum Hall transitions
Wave functions at Anderson localization-delocalization (LD) transitions are known to exhibit complicated scale-invariant behavior best characterized by an infinite set of multifractal exponents. I will present our recent study of multifractal spectra of critical wave functions at various Anderson transitions, focusing on finite systems with boundaries. In two dimensions the boundary behavior of the critical wave functions provides a way to answer the question of conformal invariance at these transitions. For the integer quantum Hall transition the multifractal spectra were conjectured to be exactly parabolic in a number of proposals of critical field theories for the transition. Our numerical results for the Chalker-Coddington network model firmly rule out the exact parabolicity. In addition, we provide an exact result for surface multifractal exponents for a related LD critical point in the BDI (chiral orthogonal) class.