The physics of localization transitions, and especially quantum critical behavior at the transitions, is rich and nontrivial. Supersymmetry (SUSY) is one of several methods that have been employed to carry out averages over different realizations of quenched disorder. The resulting many-body Hamiltonians can then be studied analytically and also numerically with the use of the density-matrix renormalization-group (DMRG) technique. Successes include exact proofs of non-vanishing density-of-states (DOS) and quantum criticality at the plateau transition in the integer quantum Hall effect (IQHE), and the accurate calculation of critical exponents in the spin quantum Hall effect (SQHE). However, the calculation of corresponding exponents for the IQHE transitions has proved to be much more difficult. I discuss the reasons for this difficulty, and ongoing work to circumvent the problem.

Updated 2 August 1999.