Critical Hamiltonians with long range hopping

Critical states are studied by a real space RG in the problem with strong d iagonal disorder and long range power law hopping. The RG flow of the distr ibution of coupling parameters is characterized by a family of non-trivial fix points. We consider the RG flow of the distribution of participation ra tios of eigenstates. Scaling of participation ratios is sensitive to the na ture of the RG fix point. For some fix points, scaling of participation rat ios is characterized by a distribution of exponents, rather than by a singl e exponent. The RG method can be generalized to treat certain fermionic Hamiltonians wi th disorder and long range hopping. We derive the RG for a model of interac ting two-level systems. Besides couplings, in this problem the RG includes the density of states. The density of states is renormalized so that it dev elops a singularity near zero energy.