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The localization-delocalization transition in interacting systems has been a very active area of inquiry for the last two decades. However, a comprehensive understanding of this transition remains elusive. Concurrently, there has been a renewal of interest in the understanding of localization induced by quasiperiodicity rather than random disorder. In this talk, I will describe how one can study the localization transition in both types of systems using a recursive Green's function approach. For a non-interacting quasiperiodic system, this approach along with a complementary calculation of the Thouless conductance reveals a failure of single parameter scaling to describe the transition in one and two dimensions but not in three dimensions. For an interacting system with random disorder, the recursive method can be adapted to Fock space to study the many-body localization transition. The imaginary part of the self-energy thus obtained can be employed as an order parameter. A calculation of this quantity reveals a diverging "non-ergodic" Fock space volume with an essential singularity as the transition is approached from the delocalized side and a diverging localization length with a critical exponent when the approach is from the localized side. Time permitting, I will also discuss the corresponding transition in an interacting quasiperiodic system. |