Seminars & Colloquia Calendar

Abstract: The spectrum of a self-adjoint operator is known to decompose into three parts, which are called pure point (pp), absolutely continuous (ac), and singular continuous (sc). In the traditional physics approach to Anderson (de-) localization for random Schrödinger operators, only the first two types of spectrum are featured: ac spectrum comes with spatially extended eigenstates (a.k.a. metallic regime), while the eigenstates for eigenvalues in the pp spectrum are localized (a.k.a. insulating regime). Now, over the last few years there have been various predictions of a possible third regime, called NEE (for non-ergodic extended), where the eigenstates are fractal, matching the phenomenology expected for the case of sc spectrum. There exists, however, an ongoing debate as to whether NEE/sc can be a true thermodynamic phase (instead of just a finite-size effect or an exotic feature that needs fine-tuning to a critical point). In this colloquium talk, I will first review the standard field-theory approach, developed by Wegner, Efetov and others, for random Schrödinger operators in the metallic and insulating regimes. Motivated by a recent proposal for the conformal field theory of the integer quantum Hall transition, I will then put forward a field-theoretical scenario for the elusive case of random Schrödinger operators with sc spectrum (NEE phase). Distinct from the usual sigma model, the proposed formulation is supported by an exact solution of Wegner´s (N=1)-orbital model on a Bethe lattice. It is expected to be generic for moderately strong disorder in high space dimension.