Disordered Systems and Stochastic Growth Phenomena

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TECHNICAL SUMMARYThe Division of Materials Research and the Division of Mathematical Sciences contribute funds to this award. This award supports research and education in theoretical condensed matter physics in two related subjects: disordered systems and stochastic growth phenomena.The projects the PI will engage on disordered systems include: the study of conformal invariance, multifractal critical functions, and symmetry of multifractal spectra at Anderson transitions; critical states and the integer quantum Hall effect in low magnetic fields; general network models for Anderson localization in all symmetry classes and the supersymmetry method for them; the theory of quantum Hall transitions and other disordered critical points in two dimensions based on conformal restriction theory and Schramm-Loewner evolution; localization and Anderson transitions in class D, including the random bond Ising model; network models with structural disorder, and other disordered systems on random surfaces.In the area of stochastic growth the specific projects will be the Schramm-Loewner evolution for critical systems with extended chiral symmetries: Wess-Zumino-Witten models and parafermionc theories; theoretical and mathematical aspects of conformal restriction in the bulk; models of stochastic growth interpolating between deterministic Laplacian growth and stochastic diffusion-limited aggregation and similar processes; stochastic perturbations of general classical integrable systems.The proposed theoretical developments connect with experimental studies of quantum Hall transitions and of various driven systems exhibiting unstable interfacial motion. The research will also make contact with direct computer simulations through collaborators who are experts in large-scale numerical simulations. The research has a strong education component involving the training of graduate students, and involves substantial international collaboration with research teams in France, Germany, and Japan, which will enrich the research enterprise in the physical sciences in the US.NON-TECHNICAL SUMMARYThe Division of Materials Research and the Division of Mathematical Sciences contribute funds to this award. This award supports research and education in theoretical condensed matter physics in two related subjects: disordered systems and stochastic growth phenomena. Impurities, lattice imperfections, and other forms of disorder crucially affect properties of electronic and other materials. Disorder alone can prevent electric current from flowing, turning a metal into an insulator. This is a consequence of the wave nature of the electron and the interference of electron waves scattered by impurities. The PI will use sophisticated theoretical concepts and mathematical methods to advance understanding of this transformation between a metal and an insulator as the amount of disorder is varied. The PI will also study how this transformation takes place in a quantum Hall system where electron conduction is richly complex. A quantum Hall system is a gas of electrons confined to a plane in an artificial semiconductor structure with an applied magnetic field perpendicular to the plane. The conduction of electrons through a quantum Hall system varies in interesting ways depending on the strength of the magnetic field. Of particular interest is the effect of the interplay of the magnetic field with disorder on the conduction of electrons through a quantum Hall system. The PI will also use sophisticated mathematical methods to advance understanding of random patterns that arise in growth. Examples of such patterns are evident in soot particles, bacterial colonies grown in a Petri dish, fingered patterns of minerals deposited by water seeping through porous rock, and vortices in turbulent fluid flows. The shapes that arise are generally rough and fractal - when examined more closely, a magnified image looks the same as the unaided image. These fractals are often driven by random forces, requiring their characterization in terms of probabilities. The research has a strong education component involving the training of graduate students, and involves substantial international collaboration with research teams in France, Germany, and Japan, which will enrich the research enterprise in the physical sciences in the US.