Disordered Systems and Stochastic Growth Phenomena

技术摘要材料研究部和数学科学部为该奖项提供资金。该奖项支持理论凝聚态物理在两个相关学科的研究和教育：无序系统和随机增长现象。PI将参与的无序系统研究项目包括：共形不变性、多重分形临界函数和多重分形谱在安德森跃迁处的对称性研究；低磁场下的临界态和整数量子霍尔效应；所有对称类中Anderson局域化的一般网络模型及其超对称方法；基于共形约束理论和Schramm-Loewner演化的二维量子霍尔跃迁和其他无序临界点理论；D类的局域化和安德森跃迁，包括随机键Ising模型；具有结构无序的网络模型，以及随机表面上的其他无序系统。在随机增长领域，具体项目将是具有扩展手性对称性的临界系统的Schramm-Loewner演化：Wess-Zumino-Witten模型和参数理论；体形约束的理论和数学研究确定性拉普拉斯增长与随机有限扩散聚集及类似过程之间的随机增长插值模型；一般经典可积系统的随机扰动。提出的理论发展与量子霍尔跃迁和各种表现不稳定界面运动的驱动系统的实验研究有关。该研究还将通过大规模数值模拟专家的合作者直接接触计算机模拟。这项研究有很强的教育成分，包括研究生的培训，并涉及与法国、德国和日本的研究团队的大量国际合作，这将丰富美国的物理科学研究事业。材料研究部和数学科学部为本奖项提供资金。该奖项支持理论凝聚态物理在两个相关学科的研究和教育：无序系统和随机增长现象。杂质、晶格缺陷和其他形式的无序严重影响电子和其他材料的性质。无序本身就能阻止电流流动，使金属变成绝缘体。这是电子的波动性质和杂质散射的电子波的干扰的结果。PI将使用复杂的理论概念和数学方法来推进对金属和绝缘体之间的这种转变的理解，因为无序的数量是不同的。PI还将研究这种转变如何在电子传导非常复杂的量子霍尔系统中发生。量子霍尔系统是一种在人造半导体结构中被限制在一个平面上的电子气体，外加垂直于该平面的磁场。电子通过量子霍尔系统的传导会根据磁场的强度以有趣的方式变化。特别令人感兴趣的是磁场与无序的相互作用对电子通过量子霍尔系统传导的影响。PI还将使用复杂的数学方法来促进对生长过程中出现的随机模式的理解。这种模式的例子在煤烟颗粒、培养皿中生长的细菌菌落、渗透多孔岩石的水沉积的矿物的指状模式以及湍流中的涡流中都很明显。出现的形状通常是粗糙的和分形的——当更仔细地观察时，放大的图像看起来和没有辅助的图像一样。这些分形通常是由随机力驱动的，需要用概率来描述它们。这项研究有很强的教育成分，包括研究生的培训，并涉及与法国、德国和日本的研究团队的大量国际合作，这将丰富美国的物理科学研究事业。

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.