Systems with Competing Periodicities

一般势和横向磁场中的二维电子气表现出非常有趣的具有各种交叉点的分形谱，并且避免了能带结构中的交叉点。这些交叉用积分拓扑不变量来刻画。这些特征是理解有机导体中最近的一些实验的关键，这些实验展示了场诱导的自旋密度波的级联，并将在涉及量子点阵列的实验中产生新的结果。在无公度极限下，二维电子气呈现出金属-绝缘体相变的级联，在零温下有一个“魔鬼叉子”相图。我们将研究一大类模型，以证明这种分形相图是普遍存在的。我们将研究一些量子混沌模型，并探索它们的实验结果。此外，还将研究量子混沌的拓扑学方面，以获得一大类具有竞争周期的系统的统一图景。建立了研究竞争长度系统有限温度性质的一般形式。该方法将用于计算场致自旋密度波系统和磁性斐波那契等周期超晶格的热力学性质。斐波那契晶格的一个新方面是热力学性质的调制下的幂定律行为。将对具有竞争周期的各种物理和模型系统进行理论研究。系统不同自然周期之间的这种竞争导致了高度的非线性行为。需要应用特殊的数学技巧来理解产生的行为。要研究的物理系统包括在强磁场下在半导体界面上形成的二维电子气、量子点阵列和有机导体中感应的磁波。混沌极限也将被研究。这项工作的结果将产生处理这些类型的系统的通用技术，并将有助于解释这些具有重要技术意义的材料的行为。

A two-dimensional electron gas in a generic potential and a transverse magnetic field exhibits very intriguing fractal spectra with various crossings and avoided crossings in the band structure. These crossings are characterized by integral topological invariants. These features are key towards understanding some recent experiments in the organic conductors which exhibit cascades of field-induced spin density waves and will have novel consequences in experiments involving quantum dot arrays. In the incommensurate limit, a two-dimensional electron gas exhibits a cascade of metal-insulator transitions which at zero temperature has a "devil's fork" phase diagram. A large class of models will be studied to demonstrate that this fractal phase diagram is universal. Some of the quantum chaos models will be examined and their experimental consequences will be explored. In addition, topological aspects of quantum chaos will be studied in order to obtain a unified picture of a large class of systems with competing periodicities. A general formalism to study the finite temperature properties of systems with competing lengths will be developed. This method will be applied to calculate thermodynamical properties of field-induced spin density waves systems and the magnetic Fibonacci and other periodic superlattices. A novel aspect of the Fibonacci lattices is the power law behavior with modulations in the thermodynamic properties. %%% Theoretical research will be conducted on a variety of physical and model systems which have competing periodicities. This competition between various natural periods of the systems results in highly nonlinear behavior. Special mathematical techniques need to be applied to understand the resulting behavior. Physical systems to be studied include the two-dimensional electron gas which forms at semiconductor interfaces exposed to high magnetic fields, quantum dot arrays, and magnetic waves induced in organic conductors. The chaotic limit will also be studied. The results of this work will yield generic techniques to handle these type of systems and will assist in interpreting the behavior of these technologically important materials.