On Relativistic and Non-Relativistic Fermi Systems

关于相对论性和非相对论性费米系统

• 批准号: 0800906
• 负责人: Rudi Weikard
• 金额: $ 11.61万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800906&HistoricalAwards=false
• 关键词: Relativistic; Non; Fermi; Systems

In recent years, in condensed matter physics, significant attention has been devoted to the study of ultra-cold atomic gases. In some remarkable experiments, the experimentalists have learned to tune the inter-atomic interaction from very weak to very strong, observing thereby different phases of the gas. Within our first project we intend to study the weak-coupling regime, which is usually called the BCS (Bardeen-Cooper-Schrieffer)-regime. Mathematically, this regime can be quite accurately described by the famous BCS gap-equation. This equation describes the wavefunction of, so called, Cooper pairs. It is highly non-linear, nonetheless, in a recent work we were able to link the BCS gap-equation to the spectral analysis of a linear, pseudo-differential, operator. This allowed a precise characterization of the class of potentials giving rise to superfluidity. In further projects we plan to study the energy gap, continuity properties of the momentum distribution, and higher order calculations of the critical temperature, as well as systems with where the two spin states are unequally populated, all questions that are particularly important for applications, such as the classification of different types of superfluids. Our second project concerns the study of relativistic particles, described by Dirac's operator. In recent works with Lewin, Solovej and Sere we established a framework within mean-field approximation which allows us to obtain ground states via a minimization principle, which is a major departure from prior results. Although originally developed in the context of QED these methods can be applied to quite arbitrary infinite quantum systems. All our previous results hold in the case of zero temperature. In near future we plan to extendour results to arbitrary positive temperature. The second part of our project on Dirac particles is associated with gravitational forces. First, we consider classical Newtonian forces and study a time-evolution which is supposed to describe the dynamical collapse of white dwarfs. Second, we plan a more innovative approach, incorporating general relativity. Broader Impact: The goal is to develop new mathematical tools for superfluid systems and Dirac systems. Such methods lead to different points of views and increase the understanding of the underlying physical systems. For example, in the long run, such methods could help to understand the mechanism of high T_c-superconductors. These have all kind of applications such as ultrafast computers, magnetic levitation, loss-less powergrid, etc. The work will include multiple collaborative efforts and contribute to the training of PhD students.

近年来，在凝聚态物理领域，超冷原子气体的研究受到了极大的关注。在一些引人注目的实验中，实验学家学会了将原子间相互作用从非常弱调整到非常强，从而观察气体的不同相。在我们的第一个项目中，我们打算研究弱耦合机制，通常称为 BCS（Bardeen-Cooper-Schrieffer）机制。从数学上讲，这种状态可以通过著名的 BCS 间隙方程来相当准确地描述。该方程描述了所谓的库珀对的波函数。尽管如此，它是高度非线性的，在最近的一项工作中，我们能够将 BCS 间隙方程与线性伪微分算子的谱分析联系起来。这使得能够精确表征引起超流动性的电位类别。在进一步的项目中，我们计划研究能隙、动量分布的连续性、临界温度的高阶计算，以及两种自旋态分布不均的系统，所有这些对于应用特别重要的问题，例如不同类型超流体的分类。我们的第二个项目涉及狄拉克算子所描述的相对论粒子的研究。在最近与 Lewin、Solovej 和 Sere 的合作中，我们在平均场近似中建立了一个框架，该框架允许我们通过最小化原理获得基态，这与之前的结果有很大的不同。 尽管最初是在 QED 背景下开发的，但这些方法可以应用于相当任意的无限量子系统。我们之前的所有结果都在零温度的情况下成立。在不久的将来，我们计划将我们的结果扩展到任意正温度。我们狄拉克粒子项目的第二部分与引力有关。 首先，我们考虑经典牛顿力并研究时间演化，该演化应该描述白矮星的动力学塌缩。其次，我们计划采用更具创新性的方法，结合广义相对论。更广泛的影响：目标是为超流体系统和狄拉克系统开发新的数学工具。这些方法会产生不同的观点并增加对底层物理系统的理解。例如，从长远来看，此类方法可以帮助理解高 T_c 超导体的机制。这些具有各种应用，例如超高速计算机、磁悬浮、无损电网等。这项工作将包括多种协作努力，并有助于博士生的培养。