Interacting, Disordered, Electrons: Two Tractable Limits

这项理论研究的重点是在两个可处理的极限中调查相互作用和无序的相互作用，并对更一般的问题有明确的影响。第一个容易处理的病例是中视眼病例。这里有两个能量尺度，单粒子能级间距和无休止能量，它由测不准原理与电子采样系统的时间联系在一起。这两者的比值是无量纲电导g。在大g的极限下，无序和(费米-液体)相互作用问题在与传统的大N理论相同的意义上是完全可解的。这种方法在无序和相互作用中都是非微扰的，这要归功于小参数1/g。有趣的相变和慢集体模也出现在这个新的框架中，这使得人们能够理解关于量子点中大的库仑阻塞涨落的实验和数值结果，以及潜在的介观环中持续电流的符号和大小。第二个易于处理的限制是在整数/分数量子霍尔电子中，其中强烈的相互作用导致禁带，使得能够控制结合无序的影响。分数量子霍尔区的一种新的哈密顿方法使使用非常简单的近似计算物理量成为可能(因为准粒子的非微扰性质，如它们的分数电荷，已经被纳入理论)。这一理论将被用来处理半填充朗道能级中存在无序的有隙分数量子霍尔态和非常有趣的费米液态场。该项目将对博士后研究助理的培训产生广泛的影响；在激发实验活动以验证该理论的预测方面；它可能对量子点在量子计算中的使用产生影响。%这项理论研究的重点是在两个可处理的极限中研究相互作用和无序的相互作用，对更一般的问题具有明显的影响。第一个容易处理的情况是中视情况，其中包括量子点。第二个案例是量子霍尔方案。该项目将对博士后研究助理的培训产生广泛影响，在刺激实验活动以验证理论预测方面产生广泛影响，并可能对量子点在量子计算中的使用产生影响。

The focus of this theoretical research is to investigate the interplay of interactions and disorder in two tractable limits, with clear implications for the more generic problem. The first tractable case is the mesocopic one. Here there are two energy scales, the single-particle level spacing and the Thouless energy, which is connected by the uncertainty principle to the time for an electron to sample the system. The ratio of these two is the dimensionless conductance, g. In the limit of large g, the problem of disorder and (Fermi-liquid) interactions is completely solvable in the same sense as a conventional large-N theory. This approach is nonperturbative in both disorder and interactions, thanks to the small parameter 1/g. Interesting phase transitions and slow collective modes also emerge in this new framework, which allows one to understand experimental and numerical results on large Coulomb blockade fluctuations in quantum dots, and potentially the sign and magnitude of persistent currents in mesoscopic rings.The second tractable limit is in the integer/fractional quantum Hall elects, where strong interaction results in gaps which enable a controlled incorporation of the effects of disorder. A new Hamiltonian approach to the fractional quantum Hall regime makes it possible to calculate physical quantities using very simple approximations (because the nonperturbative properties of the quasiparticles, such as their fractional charge, have been incorporated into the theory). This formalism will be used to treat the gapped fractional quantum Hall states and the very interesting Fermi liquid state in the half-filled Landau level in the presence of disorder.The project will have a broad impact in training of a postdoctoral research associate; in stimulating experimental activity to verify the predictions of the theory; and it may have implications for the use of quantum dots in quantum computation.%%% The focus of this theoretical research is to investigate the interplay of interactions and disorder in two tractable limits, with clear implications for the more generic problem. The first tractable case is the mesocopic one, which includes quantum dots. The second case is in the quantum Hall regime.The project will have a broad impact in training of a postdoctoral research associate; in stimulating experimental activity to verify the predictions of the theory; and it may have implications for the use of quantum dots in quantum computation.***