Interacting, Disordered, Electrons: Two Tractable Limits

这个理论研究的重点是调查的相互作用和混乱的两个易于处理的限制，更一般的问题有明确的含义。 第一个容易处理的例子是中观的。 这里有两个能量尺度，单粒子能级间距和无粒子能量，它通过不确定性原理与电子对系统进行采样的时间相联系。 这两者的比值就是无量纲电导g。 在大g的极限下，无序和（费米-液体）相互作用的问题在与传统的大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。 有趣的相变和缓慢的集体模式也出现在这个新的框架中，它允许人们理解量子点中大库仑阻塞波动的实验和数值结果，以及介观环中持续电流的潜在符号和大小。其中强相互作用导致间隙，其能够控制无序效应的结合。 分数量子霍尔体系的一种新的哈密顿方法使得使用非常简单的近似来计算物理量成为可能（因为准粒子的非微扰性质，例如它们的分数电荷，已经被纳入理论）。 该方法将用于处理无序情况下的带隙分数量子霍尔态和半满朗道能级的费米液态。该项目将在培养博士后研究人员、激发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应用产生影响。这个理论研究的重点是调查的相互作用和混乱的两个易于处理的限制，更一般的问题有明确的含义。 第一个容易处理的例子是介观的，其中包括量子点。 第二个例子是量子霍尔机制。该项目将在培养博士后研究助理方面产生广泛的影响;刺激实验活动以验证理论的预测;它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.***