Interacting, Disordered, Electrons: Two Tractable Limits

本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，这对更一般的问题具有明确的含义。第一种可处理的情况是中观的情况。这里有两种能量尺度，单粒子级间距和索利斯能量，它通过不确定性原理与电子进入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）相互作用的问题在与传统大n理论相同的意义上是完全可解的。由于参数1/g很小，这种方法在无序和相互作用中都是非摄动的。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封锁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。第二个可处理的限制是在整数/分数量子霍尔选择中，其中强相互作用导致间隙，从而使无序效应的控制结合成为可能。分数量子霍尔体系的新哈密顿方法使得使用非常简单的近似计算物理量成为可能（因为准粒子的非摄动特性，例如它们的分数电荷，已被纳入理论）。这种形式将用于处理有间隙的分数量子霍尔态和非常有趣的半填充朗道能级中存在无序的费米液态。对培养博士后研究助理具有广泛影响；刺激实验活动以验证理论的预测；这可能会对量子点在量子计算中的应用产生影响。本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，这对更一般的问题具有明确的含义。第一种可处理的情况是中观的，其中包括量子点。第二种情况是在量子霍尔体系中。对培养博士后研究助理具有广泛影响；刺激实验活动以验证理论的预测；这可能会对量子点在量子计算中的应用产生影响

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.***