Strongly Correlated Quantum Systems

本课程将研究强相关多体系统的量子力学。目的是提高我们对不能用简单摄动理论解决的新现象的理解。感兴趣的系统包括整数和分数量子霍尔效应；介观装置中的输运；一维系统中无序、相互作用与超导的相互作用分数统计的问题。整数量子霍尔效应中遗留的一个重要问题涉及无序存在下安德森局域化的性质。分数霍尔效应的研究重点将集中在低填充系数下Wigner结晶的不稳定性和边缘态激发的性质及其在介观器件输运性质中的作用。霍尔效应工作的一个中心主题是量子波函数在最低朗道能级的解析性的深刻和广泛的含义。先前关于超导体-绝缘体转变的工作将扩展到一维，并将研究某些精确可溶的模型。分数统计的概念将在二维量子自旋系统的研究中得到实际应用。

Research will be undertaken on the quantum mechanics of strongly- correlated many-body systems. The objectives are to enhance our understanding of novel phenomena which cannot be addressed by simple perturbation theory. Systems of interest include the integer and fractional quantum Hall effect; transport in mesoscopic devices; the interplay among disorder, interactions and superconductivity in one-dimensional systems; and, the problem of fractional statistics. An important question remaining in the integer quantum Hall effect concerns the nature of Anderson localization in the presence of disorder. The focus of the investigation on the fractional Hall effect will be on the instability to Wigner crystallization in the low filling factor regime and on the nature of edge-state excitations and their role in the transport properties of mesoscopic devices. A central theme of the Hall effect work is the deep and wide-ranging implications of the analyticity of the quantum wave functions in the lowest Landau level. Previous work on the superconductor-insulator transition will be extended to one-dimension and certain exactly soluble models will be investigated. The notions of fractional statistics will be put to practical application in the study of quantum spin systems in two-dimensions.