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Geometry and Large N Limits in Quantum Hall States

量子霍尔态中的几何和大 N 限制

基本信息

批准号：
376817586
负责人：
Professor Dr. Semyon Klevtsov
金额：
--
依托单位：
Institut de recherche mathématique avancée (IRMA)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2017
资助国家：
德国
起止时间：
2016-12-31 至 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/376817586?language=en
关键词：
Geometry Large Limits Quantum Hall

项目摘要

Quantum Hall effect (QHE) is a remarkable phenomenon, where the precise quantization of Hall conductance occurs in materials with impurities. It was originally discovered in two-dimensional electron systems at low temperatures and in strong magnetic fields, and more recently in graphene, where it was reported even at room temperatures. It is well known that the quantization phenomena in QHE have essentially geometric origin. This project addresses a number of questions in geometric theory of the quantum Hall states, focusing mainly on mathematical aspects, with an eye for potential physics applications. The main goals of the project are (i) to define and construct the Quantum Hall states, such as Laughlin state and Pfaffian state on various geometric backgrounds (ii) to give a complete classification of the geometric adiabatic transport on the moduli spaces of Riemann surfaces for these states (iii) develop effective new methods and tools to study large N asymptotics of the partition functions, correlation functions, adiabatic curvatures for the QH states, (iv) uncovering new exciting connections of QH states to Liouville quantum gravity, Bergman kernels, Quillen theory and geometry. In order to reach these goals we propose new approaches to QH states, making use of recent methods in Kaehler geometry, asymptotic analysis of Bergman kernels, and quantum field theory techniques. As the outcome of the project we expect to uncover new exciting connections of the QH states with various aspects of modern geometry and physics. The results of the project will be of potential interest to the mathematical physics, Kaehler geometry, Bergman kernel, random matrix and QHE communities.

量子霍尔效应（QHE）是一种引人注目的现象，在含有杂质的材料中会发生霍尔电导的精确量子化。它最初是在低温和强磁场下的二维电子系统中发现的，最近在石墨烯中发现，甚至在室温下也有报道。众所周知，QHE中的量子化现象本质上有几何根源。该项目解决了量子霍尔态几何理论中的一些问题，主要关注数学方面，并着眼于潜在的物理应用。该项目的主要目标是(i)定义和构建不同几何背景下的量子霍尔态，如Laughlin态和Pfaffian态；（ii）给出这些态在Riemann曲面模空间上的几何绝热输移的完整分类；（iii）开发有效的新方法和工具来研究QH态的配分函数、相关函数、绝热曲率的大N渐近性。（iv）发现QH态与Liouville量子引力、Bergman核、Quillen理论和几何之间令人兴奋的新联系。为了达到这些目标，我们提出了新的QH态方法，利用Kaehler几何中的最新方法，Bergman核的渐近分析和量子场论技术。作为项目的结果，我们希望发现QH状态与现代几何和物理的各个方面的新的令人兴奋的联系。该项目的结果将对数学物理、Kaehler几何、Bergman核、随机矩阵和QHE社区产生潜在的兴趣。