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本专题旨在为感兴趣且学有余力的本科生和低年级研究生提供一条从大学物理到当前二维、三维拓扑新材料研究前沿的学习路径。专题包含两篇，本文是第一篇。本科生在大学阶段中已修习过电动力学、量子力学、固体物理等课程，但各类教材往往停留在介绍该课程的传统内容，而缺乏向前多走一步、和领域前沿研究相衔接的努力。我们希望能稍微填补这一空当。第一篇从本科生学过的经典霍尔(Hall)效应讲起，首先介绍量子力学是如何被引入二维或(2+1)维霍尔系统研究中的，然后引导到各种量子霍尔效应(包括整数、分数、反常、自旋)及其中的拓扑不变量。
Abstract：We aim to provide a way for interested undergraduate and postgraduate students to launch from college physics to access the cutting edge of the current international research on two-dimensional and three-dimensional new topological materials. This topic contains two parts, this paper is the first part. Undergraduates have learned electrodynamics, quantum mechanics, solid state physics, and other courses in the university stage. However, many textbooks often stay in the introduction of the traditional content of the course, fail to extend the scope to the latest progress of modern quantum physics. This paper serves as an attempt to fill this gap. Starting from the classical Hall effect in college physics, we first introduce the technique how to apply quantum mechanics to the study of a Hall system in 2-dimension or(2+1) dimensions. Then various quantum Hall effects(integral, fractional, spin and anomalous) and topological invariants existence in them are introduced.
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(1)对式(46)的准确理解要用到二次量子化体系,即履行■手续,有■。这之下有■,它是式(46)取库仑(Coulomb)规范?iai=0时的解。
(2)如何从经典霍尔效应引出经典反常霍尔效应?对本科生来说一种容易理解的、普通物理的思考路径可以如下。首先,经典霍尔效应是对二维平面施加了垂直外磁场;然后,认为外磁场可由天然磁体(特别是铁磁体)提供。已知磁力线在磁体外是由N极指向S极,而在体内是由S指向N。那么可以假想把霍尔系统设置在磁体内,由体内的磁力线来提供磁场;第三,将铁磁体剖成二维薄片,于是每个薄片自身就既是“外磁场”的提供者又是磁场作用的对象,从而实现“不必外加磁场依然能够实现霍尔效应”的效果。显然,能够达到这一双重目的的晶格结构必然具有一定的特殊性,需要专门寻找或设计。
(3)关于自旋、手征性传导的相对论版本最早见于上世纪九十年代,产生于手征规范理论的数值模拟[29,30]。最简单的例子包含宇称和时间反演对称的■规范理论。该理论包含正反质量的体费米子、无质量的狄拉克表面模式,其中的“体”流是手征流而非电流,这非常类似后来的量子自旋霍尔效应。
基本信息:
DOI：
中图分类号:O413;TB30
引用信息:
[1]朱方泽,常治文,黄侯迪,等.拓扑新材料研究前沿的理论基础导引Ⅰ:量子霍尔效应[J].物理与工程,2022,32(06):104-120+126.
基金信息:
北京市自然科学基金重点项目(Z180007); 国家自然科学基金(11572005)