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Exotic Particles in Topological Insulators

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Exotic Particles in Topological Insulators EXOTIC PARTICLES IN TOPOLOGICAL INSULATORS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF PHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Rundong Li July 2010 © 2010 by Rundong Li. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/yx514yb1109 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Shoucheng Zhang, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Ian Fisher I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Steven Kivelson Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract Recently a new class of quantum state of matter, the time-reversal invariant topo- logical insulators, have been theoretically proposed and experimentally discovered. These topological quantum states of matter are insulating in the bulk, but have gap- less edge or surface states protected by the time-reversal symmetry. In particular, topological insulators in three dimensions are characterized by topological field theory which gives rise to the topological magnetoelectric effect, and is analogous to that describing the electromagnetism of the hypothetical particle called axion. In this thesis we will show that in these topologically nontrivial insulators, mag- netic monopoles and axions, which are originally postulated in elementary particle physics, may emerge. Firstly we will show that when time-reversal symmetry is bro- ken on the surface of a topological insulator, an electric charge near the surface will induce an image magnetic monopole. The composite particle consisting of an elec- tron and its image monopole forms a dyon and obeys fractional statistics. Secondly, when there is an antiferromagnetic order in the bulk of a topological insulator, the magnetic fluctuations couple to the electromagnetic fields exactly like axions. The physical effect of the dynamical axion and its detection will also be discussed. Then finally we propose transition metal oxide of corundum structure as a candidate for topological magnetic insulators which can give rise to the dynamical axion. iv Acknowledgments It is my great pleasure to express my gratitude to my advisor Shou-Cheng Zhang for his education and guidance through my PhD years. He is one of the physicists who opened up the field of topological insulator. In participating the development of this field, what I have learned from his breadth and depth of knowledge and his inexhaustible creativity is invaluable. Most of all he taught me how to identify beautiful and profound physics out of a complicated problem. I greatly appreciate Steve Kivelson for his stimulating discussions and inspiring lectures. I greatly appreciate Ian Fisher for his discussions and help in searching for materials for topological magnetic insulators, as well as for being the Chair of my oral examination. I wish to thank David Goldhaber-Gordon, Aharon Kapitulnik and Zhi-Xun Shen for being my thesis committee members and for their support. I would like to thank Xiao-Liang Qi and Taylor Hughes, they are always ready to teach me and share their knowledge, and in particular have helped me a lot to start on the topological insulator. I would like to thank my officemate Joseph Maciejko, for numerous intriguing discussions. I also wish to sincerely thank my academic colleagues, collaborators and friends: Ophir Auslaender, Suk-Bum Chung, Chao-Xing Liu, Qin Liu, Lan Luan, Srinavas Raghu, Jing Wang, Zhong Wang, Binghai Yan, Hong Yao, Jiadong Zang, Hai-Jun Zhang, Xiao Zhang. Last but not least, I wish to sincerely thank my parents, whose love supports me throughout my life. v Dedicated to my parents vi Contents Abstract iv Acknowledgments v 1 Introduction 1 1.1 Background . .1 1.2 Thesis overview . .2 1.3 Contributions . .3 2 The image magnetic monopole 4 2.1 Introduction . .4 2.2 The topological magnetoelectric effect . .5 2.3 The image magnetic monopole and its detection . .9 2.3.1 The image magnetic monopole . .9 2.3.2 The experimental detection of image monopoles . 12 2.3.3 The dyon gas with fractional statistics . 14 3 The dynamical axion in topological magnetic insulators 17 3.1 Introduction . 17 3.2 Effective model for 3D topological insulators . 19 3.3 P; T breaking terms . 21 3.3.1 Time-reversal breaking masses . 21 3.3.2 Surface states with time reversal breaking masses . 23 3.3.3 Chiral edge states and quantum anomalous Hall effect . 28 vii 3.4 The topological magnetic insulator . 30 3.4.1 The mean field Hamiltonian . 30 3.4.2 The minimization of free energy . 32 3.5 The dynamical axion field . 34 3.6 The axionic polariton . 36 3.6.1 The frequency dependence of reflectivity . 41 3.6.2 A physical interpretation and analogy to phonon . 44 3.7 Measuring the axion by microcantilever. 46 4 Topological magnetic insulators with corundum structure 48 4.1 Introduction . 48 4.2 The tight binding Hamiltonian . 49 4.3 The effect of electron correlation . 62 5 Conclusions and outlook 67 A Appendix 69 A.1 The image monopole for a point charge outside a sphere . 69 A.2 Detection of image monopoles by MFM . 76 A.3 Derivation of the expression for θ .................... 80 A.4 RPA calculation of the axion dynamics . 84 Bibliography 91 viii List of Tables ix List of Figures 2.1 Illustration of the image charge and monopole of an electric charge . 11 2.2 Experimental setting to measure the image monopole . 13 2.3 The fractional statistics induced by image monopole effect . 14 2.4 Measuring the fractional statistics of the dyons . 15 3.1 Crystal Structure of Bi(Fe)2Se3 ..................... 19 3.2 The chiral edge states of a ferromagnetic topological insulator . 29 3.3 Axionic polariton and ATR experiment . 37 3.4 The spectrum of the magnetic polariton . 41 4.1 The corundum structure . 50 4.2 One honeycomb layer of the corundum structure . 51 4.3 The Brillouin zone of the corundum structure . 52 4.4 Band structure for the corundum structure . 62 4.5 Magnetic orders of the corundum structure . 63 4.6 The phase diagram of the corundum structure . 66 A.1 The image charge and monopole of an applied electric charge . 70 A.2 The image charge densities seen from outside the sphere . 73 A.3 The image charge densities seen from inside the sphere . 74 A.4 The ratio of fields induced by line and point charge . 75 A.5 Experimental setup for testing the image monopole . 80 x Chapter 1 Introduction 1.1 Background One of the most important goals in condensed matter physics is to discover and clas- sify different states of matter. Most states of matter are classified by the symmetries they break. Since the discovery of the quantum Hall effect, new types of order called topological orders, which cannot be described by local order parameters, come into sight in condensed matter physics. These topological states of matter can not be adi- abatically connected to topologically trivial states and are robust under disorder and interaction. Recently a new class of topological states called topological insulators has been theoretically predicted and experimentally discovered [1, 2, 3, 4, 5, 6, 7, 8, 9]. These topological quantum states of matter are insulating in the bulk, but have gap- less edge or surface states protected by the time-reversal symmetry. The first example of these states is the quantum spin Hall (QSH) system predicted and realized in HgTe quantum wells. They have counter-propagating edge modes with opposite spins, ro- bust under non-magnetic impurities and interactions. Then topological insulators in three dimensions are soon discovered in Bi1−xSbx alloy, Bi2Te3 and Bi2Se3. They have topologically protected surface states described by the relativistic Dirac equation for massless fermions. One of the most striking discoveries of 3D topological insulators is that they are 1 CHAPTER 1. INTRODUCTION 2 characterized by topological field theory, which gives rise to the topological mag- netoelectric effect, where an electric field induces a magnetic field along the same direction inside a topological insulator, with a constant of proportionality given by odd multiples of the fine structure constant. Moreover, the topological field theory that describe 3D topological insulators is analogous to that describing the electro- magnetism of a hypothetical particle called axion. Axions are weakly interacting particles of low mass, and were postulated more than 30 years ago in the framework of the Standard Model of particle physics. Their existence could explain the missing dark matter of the Universe. However, despite intensive searches, axions have yet to be observed. Now it is predicted that in condensed matter system like topological insulators, an static axion field taking the value of π may emerge. 1.2 Thesis overview In this thesis we will explore the novel physical effect that take place in 3D topological insulators. Very interestingly, many particles that were originally postulated in the context of elementary particle physics, but yet to be experimentally detected, are now predicted to emerge in 3D topological insulators.
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