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Strongly Correlated Surface States STRONGLY CORRELATED SURFACE STATES By VICTOR A ALEKSANDROV A dissertation submitted to the Graduate School|New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Physics and Astronomy written under the direction of Piers Coleman and approved by New Brunswick, New Jersey October, 2014 ABSTRACT OF THE DISSERTATION Strongly correlated surface states By VICTOR A ALEKSANDROV Dissertation Director: Piers Coleman Everything has an edge. However trivial, this phrase has dominated theoretical condensed matter in the past half a decade. Prior to that, questions involving the edge considered to be more of an engineering problem rather than a one of fundamental science: it seemed self-evident that every edge is different. However, recent advances proved that many surface properties enjoy a certain universality, and moreover, are 'topologically' protected. In this thesis I discuss a selected range of problems that bring together topological properties of surface states and strong interactions. Strong interactions alone can lead to a wide spec- trum of emergent phenomena: from high temperature superconductivity to unconventional magnetic ordering; interactions can change the properties of particles, from heavy electrons to fractional charges. It is a unique challenge to bring these two topics together. The thesis begins by describing a family of methods and models with interactions so high that electrons effectively disappear as particles and new bound states arise. By invoking the AdS/CFT correspondence we can mimic the physical systems of interest as living on the surface of a higher dimensional universe with a black hole. In a specific example we investigate the properties of the surface states and find helical spin structure of emerged particles. The thesis proceeds from helical particles on the surface of black hole to a surface of ii samarium hexaboride: an f-electron material with localized magnetic moments at every site. Interactions between electrons in the bulk lead to insulating behavior, but the surfaces found to be conducting. This observation motivated an extensive research: weather the origin of conduction is of a topological nature. Among our main results, we confirm theo- retically the topological properties of SmB6; introduce a new framework to address similar questions for this type of insulators, called Kondo insulators. Most notably we introduce the idea of Kondo band banding (KBB): a modification of edges and their properties due to interactions. We study (chapter 5) a simplified 1D Kondo model, showing that the topology of its ground state is unstable to KBB. Chapter 6 expands the study to 3D: we argue that not only KBB preserves the topology but it could also explain the experimentally observed anomalously high Fermi velocity at the surface as the case of large KBB effect. iii Acknowledgments First and foremost, I am very happy to have had Piers Coleman as my adviser. I am thankful for his support and collaboration. I only hope I have acquired some of his vision and enthusiasm (not to mention his British accent). During my thesis I have had two collaborators: both Maxim Dzero and Onur Erten were very helpful, and as we dealt with Piers' absences for his numerous invited talks (that's a downside of being famous) we established a very healthy working atmosphere. I would like to thank Matt Strassler, who taught me a great deal of advanced quantum field theory and was guiding me through the hurdles of becoming a physicist, assisting with choices I made for my career, he recommended and secured my participation in TASI summer school (Boulder, Colorado), which turns out to be indeed one of the biggest growing evens in my life. In recent months of my PhD I have a pleasure talking to Pouyan Ghaemi, who also agreed to be an external member on my committee. I would like to thank Rutgers Physics and Astronomy Department, and all members of my PhD thesis committee for their patience, encouragement and support in allowing me to switch fields during graduate program. This extends also to a great atmosphere in the physics department and I would like to thank our Graduate directors: Ron Ransome, Ted Williams and Ron Gilman who worked so hard to maintain atmosphere of excitement feeling for all of us, graduate students. I am indebted to many students and postdocs, whose inspirational and brain-nourishing discussions defined my life here: Dima Krotov, Simon Knapen, Michael Manhart, Sanjay Arora, Bryan Leung, Sergey Aryukhin, Michael Solway, Aline Ramires, Onur Erten, Turan Birol, Lucian Pascut, Anthony Barker, Wenhu Xu, Eliav Endrey, Aleksej Mialitsin, Rishi Patel and many more. There was a time when I was organizing/(participating in) some journal clubs, to my iv great benefit. I would like to acknowledge: Matt Foster, Francesco Benini, Tzen Ong, Matthias Kaminski and Yue Zhao (AdS/CMT club), Chuck Yee, Aline Ramires, Sebastian Reyes, Sergey Aryukhin, Adina Luican, Sinisa Coh, Tahir Yusufaly, Maryam Taherinejad and many more who participated in CMT journal club. I have greatly enjoyed learning from the Rutgers faculty, especially Piers Coleman, Matt Strassler, Sasha Zamolodchikov, David Vanderbilt, Natan Andrei, Tom Banks, Kristjan Haule and Emil Yuzbashyan. I would like to acknowledge the financial support from Department of Energy Grant No. DE-FG02-99ER45790 and National Science Foundation Grant No. DMR 0907179. In addition, Rutgers Fellowship and TA for the beginning of my PhD and Department Gradate assistantship for a partial support of my study. v Dedication To my family, obviously. Most of all to my grandfather, who was the only physicist in my household but who could never speak a word about physics as he was a nuclear physicist... in the USSR. To Valentina Fedorovna, my physics teacher. vi Table of Contents Abstract :::::::::::::::::::::::::::::::::::::::::: ii Acknowledgments :::::::::::::::::::::::::::::::::::: iv Dedication ::::::::::::::::::::::::::::::::::::::::: vi 1. Motivation: Challenge of many body physics ::::::::::::::::: 1 1.1. Challenge of many body physics . .1 1.2. Advances of non-interacting physics . .2 1.3. Beyond the non-interacting limit: local physics . .4 1.4. Outline of the thesis . .5 2. Introduction and methods ::::::::::::::::::::::::::::: 6 2.1. Topological Insulators . .6 2.2. Kondo Insulators . 12 2.3. Quantum Critical Kondo metal . 22 3. Spin structure in a holographic metal ::::::::::::::::::::: 25 3.1. Background Formalism . 30 3.2. Reflection Approach . 35 3.3. Boundary terms . 37 3.4. Spin structure . 38 3.5. Discussion . 43 3.6. Evolution of the dispersion curves . 45 4. Cubic Kondo Topological Insulator ::::::::::::::::::::::: 50 vii 4.1. Review the model with tetragonal crystal symmetry . 50 4.2. Motivation for my study . 51 4.3. The model . 53 4.4. Some additional details on the mean field . 59 5. End states of 1D Kondo topological Insulator :::::::::::::::: 68 5.1. "Too-fast" surface states . 68 5.2. One dimensional model . 70 5.3. Goals . 71 5.4. The model . 73 5.5. Mean field solution . 80 5.6. Results and Discussion . 85 6. Chapter 6: S-P Model ::::::::::::::::::::::::::::::: 90 6.1. Current explanations of light surface states . 91 6.2. 2 band model with decoupled f-electron at the edges . 93 viii 1 Chapter 1 Motivation: Challenge of many body physics In this chapter I will give a brief overview of my research field. The aim is for an incoming graduate student to create a general impression. A sophisticated reader can skip this section and go straight to the next chapter. 1.1 Challenge of many body physics The purpose of condensed matter theory is to provide a predictive model for the behavior of realistic materials. On the fundamental level the problem is well understood to astonishing level of precision: all the interactions are of charged objects with spin and ar described by quantum electrodynamics. The challenge is however in the number of interacting parts of the system. Solving such a many-body problem posed a formidable challenge for the most of 20-th century and even with ever-growing computing powers will likely to remain so for a long time. This statement can actually be made quantitative: The Hilbert space of a quantum system grows exponentially, leading to the 'Van Vleck' catastrophe. One should not seek an exact solution of systems larger than several hundred electrons can not be solved exactly. For example, 150 electrons would require the total number of variables to define the wave- function to be more than all the particles in the universe. "Such a wavefunction therefore is not a sensible scientific concept" (see Walter Kohns analysis of this problem, in his Nobel lecture Reviews of Modern Physics [1]). It should be noted that there are many theoretical models (especially in one spatial dimension) that can be solved exactly for infinitely large systems. There are however many successes. In particular the theory of metals delivers many 2 predictions. With the exception of magnetic properties we can describe even "dirty" ma- terials. In a successful theory a small number of degrees of freedom (d.o.f.) should emerge from a vast number of individual characteristics. "Emergence" became a 'buzz' word for the field of theoretical condensed matter due to a celebrated Anderson's paper [2]1. I would like to touch on some of this advances in the next section, it will also be our first example of the Mean Field (MF) approximation methods which will be used extensively in the main part of this thesis. 1.2 Advances of non-interacting physics What are the methods to approach realistic systems? What can we describe and what we can not? We first limit our discussion to the periodic arrangements of nuclei (forming a crystal) for which we would like to add electrons. And the first question is what are the properties of the ground state (state that could be measured at low temperature)? By non-interacting we do not actually mean that particle must not feel each other pres- ence; rather, the main effect of interaction can be described by an effective Hamiltonian, which is bilinear in elementary fields (no scattering).
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