Explorations in Dirac Fermions and Spin Liquids A dissertation presented by Rudro Rana Biswas to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts May 2011 c 2011 - Rudro Rana Biswas All rights reserved. Thesis advisor Author Subir Sachdev Rudro Rana Biswas Explorations in Dirac Fermions and Spin Liquids Abstract A significant portion of this dissertation is devoted to the study of the effects of im- purities in substances whose low energy modes can be described by fermions obeying the gapless Dirac equation in 2+1 dimensions. First, we examine the case of a spin vacancy in the staggered flux spin liquid whose excitations are Dirac fermions coupled to a U(1) gauge field. This vacancy leads to an anomalous Curie susceptibility and does not induce any local orders. Next, a Coulomb charge impurity placed on clean graphene is considered. We find that the Dirac quasiparticles in graphene do not screen the impurity charge, to all orders in perturbation theory. However, electronic correlations are found to induce a cloud of charge having the same sign as the impurity charge. We also analyze the case of a local impurity in graphene in the presence of a magnetic field and derive the spatial fourier transform of tunneling spectroscopy data obtained on an almost-clean graphene sheet in a magnetic field. We then move on to Strong Topological Insulators (STI) whose surfaces are inhabited by an odd number of Dirac fermion species. Local impurities on the STI surface are found to induce resonance(s) in the local density of states (LDOS) in their vicinity. iii Abstract iv In the case of magnetic impurities that may be approximated by classical spins, we show the existence of RKKY interactions that favor ferromagnetic aligning of randomly placed impurity spins perpendicular to the STI surface when the doping is insignificant. Finally we also consider the effects of a step edge on the STI surface and calculate the spatial decay of LDOS perturbations away from it. The last chapter of this dissertation contains the theory for a new kind of spin liquid in a system of spin halves arranged in a triangular lattice, whose excitations are spin one Majorana fermions. This gapless spin liquid is found to possess certain properties that are exhibited by the recently discovered spin liquid state in the organic charge transfer salt EtMe3Sb(Pd(dmit)2)2. Contents Title Page . .i Abstract . iii Table of Contents . .v Citations to Previously Published Work . viii Acknowledgments . ix Dedication . xiii 1 Introduction 1 1.1 Dirac Fermions . .1 1.1.1 Graphene . .3 1.1.2 Strong Topological Insulators . .8 1.2 Spin Liquids . 16 1.2.1 The staggered flux spin liquid . 17 1.2.2 The Majorana Spin Liquid on the triangular lattice . 20 2 Theory of quantum impurities in the staggered flux spin liquid 24 2.1 Synopsis . 24 2.2 Introduction . 25 2.3 Theory and Results . 32 2.3.1 Bulk theory . 33 2.3.2 Relationship to earlier work . 34 2.3.3 Impurity exponents . 37 2.3.4 Linear response to a uniform applied field . 39 2.4 Conclusions . 45 3 Coulomb impurity in graphene 48 3.1 Synopsis . 48 3.2 Introduction . 49 3.3 Non-interacting electrons . 52 3.4 Interacting electrons . 56 3.5 Conclusions . 62 v Contents vi 4 Quasiparticle Interference and Landau Level Spectroscopy in Graphene in the presence of a Strong Magnetic Field 63 4.1 Synopsis . 63 4.2 Introduction . 64 4.3 Theory . 65 4.4 Results . 68 4.5 Conclusion . 72 5 Impurity-induced states on the surface of 3D topological insulators 75 5.1 Synopsis . 75 5.2 Introduction . 76 5.3 Theory . 78 5.4 Results . 81 5.5 Conclusion . 85 6 Scattering from Surface Step Edges in Strong Topological Insulators 87 6.1 Synopsis . 87 6.2 Introduction . 88 6.3 Theory . 91 6.4 Results . 95 6.4.1 The perfect reflector . 95 6.4.2 Discussion of recent experiments . 96 6.4.3 Generic step edges . 98 6.5 Conclusion . 99 7 SU(2)-invariant spin liquids on the triangular lattice with spinful Majorana excitations 100 7.1 Synopsis . 100 7.2 Introduction . 101 7.2.1 Low energy theory . 103 7.3 The mean field Majorana Hamiltonian on a triangular lattice . 107 7.3.1 Majorana mean field theory from a spin Hamiltonian: an example107 7.3.2 The general low energy effective theory on the lattice . 110 7.3.3 The low energy effective theory in the presence of a perpendic- ular magnetic field . 112 7.3.4 The spectrum in the presence of the Zeeman coupling . 114 7.4 Properties of the clean Majorana spin liquid . 114 7.4.1 The low energy density of states (DOS) . 115 7.4.2 Specific Heat . 116 7.4.3 Magnetic susceptibility . 116 7.4.4 The Wilson ratio { comparison with a 2DEG . 117 7.4.5 Static Structure Factor . 118 Contents vii 7.4.6 Effect of a perpendicular magnetic field . 119 7.5 Effects of weak disorder . 121 7.5.1 The bond impurity potential . 121 7.5.2 The disorder-averaged self energy in the Born approximation . 121 7.5.3 The disorder-averaged self energy in the self-consistent Born approximation (SCBA) . 123 7.5.4 The disorder-averaged single particle density of states . 124 7.5.5 The specific heat in the presence of impurities . 125 7.5.6 The spin susceptibility in the presence of impurities . 126 7.5.7 The Wilson ratio in the presence of impurities { comparison with a 2DEG . 127 7.5.8 The thermal conductivity . 128 7.6 Conclusions . 130 Bibliography 132 A The relation between the Majorana and spin 1/2 Hilbert spaces 148 Citations to Previously Published Work Chapter 2 appears in its entirety in the paper: \Theory of quantum impurities in spin liquids", A. Kolezhuk, S. Sachdev, R. R. Biswas and P. Chen, Phys. Rev. B, 74, 165114 (2006) Chapter 3 appears in its entirety in the paper: \Coulomb impurity in graphene", R. R. Biswas, S. Sachdev and D. T. Son, Phys. Rev. B, 76, 205122 (2007) Chapter 4 appears in its entirety in the paper: \Quasiparticle interference and Landau level spectroscopy in graphene in the presence of a strong magnetic field”, R. R. Biswas and A. Balatsky, Phys. Rev. B, 80, 081412(R) (2009) Chapter 5 appears in its entirety in the paper: \Impurity-induced states on the surface of three-dimensional topological insulators", R. R. Biswas and A. Balatsky, Phys. Rev. B, 81, 233405 (2010) Chapter 6 appears in its entirety in the paper: \Scattering from surface step edges in strong topological insulators", R. R. Biswas and A. Balatsky, Phys. Rev. B, 83, 075439 (2011) Chapter 7 as well as Appendix A appear in their entirety in the 2011 preprint: \SU(2)-invariant spin liquids on the triangular lattice with spinful Majo- rana excitations", R. R. Biswas, L. Fu, C. R. Laumann and S. Sachdev, arXiv:1102.3690. Electronic preprints (shown in typewriter font) are available on the Internet at the following URL: http://arXiv.org viii Acknowledgments It seems like yesterday that I walked past a snow filled Harvard Yard and into Jefferson Laboratory for the first time. As I look back at this journey from my vantage point that is now, I realize that a large part of what I was able to accomplish during this period is due to the interactions that I was able to have with some truly remarkable people. I am happy to use this opportunity to thank some of these incredible people. I thank my advisor, Prof. Subir Sachdev, for his guidance through my graduate years, for letting me work on new materials and problems that excited me, even if that meant that he would have to invest in helping me chart a course that is different from the usual. Time and again, I have had the privilege of learning from watching Subir make symmetry and scaling arguments whose extraordinary elegance was matched only by how inevitably predictive they were, completely unlike the hand-waving, uncertain and jaded arguments elsewhere. I also thank him for relentlessly inspiring me with his work ethic { I continue to be humbled and inspired by how much he accomplishes in any given day. Much of my mindset as a physicist has been influenced by my many discussions with Bert Halperin, an incredible physicist and my hero. I thank Bert for co-advising me at every step and for every physics discussion we have had over these years. Everything that I have done right as a physicist during my years at Harvard, I owe in large part to him. I thank him especially for reinforcing my belief that good Physics is lucid and crisp, that the non-trivial need not be obscure. I also thank him for nurturing me, for humoring me, for challenging me, for encouraging me and above all, for always being there for me when I knocked on his door. ix Acknowledgments x I thank the many distinguished experimentalists at Harvard who educated me and helped me have my feet more firmly planted in ground reality, being as I was, a theorist. Specifically, I wish to thank Profs. Amir Yacobi, Jenny Hoffman, Charlie Markus, Misha Lukin, Markus Greiner as well as Dr. Joe Peidle. I would also like to thank Yang Qi, Liang Fu, Lars Fritz, Cenke Xu, Chris Laumann, Jimmy Williams as well as all my co-graduate students for the many enriching conversations.