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Fractionalization and Interaction in Topological Superconductors and Insulators Sharmistha Sahoo Cuttack, Odisha, India

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Fractionalization and Interaction in Topological Superconductors and Insulators Sharmistha Sahoo Cuttack, Odisha, India Fractionalization and Interaction in Topological Superconductors and Insulators Sharmistha Sahoo Cuttack, Odisha, India Integrated M.Sc. , National Institute of Science Education and Research, Bhubaneswar, Odisha, India, 2012 A Dissertation presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy Department of Physics University of Virginia May, 2018 i Acknowledgement My deepest gratitude goes to my advisor Jeffrey Teo for his guidance and support. My knowledge on the physics of topological phases comes directly from our discussion over the years. He was always patient with me and was very approachable when I was stuck in research with any problem or idea. I would like to thank Gia-Wei Chern who has been a great mentor to me over the years. Discussions with him brought many interesting topics into my attention and helped me to put my own research in a broader perspective. I am grateful to Israel Klich and Paul Fendley, for their lectures in advanced con- densed matter courses at University of Virginia(UVa) provided me with the requisite background to carry out research in later years. Conversation with both of them always inspired me. I am also thankful to Roger Melko and his team at University of Waterloo who hosted me for a semester which gave me the opportunity to learn about the entanglement entropies and other interesting topics. I appreciate Israel Klich, Seunghun Lee, Gia-Wei Chern and Thomas Mark for putting their time and effort to review my research/ theses and giving me their feedback which were really important to me. My sincere thanks goes to the physics department at UVa, who have been really supportive in my journey as a PhD student. I very much enjoyed discussing physics with other graduate students in the physics department, especially Alex Sirota, Syed Raza, Meng Hua, Zhao Zhang, Yifei Shi and Jing Luo. I thank all my friends in the department and in the university, because of whom I had a great time in Char- lottesville. I have been fortunate to have made friends like Abha Rajan and Nguyen Ton, who have always been there for me. At last, but most importantly, I could not have completed this thesis without my family's support. I am always thankful to them especially to Ajinkya Kamat who has been a constant source of encouragement. ii Declaration This thesis is adapted from the following publications • Sharmistha Sahoo, Zhao Zhang, Jeffrey Teo, Coupled wire model of Symmetric Majorana surfaces of topological superconductors, Phys Rev B, Vol. 94, 165142 (2016). • Sharmistha Sahoo, Alexander Sirota, Gil Young Cho, Jeffrey C. Y. Teo, Sur- faces and Slabs of Fractional Topological Insulator Heterostructures, Phys Rev B, Vol. 96, 161108 (2017). Thesis has been reviewed and approved by the following committee • Jeffrey Teo, Assistant Professor, Physics department • Israel Klich, Associate Professor, Physics department • Gia-Wei Chern, Assistant Professor, Physics department • Thomas Mark, Associate Professor, Mathematics department iii Abstract We present a study of novel topological order in three dimensional (3+1D) topological superconductors and fractional topological insulators. Such topological order occurs when the surface is gapped due to the presence of many-body interactions that respect the time-reversal symmetry. 3+1D time reversal symmetric topological superconductors are characterized by gapless (massless) Majorana fermions on its surface. They are robust against any time reversal symmetric single-body perturbation weaker than the bulk energy gap. We mimic this gapless surface by coupled wire models in two spatial dimensions. We show modified models with additional time-reversal symmetric many-body interaction, that gives energy gaps to all low energy degrees of freedom. We used the embedding trick using Wess-Zumino-Witten conformal field theory to find such interacting model Hamiltonian. We show the gapped models generically carry non-trivial topological order and support anyons. Using these anyons and their condensation process, we show the topological order has a 32-fold periodicity. Fractional topological insulators (FTI) are electronic topological phases in 3+1D enriched by the time reversal and charge U(1) conservation symmetries. We focus on the simplest series of fermionic FTI, whose bulk quasiparticles consist of deconfined partons. Theses partons carry fractional electric charges in integral units of e∗ = e=(2n+1) and couple to a discrete Z2n+1 gauge theory. We propose massive symmetry preserving or breaking FTI surface states. Combining the long-ranged entangled bulk with these topological surface states, we deduce the novel topological order of quasi- (2 + 1) dimensional FTI slabs as well as their corresponding edge conformal field theories. iv Contents 1 Introduction2 1.1 Background................................3 1.1.1 Classification of topological insulators and superconductors..3 1.1.2 Fractional topological insulators................8 1.1.3 Topological order.........................9 1.1.4 Conformal field theory and bosonization............ 15 1.2 Outline of the thesis........................... 21 2 Three dimensional (3D) topological superconductor (TSC) 24 2.1 Coupled wire construction of TSC surface............... 26 2.1.1 The so(N)1 current algebra................... 31 2.1.2 Bosonizing even Majorana cones................. 34 3 Interaction and Surface Topological Order 38 3.1 Many-body interaction in coupled wire model............. 38 3.1.1 Gapping even Majorana cones.................. 42 3.1.2 Gapping odd Majorana cones.................. 53 3.1.3 Gapping by fractional quantum Hall stripes.......... 65 3.2 Surface topological order......................... 67 3.2.1 Summary of anyon contents................... 71 3.2.2 The 32-fold tensor product structure.............. 74 v 3.3 Other possibilities............................. 85 3.3.1 Consequence of the emergent E8 when N = 16......... 86 3.3.2 Alternative conformal embeddings................ 89 4 3D fractional topological insulators (FTI) 91 4.1 Ferromagetic heterostructure....................... 93 4.2 Superconducting heterostructure..................... 95 4.3 Generalized -Pfaffian∗ surface..................... 97 T 4.4 Gluing -Pfaffian∗ surfaces....................... 103 T 5 Conclusion 107 A The so(N) Lie algebra and its representations 111 B Bosonizing the so(2r)1 current algebra 115 C Bosonizing the so(2r + 1)1 current algebra 122 D Z6 parafermion model 125 E The S-matrices of the GN state 128 F Abelian Chern-Simons theory of dyons 131 vi List of Figures 1.1 A poly-acetelyn molecule with alternating bond-strength.......4 1.2 In top electron band dispersion for the SSH model with different values of hopping strength v and w. In bottom plots in dx-dy plane indicating whether the Berry's phase is 0 or π. For v < w the chain is gapped and the Berry's phase is π as the circle encloses the origin making it a \topologically non-trivial" insulator...................5 1.3 (1) Ribbon diagram showing topological spin of an anyon from the 2π twist. (2) Monodromy phase or the double exchange phase from ribbon twist phase................................. 11 1.4 Band dispersion for free electron in 1D illustrating left and right Fermi points and the linear dispersion near these points............ 16 2.1 (Left ) A coupled wire model description of the 2D quantum Hall insu- lator where the magnetic fields point into the plane. (Right) the energy dispersions for the wires before and after the coupling are shown re- spectively on top and bottom....................... 25 2.2 (Left) Coupled wire model (2.4) of N gapless surface Majorana cones. (Right) Fractionalization (3.4) and couple wires construction (3.8) of gapped anomalous and topological surface state............. 27 2.3 The energy spectrum of the coupled Majorana wire model (2.4)... 28 vii 2.4 Coupled Majorana wire model on the surface of (a) a stack of alternat- ing p ip superconductors, and (b) a class DIII topological super- x ± y conductor (TSC) with alternating TR breaking surface domains. (c) A dislocation................................ 30 3.1 Energy spectrum of the N = 2 coupled Majorana wire model with inter-flavor mixing............................. 40 3.2 Interwire gapping terms (3.8) (green rectangular boxes) between chiral R; L; fractional ±; ± sectors (resp. ; ) in opposite direction..... 41 GN GN ⊗ 3.3 Gapping N surface Majorana cones by inserting (2 + 1)D GN stripe state and removing edge modes by current-current backscattering in- teraction.................................. 65 3.4 Chiral interface (highlighted line) between a time reversal breaking gapped region and a TR symmetric topologically ordered gapped region. 68 3.5 The GN topological order of a quasi-2D slab with time reversal sym- metric gapped top surface and time reversal breaking gapped bottom surface................................... 68 4.1 Summary of the quasi-particles and gauge flux content in FTI slabs. A pair of Pf∗ FTI slabs are merged into a fractional Chern FTI slab F by gluing the two TR symmetric -Pf∗ surfaces. Directed bold lines T on the front surface are chiral edge modes of the Pf∗ and FTI slabs. 95 F 4.2 Gapped surfaces on TI. Splitting of central charge showing TO with c = 1=2 boundary CFT on the TR symmetric gapped surface..... 99 1 List of Tables 3.1 The \angular momenta" s, conformal dimensions hs and quantum di- mensions ds of primary fields Vs of so(3)3 = su(2)6........... 57 2πihx 3.2 The exchange phase θx = e and quantum dimensions of anyons x in a (2 + 1)D SO(r)1 topological phase.................. 71 2πihx 3.3 The exchange phase θx = e and quantum dimensions of anyons x in a (2 + 1)D SO(3)3 b SO(r)1 topological phase............ 72 3.4 Identification of the seven anyon types in table 3.3 as tensor products. 76 4.1 Charge and spin for 24 possible quasi-particles (anyons) in Moore-Read Pfaffian state................................ 98 2 Chapter 1 Introduction Condensed matter physics explores microscopic and macroscopic properties of matter.
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