Non-Abelian Anyons and Topological Quantum Computation
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Lecture Notes: BCS Theory of Superconductivity
Lecture Notes: BCS theory of superconductivity Prof. Rafael M. Fernandes Here we will discuss a new ground state of the interacting electron gas: the superconducting state. In this macroscopic quantum state, the electrons form coherent bound states called Cooper pairs, which dramatically change the macroscopic properties of the system, giving rise to perfect conductivity and perfect diamagnetism. We will mostly focus on conventional superconductors, where the Cooper pairs originate from a small attractive electron-electron interaction mediated by phonons. However, in the so- called unconventional superconductors - a topic of intense research in current solid state physics - the pairing can originate even from purely repulsive interactions. 1 Phenomenology Superconductivity was discovered by Kamerlingh-Onnes in 1911, when he was studying the transport properties of Hg (mercury) at low temperatures. He found that below the liquifying temperature of helium, at around 4:2 K, the resistivity of Hg would suddenly drop to zero. Although at the time there was not a well established model for the low-temperature behavior of transport in metals, the result was quite surprising, as the expectations were that the resistivity would either go to zero or diverge at T = 0, but not vanish at a finite temperature. In a metal the resistivity at low temperatures has a constant contribution from impurity scattering, a T 2 contribution from electron-electron scattering, and a T 5 contribution from phonon scattering. Thus, the vanishing of the resistivity at low temperatures is a clear indication of a new ground state. Another key property of the superconductor was discovered in 1933 by Meissner. -
Anyon Theory in Gapped Many-Body Systems from Entanglement
Anyon theory in gapped many-body systems from entanglement Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Bowen Shi, B.S. Graduate Program in Department of Physics The Ohio State University 2020 Dissertation Committee: Professor Yuan-Ming Lu, Advisor Professor Daniel Gauthier Professor Stuart Raby Professor Mohit Randeria Professor David Penneys, Graduate Faculty Representative c Copyright by Bowen Shi 2020 Abstract In this thesis, we present a theoretical framework that can derive a general anyon theory for 2D gapped phases from an assumption on the entanglement entropy. We formulate 2D quantum states by assuming two entropic conditions on local regions, (a version of entanglement area law that we advocate). We introduce the information convex set, a set of locally indistinguishable density matrices naturally defined in our framework. We derive an isomorphism theorem and structure theorems of the information convex sets by studying the internal self-consistency. This line of derivation makes extensive usage of information-theoretic tools, e.g., strong subadditivity and the properties of quantum many-body states with conditional independence. The following properties of the anyon theory are rigorously derived from this framework. We define the superselection sectors (i.e., anyon types) and their fusion rules according to the structure of information convex sets. Antiparticles are shown to be well-defined and unique. The fusion rules are shown to satisfy a set of consistency conditions. The quantum dimension of each anyon type is defined, and we derive the well-known formula of topological entanglement entropy. -
First-Principles Calculations and Model Hamiltonian Approaches to Electronic and Optical Properties of Defects, Interfaces and Nanostructures
First-principles calculations and model Hamiltonian approaches to electronic and optical properties of defects, interfaces and nanostructures by Sangkook Choi A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Grauduate Division of the University of California, Berkeley Committee in charge: Professor Steven G. Louie, Chair Professor John Clarke Professor Mark Asta Fall 2013 First-principles calculations and model Hamiltonian approaches to electronic and optical properties of defects, interfaces and nanostructures Copyright 2013 by Sangkook Choi Abstract First principles calculations and model Hamiltonian approaches to electronic and optical properties of defects, interfaces and nanostructures By Sangkook Choi Doctor of Philosophy in Physics University of California, Berkeley Professor Steven G. Louie, Chair The dynamics of electrons governed by the Coulomb interaction determines a large portion of the observed phenomena of condensed matter. Thus, the understanding of electronic structure has played a key role in predicting the electronic and optical properties of materials. In this dissertation, I present some important applications of electronic structure theories for the theoretical calculation of these properties. In the first chapter, I review the basics necessary for two complementary electronic structure theories: model Hamiltonian approaches and first principles calculation. In the subsequent chapters, I further discuss the applications of these approaches to nanostructures (chapter II), interfaces (chapter III), and defects (chapter IV). The abstract of each section is as follows. ● Section II-1 The sensitive structural dependence of the optical properties of single-walled carbon nanotubes, which are dominated by excitons and tunable by changing diameter and chirality, makes them excellent candidates for optical devices. -
Φ0-Magnetic Force Microscopy for Imaging and Control of Vortex
Imaging and controlling vortex dynamics in mesoscopic superconductor-normal-metal-superconductor arrays Tyler R. Naibert1, Hryhoriy Polshyn1,2,*, Rita Garrido-Menacho1, Malcolm Durkin1, Brian Wolin1, Victor Chua1, Ian Mondragon-Shem1, Taylor Hughes1, Nadya Mason1,*, and Raffi Budakian1,3 1Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green St., Urbana, IL 61801-3080, USA 2Department of Physics, University of California, Santa Barbara, CA 93106, USA 3Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada, N2L3G1 Department of Physics, University of Waterloo, Waterloo, ON, Canada, N2L3G1 Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada, N2L2Y5 Canadian Institute for Advanced Research, Toronto, ON, Canada, M5G1Z8 *Corresponding authors. Email: [email protected]; [email protected] Harnessing the properties of vortices in superconductors is crucial for fundamental science and technological applications; thus, it has been an ongoing goal to locally probe and control vortices. Here, we use a scanning probe technique that enables studies of vortex dynamics in superconducting systems by leveraging the resonant behavior of a raster-scanned, magnetic-tipped cantilever. This experimental setup allows us to image and control vortices, as well as extract key energy scales of the vortex interactions. Applying this technique to lattices of superconductor island arrays on a metal, we obtain a variety of striking spatial patterns that encode information about the energy landscape for vortices in the system. We interpret these patterns in terms of local vortex dynamics and extract the relative strengths of the characteristic energy scales in the system, such as the vortex-magnetic field and vortex-vortex interaction strengths, as well as the vortex chemical potential. -
8.7 Hubbard Model
OUP CORRECTED PROOF – FINAL, 12/6/2019, SPi 356 Models of Strongly Interacting Quantum Matter which is the lower edge of a continuum of excitations whose upper edge is bounded by ω(q) πJ cos (q/2). (8.544) = The continuum of excitations develops because spinons are always created in pairs, and therefore the momentum of the two spinons can be distributed in a continuum of different ways. Neutron scattering experiments on quasi-one-dimensional materials like KCuF3 have corroborated the picture outlined here (see, e.g. Tennant et al.,1995). In dimensions higher than one, separating a !ipped spin into a pair of kinks, or a magnon into a pair of spinons, costs energy, which thus con"nes spinons in dimensions d 2. ≥The next section we constructs the Hubbard model from "rst principles and then show how the Heisenberg model can be obtained from the Hubbard model for half- "lling and in the limit of strong on-site repulsion. 8.7 Hubbard model The Hubbard model presents one of the simplest ways to obtain an understanding of the mechanisms through which interactions between electrons in a solid can give rise to insulating versus conducting, magnetic, and even novel superconducting behaviour. The preceding sections of this chapter more or less neglected these interaction or correlation effects between the electrons in a solid, or treated them summarily in a mean-"eld or quasiparticle approach (cf. sections 8.2 to 8.5). While the Hubbard model was "rst discussed in quantum chemistry in the early 1950s (Pariser and Parr, 1953; Pople, 1953), it was introduced in its modern form and used to investigate condensed matter problems in the 1960s independently by Gutzwiller (1963), Hubbard (1963), and Kanamori (1963). -
Scanning Hall Probe Microscopy of Vortex Matter in Single-And Two-Gap Superconductors
ARENBERG DOCTORAL SCHOOL Faculty of Science Scanning Hall probe microscopy of vortex matter in single-and two-gap superconductors - Bart Raes Promotors: Dissertation presented in partial Prof.Dr. Victor V. Moshchalkov fulfillment of the requirements for the Prof.Dr. Jacques Tempère PhD degree July 2013 Scanning Hall probe microscopy of vortex matter in single-and two-gap superconductors Bart RAES Dissertation presented in partial fulfillment of the requirements for the PhD degree Members of the Examination committee: Prof. Dr. Victor V. Moshchalkov KU Leuven (Promotor) Prof. Dr. Jacques Tempère Universiteit Antwerpen (Co-promotor) Prof. Dr. J. Van de Vondel KU Leuven (Secretary) Prof. Dr. L. Chibotaru KU Leuven (President) Prof. Dr. J. Vanacken KU Leuven Dr. J. Gutierrez Royo KU Leuven Prof. Dr. A.V. Silhanek Université de Liège Prof. Dr. S. Bending University of Bath Prof. Dr. G. Borghs KU Leuven, IMEC July 2013 © 2013 KU Leuven, Groep Wetenschap & Technologie, Arenberg Doctoraatsschool, W. de Croylaan 6, 3001 Leuven, België Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotocopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaande schriftelijke toestemming van de uitgever. All rights reserved. No part of the publication may be reproduced in any form by print, photoprint, microfilm or any other means without written permission from the publisher. ISBN 978-90-8649-638-9 D/2013/10.705/53 Dankwoord-Acknowledgements Doctoreren is niet alleen het resultaat van bijna vier jaar wetenschappelijk onderzoek, het is een lange weg met veel hindernissen maar zeker ook enkele hoogtepunten. -
Arxiv:2007.11161V3 [Cond-Mat.Str-El] 13 Mar 2021
Topological phase transition and single/multi anyon dynamics of /2 spin liquid Zheng Yan,1 Yan-Cheng Wang,2 Nvsen Ma,3 Yang Qi,4, 5, 6, ∗ and Zi Yang Meng1, y 1Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong 2School of Materials Science and Physics, China University of Mining and Technology, Xuzhou 221116, China 3School of Physics, Key Laboratory of Micro-Nano Measurement-Manipulation and Physics, Beihang University, Beijing 100191, China 4State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China 5Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China 6Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China Among the quantum many-body models that host anyon excitation and topological orders, quantum dimer models (QDM) provide a unique playground for studying the relation between single-anyon and multi-anyon continuum spectra. However, as the prototypical correlated system with local constraints, the generic solution of QDM at different lattice geometry and parameter regimes is still missing due to the lack of controlled methodologies. Here we obtain, via the newly developed sweeping cluster quantum Monte Carlo algorithm, the excitation spectra in different phases of the triangular lattice QDM. Our resultsp revealp the single vison excitations inside the /2 quantum spin liquid (QSL) and its condensation towards the 12 × 12 valence bond solid (VBS), and demonstrate the translational symmetry fractionalization and emergent O(4) symmetry at the QSL-VBS transition. We find the single vison excitations, whose convolution qualitatively reproduces the dimer spectra, are not free but subject to interaction effects throughout the transition. -
Introduction to Abelian and Non-Abelian Anyons
Introduction to abelian and non-abelian anyons Sumathi Rao Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019, India. In this set of lectures, we will start with a brief pedagogical introduction to abelian anyons and their properties. This will essentially cover the background material with an introduction to basic concepts in anyon physics, fractional statistics, braid groups and abelian anyons. The next topic that we will study is a specific exactly solvable model, called the toric code model, whose excitations have (mutual) anyon statistics. Then we will go on to discuss non-abelian anyons, where we will use the one dimensional Kitaev model as a prototypical example to produce Majorana modes at the edge. We will then explicitly derive the non-abelian unitary matrices under exchange of these Majorana modes. PACS numbers: I. INTRODUCTION The first question that one needs to answer is why we are interested in anyons1. Well, they are new kinds of excitations which go beyond the usual fermionic or bosonic modes of excitations, so in that sense they are like new toys to play with! But it is not just that they are theoretical constructs - in fact, quasi-particle excitations have been seen in the fractional quantum Hall (FQH) systems, which seem to obey these new kind of statistics2. Also, in the last decade or so, it has been realised that if particles obeying non-abelian statistics could be created, they would play an extremely important role in quantum computation3. So in the current scenario, it is clear that understanding the basic notion of exchange statistics is extremely important. -
Fermion Condensation and Gapped Domain Walls in Topological Orders
Fermion Condensation and Gapped Domain Walls in Topological Orders Yidun Wan1,2, ∗ and Chenjie Wang2, † 1Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China 2Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada (Dated: September 12, 2018) We propose the concept of fermion condensation in bosonic topological orders in two spatial dimensions. Fermion condensation can be realized as gapped domain walls between bosonic and fermionic topological orders, which are thought of as a real-space phase transitions from bosonic to fermionic topological orders. This generalizes the previous idea of understanding boson conden- sation as gapped domain walls between bosonic topological orders. We show that generic fermion condensation obeys a Hierarchy Principle by which it can be decomposed into a boson condensation followed by a minimal fermion condensation, which involves a single self-fermion that is its own anti-particle and has unit quantum dimension. We then develop the rules of minimal fermion con- densation, which together with the known rules of boson condensation, provides a full set of rules of fermion condensation. Our studies point to an exact mapping between the Hilbert spaces of a bosonic topological order and a fermionic topological order that share a gapped domain wall. PACS numbers: 11.15.-q, 71.10.-w, 05.30.Pr, 71.10.Hf, 02.10.Kn, 02.20.Uw I. INTRODUCTION to condensing self-bosons in a bTO. A particularly inter- esting question is: Is it possible to condense self-fermions, A gapped quantum matter phase with intrinsic topo- which have nontrivial braiding statistics with some other logical order has topologically protected ground state anyons in the system? At a first glance, fermion conden- degeneracy and anyon excitations1,2 on which quantum sation might be counterintuitive; however, in this work, computation may be realized via anyon braiding, which is we propose a physical context in which fermion conden- robust against errors due to local perturbation3,4. -
Electron-Phonon Interaction in Conventional and Unconventional Superconductors
Electron-Phonon Interaction in Conventional and Unconventional Superconductors Pegor Aynajian Max-Planck-Institut f¨ur Festk¨orperforschung Stuttgart 2009 Electron-Phonon Interaction in Conventional and Unconventional Superconductors Von der Fakult¨at Mathematik und Physik der Universit¨at Stuttgart zur Erlangung der W¨urde eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung vorgelegt von Pegor Aynajian aus Beirut (Libanon) Hauptberichter: Prof. Dr. Bernhard Keimer Mitberichter: Prof. Dr. Harald Giessen Tag der m¨undlichen Pr¨ufung: 12. M¨arz 2009 Max-Planck-Institut f¨ur Festk¨orperforschung Stuttgart 2009 2 Deutsche Zusammenfassung Die Frage, ob ein genaueres Studium der Phononen-Spektren klassischer Supraleiter wie Niob und Blei mittels inelastischer Neutronenstreuung der M¨uhe wert w¨are, w¨urde sicher von den meisten Wissenschaftlern verneint werden. Erstens erk¨art die ber¨uhmte mikroskopische Theorie von Bardeen, Cooper und Schrieffer (1957), bekannt als BCS Theorie, nahezu alle Aspekte der klassischen Supraleitung. Zweitens ist das aktuelle Interesse sehr stark auf die Hochtemperatur-Supraleitung in Kupraten und Schwere- Fermionen Systemen fokussiert. Daher waren die ersten Experimente dieser Arbeit, die sich mit der Bestimmung der Phononen-Lebensdauern in supraleitendem Niob und Blei befaßten, nur als ein kurzer Test der Aufl¨osung eines neuen hochaufl¨osenden Neutronen- spektrometers am Forschungsreaktor FRM II geplant. Dieses neuartige Spektrometer TRISP (triple axis spin echo) erm¨oglicht die Bestimmung von Phononen-Linienbreiten uber¨ große Bereiche des Impulsraumes mit einer Energieaufl¨osung im μeV Bereich, d.h. zwei Gr¨oßenordnungen besser als an klassische Dreiachsen-Spektrometern. Philip Allen hat erstmals dargelegt, daß die Linienbreite eines Phonons proportional zum Elektron-Phonon Kopplungsparameter λ ist. -
Superselection Sectors in Quantum Spin Systems∗
Superselection sectors in quantum spin systems∗ Pieter Naaijkens Leibniz Universit¨atHannover June 19, 2014 Abstract In certain quantum mechanical systems one can build superpositions of states whose relative phase is not observable. This is related to super- selection sectors: the algebra of observables in such a situation acts as a direct sum of irreducible representations on a Hilbert space. Physically, this implies that there are certain global quantities that one cannot change with local operations, for example the total charge of the system. Here I will discuss how superselection sectors arise in quantum spin systems, and how one can deal with them mathematically. As an example we apply some of these ideas to Kitaev's toric code model, to show how the analysis of the superselection sectors can be used to get a complete un- derstanding of the "excitations" or "charges" in this model. In particular one can show that these excitations are so-called anyons. These notes introduce the concept of superselection sectors of quantum spin systems, and discuss an application to the analysis of charges in Kitaev's toric code. The aim is to communicate the main ideas behind these topics: most proofs will be omitted or only sketched. The interested reader can find the technical details in the references. A basic knowledge of the mathematical theory of quantum spin systems is assumed, such as given in the lectures by Bruno Nachtergaele during this conference. Other references are, among others, [3, 4, 15, 12]. Parts of these lecture notes are largely taken from [12]. 1 Superselection sectors We will say that two representations π and ρ of a C∗-algebra are unitary equiv- alent (or simply equivalent) if there is a unitary operator U : Hπ !Hρ between the corresponding Hilbert spaces and in addition we have that Uπ(A)U ∗ = ρ(A) ∼ ∗ for all A 2 A. -
Probability and Statistics, Parastatistics, Boson, Fermion, Parafermion, Hausdorff Dimension, Percolation, Clusters
Applied Mathematics 2018, 8(1): 5-8 DOI: 10.5923/j.am.20180801.02 Why Fermions and Bosons are Observable as Single Particles while Quarks are not? Sencer Taneri Private Researcher, Turkey Abstract Bosons and Fermions are observable in nature while Quarks appear only in triplets for matter particles. We find a theoretical proof for this statement in this paper by investigating 2-dim model. The occupation numbers q are calculated by a power law dependence of occupation probability and utilizing Hausdorff dimension for the infinitely small mesh in the phase space. The occupation number for Quarks are manipulated and found to be equal to approximately three as they are Parafermions. Keywords Probability and Statistics, Parastatistics, Boson, Fermion, Parafermion, Hausdorff dimension, Percolation, Clusters There may be particles that obey some kind of statistics, 1. Introduction generally called parastatistics. The Parastatistics proposed in 1952 by H. Green was deduced using a quantum field The essential difference in classical and quantum theory (QFT) [2, 3]. Whenever we discover a new particle, descriptions of N identical particles is in their individuality, it is almost certain that we attribute its behavior to the rather than in their indistinguishability [1]. Spin of the property that it obeys some form of parastatistics and the particle is one of its intrinsic physical quantity that is unique maximum occupation number q of a given quantum state to its individuality. The basis for how the quantum states for would be a finite number that could assume any integer N identical particles will be occupied may be taken as value as 1<q <∞ (see Table 1).