DOCSLIB.ORG

Towards Topological Quantum Computing with Kitaev Materials

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Towards Topological Quantum Computing with Kitaev Materials Towards topological quantum computing with Kitaev materials Nature 559, 227 (2018) Jason Alicea, Caltech Topological quantum computation overview Non-Abelian anyons Locally Inherently indistinguishable fault-tolerant ground states qubits! 000 , 001 ,... Anyons carry exotic | i | i zero-energy modes Alexei Kitaev Non-Abelian anyons Locally Inherently indistinguishable fault-tolerant ground states qubits! 000 , 001 ,... Anyons carry exotic | i | i zero-energy modes Non-Abelian Inherently statistics fault-tolerant gates! U i ! ij j How to build the hardware? Alexei Kitaev “Designer” realizations 1D p-wave 2D p+ip superconductor superconductor “Designer” realizations Mourik et al. Science 336, 1003 (2012) Albrecht et al. Nature 531, 206 (2016) ..and many more Nadj-Perge et al. Science 346, 602 (2014) Ren et al. arXiv:1809.03076 “Intrinsic” realizations Fractional quantum Hall Quantum spin liquids http://manfragroup.org/non-abelian-phases-in-the-fractional-quantum-hall-regime/ https://neutronsources.org/news/scientific-highlights/neutrons-exotic-particles.html Non-Abelian spin liquids & “Kitaev materials” h Frustration x x SySy SzSz H = − J ∑ Sr Sr′" + ∑ r r′" + ∑ r r′" (⟨rr′⟩" ⟨rr′⟩" ⟨rr′⟩" ) −∑ h ⋅ Sr r 0 Sr = iγr γr⃗ → ~free Majorana fermions Gapless Gapped, non-Abelian spin liquid spin liquid! Chiral Majorana edge state Fermion x x SySy SzSz ψ H = − J ∑ Sr Sr′" + ∑ r r′" + ∑ r r′" (⟨rr′⟩" ⟨rr′⟩" ⟨rr′⟩" ) σ − h ⋅ S ‘Ising’ non-Abelian anyons ∑ r r σ 0 Sr = iγr γr⃗ → ~free Majorana fermions Gapless Gapped, non-Abelian spin liquid spin liquid! Chiral Majorana edge state Chiral Majorana edge state Fermion Fermion Emergent fermions Built from physical in a bosonic system microscopic fermions ψ ψ σ ≠ σ ‘Ising’ non-Abelian anyons ‘Ising’ non-Abelian anyons σ σ non-Abelian spin liquid p+ip superconductor Towards experimental reality h x x SySy SzSz H = − J ∑ Sr Sr′" + ∑ r r′" + ∑ r r′" (⟨rr′⟩" ⟨rr′⟩" ⟨rr′⟩" ) −∑ h ⋅ Sr +⋯ r Focus here: ⍺-RuCl3 Plumb et al., PRB 90, 041112 (2014) Yields quantized thermal Hall conductance Chiral Majorana edge state Quantized value Fermion !!! ψ σ ‘Ising’ non-Abelian anyons conductivity Thermal Hall σ Magnetic field [T] First experimental evidence of a non-Abelian spin liquid! Contender for topological quantum computing hardware? Virtues: energy scales, fabrication flexibility Challenges: manipulation & detection of anyons Nature 559, 227 (2018) So far withstood theoretical scrutiny! Vinkler-Aviv & Rosch PRX; Ye, Halasz, Savary, Balents PRL Punchline I Detectable with electrical Chiral Majorana edge state probes—even though RuCl3 is an electrical insulator! Fermion ψ σ ‘Ising’ non-Abelian anyons Dave Roger Ben David σ Aasen Mong Hunt Mandrus Phys. Rev. X 10, 031014 (2020) Electrical probes of non-Abelian spin liquids Converting physical and emergent fermions Integer quantum RuCl3 Hall Superconductor Turn on interactions… arXiv:1709.01941 Circuit designs V Edge-state detection e2 G(V → 0,T → 0) = 2h IQH RuCl3 IQH Drain Source (Floating) superconductor Ising-anyon detection Ising-anyon/fermion detection V Gσ ≠ GI = Gψ V GI,σ,ψ all differ ψ IQH RuCl3 IQH RuCl3 Source Source σ σ (Grounded) (Grounded) superconductor superconductor Experimental status Punchline II Detectable with simpler time-domain probes— no QH/SC needed! Chiral Majorana edge state Fermion ψ σ ‘Ising’ non-Abelian anyons Kai Dave Roger Eugene σ Klocke Aasen Mong Demler Available soon… Majorana edge-mode detection Excited Ground state state RuCl3 Probes chiral edge state velocity x H(t) = − iv γ∂xγ + h ⋅ sj + iλj(t)s γ∂xγ| ∫ ∑ [ j xj ] x j=1,2 time Time-domain anyon interferometry Excited state Fermion RuCl3 ψ σ Ground state Time-domain anyon interferometry Excited state Fermion RuCl3 ψ σ Ground state Time-domain anyon interferometry Fermion RuCl3 ψ σ Time-domain anyon interferometry Paths can interfere in a way that depends on enclosed quasiparticle type Fermion RuCl3 ψ σ -0.84 1 ∆ = v⌧ = La/2 a = | | sz(t) h 2 i a = σ a = Detects individual -1 bulk anyons and their exotic statistics! λj(t) λ1(t) λ2(t) 2⌧ energy arrives 0 t1 t t2 time Summary & Outlook Non-Abelian spin liquids in “Kitaev materials” appear to be experimental reality, providing new horizon in topological quantum computation. Key features detectable with Chiral Majorana edge state electrical probes and simpler time-domain measurements! RuCl3 Fermion ψ σ ‘Ising’ non-Abelian anyons σ Practical anyon manipulation schemes needed (Ising anyons particularly subtle). Warm-up: stitching p+ip superconductors Chiral Majorana edge state p+ip p+ip superconductor superconductor ℋ = − ivγR∂xγR + ivγL∂xγL +imγRγL Warm-up: stitching p+ip superconductors Chiral Majorana edge state p+ip SC’s p+ip superconductor sewn up superconductor ℋ = − ivγR∂xγR + ivγL∂xγL +imγRγL Warm-up: stitching non-Abelian spin liquids Chiral Majorana edge state RuCl3 RuCl3 ℋ = − ivγR∂xγR + ivγL∂xγL +imγRγL Unphysical! (because fermions are emergent) Warm-up: stitching non-Abelian spin liquids Chiral Majorana edge state Spin liquids RuCl3 sewn up RuCl3 Assume “large”… ℋ = − ivγR∂xγR + ivγL∂xγL +λγRγRγLγL → λ⟨iγRγL⟩iγRγL ≡ imeffγRγL …mass generated spontaneously!.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Equation of State of an Anyon Gas in a Strong Magnetic Field
    EQUATION OF STATE OF AN ANYON GAS IN A STRONG MAGNETIC FIELD Alain DASNIÊRES de VEIGY and Stéphane OUVRY ' Division de Physique Théorique 2, IPN, Orsay Fr-91406 Abstract: The statistical mechanics of an anyon gas in a magnetic field is addressed. An har- monic regulator is used to define a proper thermodynamic limit. When the magnetic field is sufficiently strong, only exact JV-aiiyon groundstates, where anyons occupy the lowest Landau level, contribute to the equation of state. Particular attention is paid to the inter- \ val of definition of the statistical parameter a 6 [-1,0] where a gap exists. Interestingly J enough, one finds that at the critical filling v = — 1 /a where the pressure diverges, the t external magnetic field is entirely screened by the flux tubes carried by the anyons. PACS numbers: 03.65.-w, 05.30.-d, ll.10.-z, 05.70.Ce IPN0/TH 93-16 (APRIL 1993) "f* ' and LPTPE, Tour 16, Université Paris 6 / electronic e-mail: OVVRYeFRCPNlI I * Unité de Recherche des Univenitéi Paria 11 et Paria 6 associée au CNRS y -Introduction : It is now widely accepted that anyons [1] should play a role in the Quantum Hall effect [2]. In the case of the Fractional Quantum Hall effect, Laughlin wavefunctions for the ground state of N electrons in a strong magnetic field with filling v - l/m provide an interesting compromise between Fermi degeneracy and Coulomb correlations. A physical interpretation is that at the critical fractional filling electrons carry exactly m quanta of flux <j>o (m odd), m - 1 quanta screening the external applied field.
    [Show full text]
  • Anyons and Lowest Landau Level Anyons
    S´eminaire Poincar´eXI (2007) 77 – 107 S´eminaire Poincar´e Anyons and Lowest Landau Level Anyons St´ephane Ouvry Laboratoire de Physique Th´eorique et Mod`eles Statistiques, Universit´eParis-Sud CNRS UMR 8626 91405 Orsay Cedex, France Abstract. Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topol- ogy of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum many-body states with a multivalued phase factor, which encodes the anyonic nature of particle windings. Bosons and fermions appear as two limiting cases. Gauging away the phase leads to the so-called anyon model, where the charge of each particle interacts ”`ala Aharonov-Bohm” with the fluxes carried by the other particles. The multivaluedness of the wave function has been traded off for topological interactions between ordinary particles. An alternative Lagrangian formulation uses a topological Chern-Simons term in 2+1 dimensions. Taking into account the short distance repulsion between particles leads to an Hamil- tonian well defined in perturbation theory, where all perturbative divergences have disappeared. Together with numerical and semi-classical studies, pertur- bation theory is a basic analytical tool at disposal to study the model, since finding the exact N-body spectrum seems out of reach (except in the 2-body case which is solvable, or for particular classes of N-body eigenstates which generalize some 2-body eigenstates). However, a simplification arises when the anyons are coupled to an external homogeneous magnetic field.
    [Show full text]
  • Collective Coordinate Quantization and Spin Statistics of the Solitons in The
    Collective coordinate quantization and spin statistics of the solitons in the CP N Skyrme-Faddeev model Yuki Amaria,∗ Paweł Klimasb,† and Nobuyuki Sawadoa‡ a Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan b Universidade Federal de Santa Catarina, Trindade, 88040-900, Florianópolis, SC, Brazil (Dated: September 26, 2018) The CP N extended Skyrme-Faddeev model possesses planar soliton solutions. We consider quantum aspects of the solutions applying collective coordinate quantization in regime of rigid body approximation. In order to discuss statistical properties of the solutions we include an Abelian Chern-Simons term (the Hopf term) in the 1 Lagrangian. Since Π3(CP ) = Z then for N = 1 the term becomes an integer. On the other hand for N > 1 N it became perturbative because Π3(CP ) is trivial. The prefactor of the Hopf term (anyon angle) Θ is not quantized and its value depends on the physical system. The corresponding fermionic models can fix value of the angle Θ for all N in a way that the soliton with N = 1 is not an anyon type whereas for N > 1 it is always an anyon even for Θ = nπ, n Z. We quantize the solutions and calculate several mass spectra for N = 2. Finally we discuss generalization∈ for N ≧ 3. PACS numbers: 11.27.+d, 11.10.Lm, 11.30.-j, 12.39.Dc I. INTRODUCTION Skyrme-Faddeev Hopfions [21, 22]. An alternative approach based on a canonical quantization method has been already examined for the baby Skyrme model [15] and the Skyrme-Faddeev Hopfions [23].
    [Show full text]
  • Relativistic Anyon Beam: Construction and Properties
    PHYSICAL REVIEW LETTERS 123, 164801 (2019) Relativistic Anyon Beam: Construction and Properties Joydeep Majhi, Subir Ghosh, and Santanu K. Maiti* Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India (Received 13 February 2019; revised manuscript received 9 August 2019; published 17 October 2019) Motivated by recent interest in photon and electron vortex beams, we propose the construction of a relativistic anyon beam. Following Jackiw and Nair [R. Jackiw and V. P. Nair, Phys. Rev. D 43, 1933 (1991).] we derive an explicit form of the relativistic plane wave solution of a single anyon. Subsequently we construct the planar anyon beam by superposing these solutions. Explicit expressions for the conserved anyon current are derived. Finally, we provide expressions for the anyon beam current using the superposed waves and discuss its properties. We also comment on the possibility of laboratory construction of an anyon beam. DOI: 10.1103/PhysRevLett.123.164801 We propose construction of a relativistic beam of anyons beam [1,2]. But, in 2D a monoenergetic wave beam will in a plane. Anyons are planar excitations with arbitrary necessary have dispersion in momentum along propagation spin and statistics. The procedure is similar to three- direction and hence cannot be completely delocalized. The dimensional (3D) optical (spin 1) and electron (spin 1=2) delocalization along propagation direction will be more for vortex beams. a narrower angle of superposition. Vortex beams, of recent interest, carrying intrinsic orbital Apart from earlier interest [6], graphene also involves angular momentum are nondiffracting wave packets in anyons [8]. Non-Abelian anyons [9] are touted as theo- motion.
    [Show full text]
  • Field Theory of Anyons and the Fractional Quantum Hall Effect 30- 50
    NO9900079 Field Theory of Anyons and the Fractional Quantum Hall Effect Susanne F. Viefers Thesis submitted for the Degree of Doctor Scientiarum Department of Physics University of Oslo November 1997 30- 50 Abstract This thesis is devoted to a theoretical study of anyons, i.e. particles with fractional statistics moving in two space dimensions, and the quantum Hall effect. The latter constitutes the only known experimental realization of anyons in that the quasiparticle excitations in the fractional quantum Hall system are believed to obey fractional statistics. First, the properties of ideal quantum gases in two dimensions and in particular the equation of state of the free anyon gas are discussed. Then, a field theory formulation of anyons in a strong magnetic field is presented and later extended to a system with several species of anyons. The relation of this model to fractional exclusion statistics, i.e. intermediate statistics introduced by a generalization of the Pauli principle, and to the low-energy excitations at the edge of the quantum Hall system is discussed. Finally, the Chern-Simons-Landau-Ginzburg theory of the fractional quantum Hall effect is studied, mainly focusing on edge effects; both the ground state and the low- energy edge excitations are examined in the simple one-component model and in an extended model which includes spin effets. Contents I General background 5 1 Introduction 7 2 Identical particles and quantum statistics 10 3 Introduction to anyons 13 3.1 Many-particle description 14 3.2 Field theory description
    [Show full text]
  • Derivation of a Hamiltonian for Formation of Particles in a Rotating System Subjected to a Homogeneous Magnetic field
    Derivation of a Hamiltonian for formation of particles in a rotating system subjected to a homogeneous magnetic field. Sergio Manzetti 1;2 and Alexander Trounev 3 1. Uppsala University, BMC, Dept Mol. Cell Biol, Box 596, SE-75124 Uppsala, Sweden. 2. Fjordforsk A/S, Bygdavegen 155, 6894 Vangsnes, Norway. 3. A & E Trounev IT Consulting, Toronto, Canada. December 17, 2018 1 1 Abstract Bose-einstein condensates have received wide attention in the last decades given their unique properties. In this study, we apply supersymmetry rules on an exist- ing Hamiltonian and derive a new model which represents an inverse scattering problem with similarities to the circuit and the harmonic oscillator equations. The new Hamiltonian is analysed and its numerical solutions are presented. The 1Please cite as: Sergio Manzetti and Alexander Trounev. (2019) "Derivation of a Hamilto- nian for formation of particles in a rotating system subjected to a homogeneous magnetic field” In: Modeling of quantum vorticity and turbulence with applications to quantum computations and the quantum Hall effect. Report no. 142020. Copyright Fjordforsk A/S Publications. Vangsnes, Norway. www.fjordforsk.no 1 results suggest that the SUSY Hamiltonian describes the formation of vortices in a homogeneous magnetic field in a rotating system. 2 Introduction Hamiltonian operators are probably the most fundamental part in quantum physics and form the basis for calculating the physical properties for elementary particles, their energy and observables. Operators possess properties, such as self-adjointness and unboundness, which allows them to give sensible physical answers to quantized systems, where operators such as the Hamiltonian in the anyon-model of Laughlin [1] are of particular appeal.
    [Show full text]
  • P Age Anyon Statistics and the Topology of Dark Matter
    Anyon Statistics and the Topology of Dark Matter Ervin Goldfain Abstract Matching current observations on non-baryonic Dark Matter (DM), Cantor Dust was recently conjectured to emerge as large-scale topological structure in the early Universe. The mechanism underlying the formation of Cantor Dust hinges on dimensional condensation of spacetime endowed with minimal fractality. It is known that anyons are quasiparticles exhibiting anomalous statistics and fractional charges in 2+1 spacetime. This brief report is a preliminary exploration of the intriguing analogy between anyons and the Cantor Dust picture of DM. Key words: Dark Matter, Cantor Dust, anyons, Topological field theory, Maxwell-Chern-Simons theory, Topological phases of matter. 1. Introduction Rooted in the Renormalization Group program of Quantum Field Theory, the minimal fractal manifold (MFM) describes a spacetime continuum having arbitrarily small deviations from four-dimensions ( ). This fine structure of spacetime is conjectured to set in near or above the Fermi scale. The emergence of the MFM sheds light on many ongoing puzzles of the Standard Model (SM), while meeting all consistency requirements mandated by effective QFT in the conformal limit . The underlying rationale, conceptual benefits and implications of the MFM for the development of Quantum Field Theory and the SM are reported elsewhere (refs. 1-12 of [1]). Recently, a proposal was put forward according to which DM amounts to Cantor Dust, a large-scale topological structure formed in the early Universe. The mechanism underlying 1 | P a g e the onset of Cantor Dust hinges on dimensional condensation of spacetime endowed with minimal fractality. Condensed matter physics views anyons as quasiparticles exhibiting anomalous statistics and fractional charges in 2+1 spacetime.
    [Show full text]
  • Anyons and Dualities in 2+1 Dimensions
    Anyons and Dualities in 2+1 Dimensions Anna Movsheva May 4, 2017 1 Contents 1 Introduction 4 2 Chern-Simons action 8 2.1 Symmetries of the action . 10 2.2 Quantization of abelian Chern-Simons . 10 2.3 Topological Interpretation of L(γi; γj) ....................... 13 2.4 The action for anyons . 17 2.5 Integrating out Chern-Simons field . 18 2.6 Derivation of Hamiltonian . 23 2.7 Discussion of spectrum in case of zero density . 25 3 Random phase approximation 26 1 e2ρ¯ 2 3.1 Spectrum of 2m Sp^ + 2 JrS .............................. 27 3.2 Anyons as deformations of bosons and fermions . 28 3.3 Anyonic Hamiltonian in second quantization formalism . 30 4 Vortices 34 4.1 Example of a simplest pair of dual theories . 34 4.2 Particle-vortex duality in 3 dimensions . 36 5 Topological insulators and superconductors 38 5.1 Back to fermions . 38 A Quantum symmetries 40 A.1 Symmetric states . 43 A.2 Topological construction of coverings and group actions. 44 A.3 Braid group . 45 B On solutions of equation d! = δ 48 C Gaussian integrals 50 D Elements of second quantization formalism 52 2 Acknowledgement I would like to thank Professor Antal Jevicki of Brown University and Professor Rostislav Matveyev of Max Planck Institute for Mathematics in the Sciences for helping me complete this thesis. 3 1 Introduction All elementary particles known to us fall into two big classes: bosons and fermions. This classification is made according to how two identical particles can coexist in the same state. Electrons, quarks, and other fields of matter are fermion.
    [Show full text]
  • Anyons Braid Statistics in Quantum Mechanics
    Anyons Braid statistics in quantum mechanics Julian Lang May 4, 2018 Abstract This report contains the information that was presented in my talk about Anyons in the Pros- eminar: Algebra, Topology and Group Theory in Physics by Prof. Matthias Gaberdiel (2018). It contains two parts. In the first part a theoretical explanation of what anyons are and why they can exist is given, in the second part the cyon is discussed as a physical representation of a system that shows fractional statistic. It thereby closely follows the ideas and derivation from the book Anyons from A. Lerda [1] 1 1 Theoretical discussion of the anyon An important idea for the discussion of anyonic statistics is the fact that particles in quantum mechan- ics are indistinguishable. This is due to the fact that there cannot be made any observable distinction between two physical states that differ only by the exchange of indistinguishable particles. This means there exists no experiment by which it is possible to determine if particles were exchanged or not. In classical physics, however, it is possible to follow the trajectories of the particles and by doing so, at least in principle distinguish the particles from one another. In quantum mechanics this is not possible since the particles positions and velocities are fundamentally not determined between measurements. It is known that in 3 dimensions of space this indistinguishability leads to the restriction that only bosons or fermions can exist. Although this result will be reproduced in this report, we will see that in two dimensions there exist additional solutions.
    [Show full text]