DOCSLIB.ORG

Exchange Symmetry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Exchange Symmetry Exchange symmetry Jan Myrheim Department of physics, NTNU April 12, 2016 Contents – Some general remarks about identical particles – Two and three anyons in a harmonic oscillator potential – Cluster and virial expansions for the anyon gas – An approximate relation to exclusion statistics Heisenberg and Dirac (1926) Definition: N particles, numbered 1 to N, are identical if all observables are symmetric under permutations of indices. A symmetric operator X representing an observable preserves the symmetry of the wave function , ! X ; S ! SS = S ; A ! SA = A : We can have two different complete theories of quantum mechanics for identical particles. (I) Using only symmetric wave functions. (II) Using only antisymmetric wave functions. Heisenberg and Dirac argue that these two possibilities exist, but make no attempt to prove that other possibilities do not exist. Statistics In case (I), symmetric wave functions, counting of states leads to Bose–Einstein statistics. In case (II), antisymmetric wave functions, the Pauli exclusion principle holds, and counting of states leads to Fermi–Dirac statistics. If we admit all wave functions, without imposing symmetry or antisymmetry, we get Maxwell–Boltzmann statistics. In the thermodynamic limit we let N ! 1 and the volume V ! 1 with constant particle density n = N=V. This limit exists for ideal gases of bosons and fermions, since their entropy is extensive (proportional to N at constant density). With Maxwell–Boltzmann statistics, the number of states grows too fast with N for the thermodynamic limit to exist. Statistics In both classical and quantum mechanics we need particles to be identical because we want a thermodynamic limit. In classical statistical mechanics the solution is to divide the phase space volume by N! In quantum statistical mechanics the solution is to symmetrize or antisymmetrize the wave functions. Except that we often do not. We antisymmetrize the wave function of the two electrons in a helium atom, but we do not antisymmetrize with the other 1026 electrons around. Two electrons at different positions are identical, but distinguishable. We need not antisymmetrize if we, for example, scatter electrons in such a way that we can follow one electron from the initial to the final state. Scattering process 1 + 2 ! 3 + 4 detector p0 p θ −p −p0 Incoming momenta p1 = −p2 = ±p. 0 Outgoing momenta p3 = −p4 = ±p . 0 Scattering angle α = θ or π − θ, definition: p1 · p3 = jpjjp j cos α. Scattering amplitude f = f (α) because of rotational invariance. f (α) = houtjFjini : Scattering p + n ! p + n As an example for illustration, consider scattering of protons on neutrons. Disregard spin (assume spin independent interactions, a bad approximation). Treating them as different particles, we define the in and out states as jini = jpnijp; −pi ; jouti = jpnijp0; −p0i : We label, e.g., the incoming proton as particle 1 and call its momentum p. To measure the cross section dσ = jf (θ)j2 dΩ we count the outgoing protons, but not the neutrons. If our detector counts both protons and neutrons, without distinction, we would have to redefine dσ 1 = jf (θ)j2 + jf (π − θ)j2 ; dΩ 2 dividing by two to compensate for the double counting. Isospin In 1932, immediately after the discovery of the neutron, Heisenberg suggested treating neutrons and protons as identical fermions. This is a natural idea because the nuclear forces do not distinguish them. Then we have to antisymmetrize the in and out states. The isospin formalism is a technical aid. We define the proton and neutron to be members of an isospin doublet. The isospin component I3 is +1=2 for protons and −1=2 for neutrons. A state of two nucleons can have total isospin I = 0 or I = 1. There are three symmetric states jI; I3i with I = 1: 1 j1; 1i = jppi ; j1; 0i = p jpni + jnpi ; j1; −1i = jnni ; 2 and one antisymmetric state with I = 0: 1 j0; 0i = p jpni − jnpi : 2 Isospin Define also symmetric and antisymmetric momentum states 1 jp+i = p jp; −pi + j − p; pi ; 2 1 jp−i = p jp; −pi − j − p; pi ; 2 and similarly for p0. Then we have two possible proton neutron in states, with I = 1 or I = 0, jini = j1; 0ijp−i or j0; 0ijp+i ; and two possible out states, jouti = j1; 0ijp0−i or j0; 0ijp0+i : Scattering Conservation of isospin, or equivalently, conservation of the symmetry of the momentum state, implies that we have two different scattering amplitudes. One with isospin I = 1, 0 f−(θ) = hp − jFjp−i 1 = hp0; −p0jFjp; −pi − hp0; −p0jFj − p; pi 2 −h−p0; p0jFjp; −pi + h−p0; p0jFj − p; pi = f (θ) − f (π − θ) : And one with isospin I = 0, 0 f+(θ) = hp + jFjp+i 1 = hp0; −p0jFjp; −pi + hp0; −p0jFj − p; pi 2 +h−p0; p0jFjp; −pi + h−p0; p0jFj − p; pi = f (θ) + f (π − θ) : Scattering We compute the cross sections as dσ 1 1 ± = jf (θ)j2 = jf (θ)j2 + jf (π − θ)j2 ± Ref (θ)f (π − θ)∗ ; dΩ 2 ± 2 dividing by two to compensate for double counting. Conclusions: Proton neutron scattering can take place via one of two channels: either symmetric in momentum (I = 0) or antisymmetric in momentum (I = 1). The cross section we compute by treating them as different particles is the average of the symmetric and the antisymmetric channel, dσ 1 1 dσ dσ = jf (θ)j2 + jf (π − θ)j2 = + + − : dΩ 2 2 dΩ dΩ Scattering Even though a proton is different from a neutron, we may choose to prepare the initial state to be symmetric or antisymmetric in momentum. Then we observe bosonic or fermionic scattering, as we choose. On the other hand, if we scatter identical particles we have no choice. They prepare themselves in symmetric or antisymmetric initial states. If there is a mystery, it is this: how do electrons know that they are fermions? And α particles that they are bosons? Photons know that they are bosons because they are quanta of harmonic oscillators. That is how we derive QED from Maxwell’s equations. Or if we regard the quantum theory as more fundamental, we want photons to be bosons because we want Maxwell’s equations as a classical limit. Is isospin only a book keeping trick? In the case of protons and neutrons, we may regard isospin as a tool for keeping track of the symmetry or antisymmetry of spatial wave functions. But isospin is a useful quantum number for all strongly interacting particles. For example, isospin conservation imposes strong constraints on the scattering of π mesons on nucleons. In the quark model, 2I3 is the number of u quarks minus the number of d quarks. Isospin conservation can be understood as an approximate SU(2) symmetry, which is broken because the u and d quarks have different electric charge and different mass. We treat all six quarks u, d, c, s, t, b as identical fermions distinguished by different values of quantum numbers, called flavours, similar to isospin. However, the SU(6) symmetry extending the SU(2) isospin symmetry is badly broken by the electroweak interaction and by the quark mass matrix. The quark masses are free parameters in the so called standard model. Colour The quark model nearly died in its infancy because it seemed to require quarks to be bosons, instead of fermions as required by the spin–statistics theorem. The most dramatic case was the baryon decuplet, for example the ∆++ resonance made up of three u quarks. In the ground state of the uuu system we expect the quarks to have zero orbital angular momentum. Therefore the spatial part of the wave function should be symmetric. The ∆++ has spin 3/2, requiring also a symmetric spin wave function. Conclusion: the u quarks seemed to be spin 1/2 bosons. One idea at the time was that quarks were neither bosons nor fermions, but satisfied parastatistics: the N-particle wave functions would belong to more general representations of the permutation group SN . Greenberg solved the problem in 1964 simply by postulating a new three-valued quantum number, later called colour, now the basis of QCD. The quarks are fermions with an antisymmetric colour wave function, then the rest of the wave function must be symmetric. Which particles are identical? Consider the Hamiltonian of the hydrogen atom, again disregarding spin, p 2 p 2 k q q H = 1 + 2 − ; k = 1 2 : 2m1 2m2 jr1 − r2j 4π0 It is symmetric under the interchange 1 $ 2, a fact which becomes even more obvious when we split off the centre of mass motion and write P2 p2 k H = + − ; 2M 2m jrj with M = m1 + m2, m = m1m2=M, m p − m p P = p + p ; r = r − r ; p = 2 1 1 2 : 1 2 1 2 M Which particles are identical? The Hamiltonian is symmetric, what about other observables? The electric dipole moment is symmetric, it is d = e(rp − re) = q1r1 + q2 r2 ; independent of how we label the electron and the proton as particle 1 and 2. The electron position is symmetric, it is re = δ1e r1 + δ2e r2 : All observables are symmetric under the interchange 1 $ 2. They have to be, because the labels are arbitrary, different labellings just create different mathematical descriptions of the same physical reality. So in this sense the electron and proton are identical particles. Why don’t we symmetrize or antisymmetrize the wave function? The configuration space particle identity 2 1 proton 1 2 electron particle position particle position There is an electron at position re and a proton at position rp. This statement describes uniquely a configuration of the system.
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]
  • 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!
    [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]