DOCSLIB.ORG

TWO-DIMENSIONAL QUANTUM GAS S

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
TWO-DIMENSIONAL QUANTUM GAS S f .'..~«rf£^.« ;»*«4 I INSTITUTO DE FÍSICA TEÓRICA 1TT/P-17/9O TWO-DIMENSIONAL QUANTUM GAS a* R. Aldr ovandl* Instituto de Física Teóricas State Untversity oi 55o Paulo - UNESP Rúa Parriplcna 145 - 01405 -São.Pauio S? - Brazil s Abstract o ícter,!;cal itnprnstfcile perlicles '.n a 2-diíiieosior4! ccnfíg-jralisn serra fcrotá stetisti», Ihí&rír.a;i3'o beUv=eri bosons end . sy.iMügiric croups fte r.ein properties of ao ij^J jis cl su-;-. ^ are presèTiiâl Itw/ CD iaterpolôte U*e prepertiçs et t5so.is erd If i fr.s but cióssicai pyticles as a special case Restriction to 2 c .; *',ens prscVjfcs l4.T»tcü points but ori-jic-sías fl pecv! - . symmetry,responsible in particular for ths'icsrj , fi - boson ft.il iernt:on specific h£c:!S. o 0 * With perttel «uppyl cí CMr « Idex 11-31 370Uii'-.r 5^ .1fi!ôf«00 55ll36i4«? U£SP <• E?JA?£SP. tíí.MET 1.Introduction Quantum mechanics of a given system is controlled Dy the group of classes of closed loops (the fundamental group) or Us configuration space ill and, of course, statistical mechanics will follow suit. The configuration space of a system of N identical particles will have a non-trivial fundamental group [2], that is, will be multiply connected Let M be the configuration space of each particle and consider the N set M • |x • (xt, x2,.... xN)), the cartesian product of manifold M by itself N tifiics. MN will be the configuration space of a gas of N distinguishable particles contained in fi n is usually a 3- dimensional Euclidean box whose volume V is taken to infinity at the thermodynamica! limit. When the particles are indistinguishable, two 1 N points x and x of M are equivalent if the sets (x,, x2,..., xN) and (x',, x'2, . , x'N) differ only by a permutation, a transformation belonging to me symmetric group SN, and tne configuration space N becomes the çjotient M /SN When M is inched Euclidean, and so N topoiogically trivial, the funaamentai group, indicated Tijln /5N), is N just SH. Space M is then the universal covering of the configuration space and tne N-particle wavefunction will belong in the earner space of some representation of 5N When the particles are also impenetrable, coincident positions for them must be excluded. This is done by defining the set 0f(- {<x,, x2,..., xN) sucn that x, = xJ for some ij) and taking its complement in MN, N FNM« M \DN . (I.I) This would u the configuration space for N distinct impenetrable particles. The fundamental group of this space, Pw • ITI[FH r,J, is the pure úraiü group 13) it is tn particular the fundamental group ícr me configuration spaa of a classical hard-sphere gas. Consider further the particles to be indistinguishable. Let B^tt be the space obtained by identifying all equivalent points, the quotient by SN: N BNM-FNM/SN-If1 \DN)/SN . (1.2) The íuuüamenuii group of this configuration space, tK •liilBNttj, is the full braid group, or simply braid group. bumming up, FNM is the configuration space for a gas of N Impenetrable particles. BNM is the configuration space for a gas of N impenetrable and identical particles and the pure and full braid groups are the respective fundamental groups of such spaces Quantization is to be performed on such highly complicated, multiply- connered space H). We shall see below that braid groups concern real, usual Draiai and that, because braids can be unwoven in spaces of three or more dimensions, all this Is of possible practical interest only for two-dimensional systems. The quotient of the full braid grojp by the pure braid group is the symmetric group: BN/PN-SN, (1.3) N When dimM>2, the braid groups are extensions by Ss of ITÍI(M)] . When M is itself topologically non-trivial, iT](ri) will be some discrete group When M is trivial, it reduces to the identity in the Euclidean 3 3N case M«E , JIi(M)-J, BN • SN, jr1lE XSNl - SN. The classes of loops, highly non-trivial on a punctured E2, become trivial in punctured E3, where each loop may be continuously wformed to a point. Consequently, for dimM i 3, statistical mechanics will be Just as usual, regulated by the symmetric group, with bosons and fermions. 3 The whole point is that this ts not forctDly the case for dim II-2. A gas at 2 dimensions may have a different statistics, governed by braid groups, a Oraid statistics. Such statistics ts expected to play a{ role in 2-dtmensional condensed matter physics phenomena such as the fractional quantum Hall effect and high temperature superconductivity 151 Braid statistics has been the object of htghy sophisticated analysts [6], in comparison to which our approach is undoubtedly naive, host treatments are concerned with 'particles* whose exotic statistics is due to their special structure, such as Wilczek s anyons [71 In reality, manifestations of such exotic statistics are to be expected as soon as Impenetrable particles are considered in two dimensions [8], whatever their internal structure (91 It is our contention that the first step In the understanding of a new statistics is to examine the corresponding ideal gas. That is what we intend to do here, taking into account the fact that impenetrability is enough to ensure the emergence of braid statistics. Such a free gas might come to model, up to interactions, the electron gas in superconductors, in which the electrons are confined to the surface. It is anyhow the starting point, to v/hich dynamical effects are to be added in a deeper analysis. Dynamics has been extensively studied, mainl/ in the line of Landau-Ginzburg models [10}. A complete knowledge of the ideal case will help to separate effects of dynamical nature from those of purely statistical origin. As the test way to introduce braid groups is through the symmetric groups, and also because these would anyhow regulate the case of interper.etrable particles, we start by recalling in 82 trie Ideal gas statistical mechanics as ruled by SN. The canonical partition function of a real gas of N particles is a certain SN-invariant polynomial, the cycle indicator polynomial. Furthermore, the symmetric group maintains a bookkeeping role even for braid statistics, as It counts the classical configurations in terms of which the quantum partition function is ultimately written. The adopted pedagogical tune seems unavoidable because we want to stress some aspects to be later adapted to the braid case, in particular, permutations are Introduced in a pictorial way specially convenient to generalization. The 2-dlmenslonal case exhibits some characteristic symmetries, leaciiny in pãUicüUr to the identity of boson and fcrmion specific heat, in thebosomc case, although ground state crowding or particles do occur at low enough temperatures, no lambda point Is found, so that transition must proceed smoothly. A more mathematical interpretation of the usual realization of wavefunctions In terms of distinguishable particles Is given in S3. It helps clarifying what happens when exchange symmetries distinct from SN are at work. We give a (necessarily very Incomplete) introduction to braids and their groups in S4 and proceed to study the corresponding statistical mechanics In SS. There is one braid statistics for each value of an angular parameter <p, with <p = O corresponding to bosons and <p c TI to fermions. As It might be expected, Intermediate values of <p leads to intermediate behavior of the physical quantities, but this "interpolation" goes through the classical case at <p • n/2, where a Boltzmmann gas comes out. Some final comments, Including a comparison to parastatlstics, are made in S 6. 2. Symmetric group statistics The role of the symmetric group ill] In Statistical Mechanics is best seen in the cluster decomposition [12) of the canonical partition function, which happens to be an Invariant polynomial of SN. in order to see it, let us start by shortly reviewing some well known facts about permutations. A general permutation P of particles labelled Xj,x2,..., xN.,,xN is usually indicated by P • I í1 Í2 "' Í""1 J1) and can always be decomposed into the product of disjoint cycles, particular tatl-bitlng permutations ot type (IJ2'.'.'.X'£)• TMS IS a cycle of length r, or at r-cycle it Is convenient [131 to attribute a variable tr to a cycle of length "r" and indicate the cycle structure of a permutation by the monomial O1?*1!3 ••• tVr • mean*n9 *hat lher* are vi 1" cycles, v2 2-cydes, etc. Permutations of the same cycle type, that Is, with the same set (Vj), go into each other under the action of any element of Sk they constitute conjugate classes. To all permutations of a fixed class will be attributed the same monomial above. In this sense, such monomials are invariants of the group 5N. The total number of permutations with such a fixed cycle configuration is N! —— . The N-variable generating function for these numbers Is the cycle Indicator polynomial [14] where the star (•) recalls that the summation takes place over the sets lv,) of ncn-negative Integers for wnicn Cycle indicator polynomials satisfy the relation [151 N n 2 ^N-mlt)) . (22) which may lead to recursion formulae when the tr's are known and will be useful later on. The canonical partition function oi a real non-relatlvlstlc gas of N particles contained in a d-dimenslonal volume V is a cycle indicator polynomial for the symmetric group 5N, with the V (combinatorially meaningless) variables tj * jbj TQ giving the contribution of the J-th order cluster integral: H CN( jb, ^ J (2.3) where p • 1/kT; X is the mean thermal wavelength, bj Is the J-th cluster Integral H6J.
Load more

Recommended publications
  • Majorana Fermions in Mesoscopic Topological Superconductors: from Quantum Transport to Topological Quantum Computation
    Majorana Fermions in Mesoscopic Topological Superconductors: From Quantum Transport to Topological Quantum Computation Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨atD¨usseldorf vorgelegt von Stephan Plugge D¨usseldorf, April 2018 Aus dem Institut f¨ur Theoretische Physik, Lehrstuhl IV der Heinrich-Heine-Universit¨at D¨usseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨at D¨usseldorf Berichterstatter: 1. Prof. Dr. Reinhold Egger 2. PD Dr. Hermann Kampermann 3. Prof. Felix von Oppen, PhD Tag der m¨undlichen Pr¨ufung: 15. Juni 2018 List of included publications S. Plugge, A. Zazunov, P. Sodano, and R. Egger, Majorana entanglement bridge,Phys. Rev. B 91, 214507 (2015). [Selected as Editors’ Suggestion] L.A. Landau, S. Plugge, E. Sela, A. Altland, S.M. Albrecht, and R. Egger, Towards real- istic implementations of a Majorana surface code, Phys. Rev. Lett. 116, 050501 (2016). S. Plugge, A. Zazunov, E. Eriksson, A.M. Tsvelik, and R. Egger, Kondo physics from quasiparticle poisoning in Majorana devices,Phys.Rev.B93, 104524 (2016). S. Plugge, L.A. Landau, E. Sela, A. Altland, K. Flensberg, and R. Egger, Roadmap to Majorana surface codes,Phys.Rev.B94, 174514 (2016). [Selected as Editors’ Suggestion] S. Plugge, A. Rasmussen, R. Egger, and K. Flensberg, Majorana box qubits, New J. Phys. 19 012001 (2017). [Fast-Track Communication; selected for Highlights of 2017 collection] T. Karzig, C. Knapp, R. Lutchyn, P. Bonderson, M. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. Marcus, and M. H. Freedman, Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes,Phys.Rev.B95, 235305 (2017).
    [Show full text]
  • LIST of PUBLICATIONS Jürg Fröhlich 1. on the Infrared Problem in A
    LIST OF PUBLICATIONS J¨urgFr¨ohlich 1. On the Infrared Problem in a Model of Scalar Electrons and Massless Scalar Bosons, Ph.D. Thesis, ETH 1972, published in Annales de l'Inst. Henri Poincar´e, 19, 1-103 (1974) 2. Existence of Dressed One-Electron States in a Class of Persistent Models, ETH 1972, published in Fortschritte der Physik, 22, 159-198 (1974) 3. with J.-P. Eckmann : Unitary Equivalence of Local Algebras in the Quasi-Free Represen- tation, Annales de l'Inst. Henri Poincar´e 20, 201-209 (1974) 4. Schwinger Functions and Their Generating Functionals, I, Helv. Phys. Acta, 47, 265-306 (1974) 5. Schwinger Functions and Their Generating Functionals, II, Adv. of Math. 23, 119-180 (1977) 6. Verification of Axioms for Euclidean and Relativistic Fields and Haag's Theorem in a Class of P (φ)2 Models, Ann. Inst. H. Poincar´e 21, 271-317 (1974) 7. with K. Osterwalder : Is there a Euclidean field theory for Fermions? Helv. Phys. Acta 47, 781 (1974) 8. The Reconstruction of Quantum Fields from Euclidean Green's Functions at Arbitrary Temperatures, Helv. Phys. Acta 48, 355-369 (1975); (more details are contained in an unpublished paper, Princeton, Dec. 1974) 9. The Pure Phases, the Irreducible Quantum Fields and Dynamical Symmetry Breaking in Symanzik-Nelson Positive Quantum Field Theories, Annals of Physics 97, 1-54 (1975) 10. Quantized "Sine-Gordon" Equation with a Non-Vanishing Mass Term in Two Space-Time Dimensions, Phys. Rev. Letters 34, 833-836 (1975) 11. Classical and Quantum Statistical Mechanics in One and Two Dimensions: Two-Compo- nent Yukawa and Coulomb Systems, Commun.
    [Show full text]
  • Attempts to Go Beyond Bose and Fermi Statistics
    Beyond Bose and Fermi Statistics O.W. Greenberg University of Maryland Spin-Statistics 2008 October 21, 2008 Trieste, Italy Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W. Greenberg Beyond Bose and Fermi Statistics Outline 1 Introduction 2 Spin-statistics vrs. spin-locality 3 What quantum mechanics allows 4 Doplicher, Haag, Roberts analysis 5 Parastatistics 6 Quons 7 Anyons 8 Experimental tests of statistics O.W.
    [Show full text]
  • An Introduction to Anyons
    An Introduction to Anyons Avinash Deshmukh University of British Columbia In this paper, we introduce anyons, particles that pick up a non-trivial phase under exchange. We start by considering the topological conditions that allow anyons to be created and construct the braid group, which defines anyonic exchange statistics. We proceed to discuss abelian and non- abelian exchange representations and review cyons, fibonacci anyons, and Ising anyons as illustrative examples. Finally, we briefly examine solved condensed matter models in which anyons are produced and review the exchange statistics of these systems. I. INTRODUCTION rigorous version of this argument. It is well known that in three-dimensions, there are two One important consequence of θ being any value is that it is iθ iθ types of particles; bosons are symmetric under exchange and not generally true that e “ e´ . We must then distinguish fermions are anti-symmetric under exchange. However, in between \counterclockwise" exchanges and \clockwise" ex- two-dimensions, it is possible to have particles that are nei- changes, which correspond to the accumulated phases θ and ther anti-symmetric nor symmetric under exchange. These ´θ respectively. particles are known as anyons. It has been shown that the exotic statistics of anyons can be used to encode information for topological quantum computation [1], and it is believed B. The Braid Group that they can be used in the construction of superconductors [2]. Earlier this year, abelian anyons were directly observed Consider a system of n particles that follow θ-statistics in a two-dimensional electronic gas collider [3] and in an elec- 2 with the positions r1; :::; rn P R in the center of mass tronic Fabry-Perot intereferometer [4], showing that these frame, described by the wave function pr1; :::; rnq.
    [Show full text]
  • Boson-Lattice Construction for Anyon Models
    Boson-Lattice Construction for Anyon Models Bel´enParedes Ludwig Maximilian University, Munich This work is an attempt to unveil the skeleton constituents. The set of anyon types emerging from a of anyon models. I present a construction to sys- certain topological order satisfies a collection of fusion tematically generate anyon models. The construc- and braiding rules [10{14], which determine the way in tion uses a set of elementary pieces or fundamental which anyons fuse with each other to give rise to other anyon models, which constitute the building blocks anyon types, and the form in which they braid around to construct other, more complex, anyon models. each other. A set of anyon types together with their fu- A principle of assembly is established that dictates sion and braiding rules define an anyon model, which per- how to articulate the building blocks, setting out the fectly mirrors its corresponding underlying physical state. global blueprint for the whole structure. Remark- What are the possible anyon models that can exist? ably, the construction generates essentially all tabu- From a purely mathematical perspective, it is known that lated anyon models. Moreover, novel anyon models the language underlying anyon models is modular ten- (non-tabulated, to my knowledge) arise. To embody sor category [15{22]. Within this beautiful (and com- the construction I develop a very physical, visual and plex) mathematical formalism the answer to this ques- intuitive lexicon. An anyon model corresponds to a tion can be simply phrased: any possible anyon model system of bosons in a lattice. By varying the num- corresponds to a unitary braided modular tensor cate- ber of bosons and the number of lattice sites, towers gory.
    [Show full text]
  • Exchange and Exclusion in the Non-Abelian Anyon Gas
    EXCHANGE AND EXCLUSION IN THE NON-ABELIAN ANYON GAS DOUGLAS LUNDHOLM AND VIKTOR QVARFORDT Abstract. We review and develop the many-body spectral theory of ideal anyons, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on N strands. Allowing for arbitrary rank (dependent on N) and non-abelian representations, and letting N ! 1, this defines the ideal non-abelian many-anyon gas. We compute exchange operators and phases for a common and wide class of representations defined by fusion algebras, including the Fibonacci and Ising anyon models. Furthermore, we extend methods of statistical repulsion (Poincar´eand Hardy inequalities) and a local exclusion principle (also implying a Lieb{Thirring inequality) developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequences of unitary representations of the braid group, for which two-anyon exchange is nontrivial. Contents 1. Introduction2 1.1. Anyons | abelian vs. non-abelian3 1.2. Some mathematically rigorous results for abelian anyons5 1.3. Main new results8 2. Algebraic anyon models 11 2.1. Fusion 11 2.2. Braiding 14 2.3. Pentagon and hexagon equations 15 2.4. Vacuum, bosons and fermions 17 2.5. Abelian anyons 18 2.6. Fibonacci anyons 18 2.7. Ising anyons 20 3. Exchange operators and phases 21 3.1. Braid group representations 22 3.2. Exchange operators 25 3.3. Abelian anyons 26 3.4. Fibonacci anyons 27 3.5. Ising anyons 29 arXiv:2009.12709v1 [math-ph] 26 Sep 2020 3.6. Clifford anyons 32 3.7.
    [Show full text]
  • Annual Report 2006
    ANNUAL REPORT 2006 Dutch Research School of Theoretical Physics (DRSTP) Landelijke Onderzoekschool voor Theoretische Natuurkunde (LOTN) Visiting address: Minnaert building Leuvenlaan 4 3584 CE Utrecht Postal address: P.O. Box 80.195 3508 TD Utrecht the Netherlands tel.: +31 (0)30 2535916 fax: +31 (0)30 2535937 e-mail: [email protected] website: http://www1.phys.uu.nl/drstp/ Preface This is the Annual Report 2006 of the Dutch Research School of Theoretical Physics (DRSTP). It provides an overview of the educational and research activities during 2006, intended for a broad spectrum of interested parties. The report also presents two research highlights written by staff members of the Research School. In addition, it offers a wealth of factual information, such as a list of the participating staff, of the PhD students, a comprehensive list of publications, as well as other relevant statistics. The annual report is not the only information that we make available throughout the year. We also publish a monthly newsletter and a yearly guide of our educational activities. Up-to-date information on the DRSTP is also readily available on internet at: http://www1.phys.uu.nl/drstp/. Finally, we should like to thank all of those who contributed to the Research School during this past year. prof. dr. B. de Wit prof. dr. K. Schoutens Scientific director Chair governing board August 2007 Contents 1 The DRSTP in 2006 7 2 Scientific highlights 11 3 PhD programme 19 3.1 Educational programme . 19 3.1.1 DRSTP postgraduate courses (AIO/OIO schools) . 19 3.1.2 Guest lecturers .
    [Show full text]
  • Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation
    DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2017 Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation VIKTOR QVARFORDT KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation VIKTOR QVARFORDT Degree Projects in Mathematics (30 ECTS credits) Degree Programme in Mathematics (120 credits) KTH Royal Institute of Technology year 2017 Supervisor at KTH: Douglas Lundholm Examiner at KTH: Kristian Bjerklöv TRITA-MAT-E 2017:17 ISRN-KTH/MAT/E--17/17--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored.
    [Show full text]
  • Thermodynamics of Non-Abelian Exclusion Statistics
    THERMODYNAMICS OF NON-ABELIAN EXCLUSION STATISTICS Wung-Hong Huang Department of Physics National Cheng Kung University Tainan,70101,Taiwan The thermodynamic potential of ideal gases described by the simplest non-abelian statistics is investigated. I show that the potential is the linear function of the element of the abelian- part statistics matrix. Thus, the factorizable property in the Haldane (abelian) fractional exclusion shown by the author [W. H. Huang, Phys. Rev. Lett. 81, 2392 (1998)] is now extended to the non-abelian case. The complete expansion of the thermodynamic potential is also given. Keywords: fractional exclusion statistics; quantum Hall effect. Classification Number: 05.30.-d, 71.10.+x E-mail: [email protected] arXiv:hep-th/0008153v1 19 Aug 2000 1 I. INTRODUCTION It is know that the quasiparticle appearing in the strong correlation system in the low dimension may obeying the Haldane fractional exclusion statistics [1]. The famous examples are those in the fractional quantized Hall effect and spin 1/2 antiferromagnetic chain [2-4]. The thermodynamics in these system had been investigated by several authors [5-8]. Since the braid group in two space dimensions may be represented by the non-commute matrix, the non-abelian braid statistics could be shown in a real world [9-16]. These include the quasiparticle in the non-abelian fractional quantum Hall state as well as those in the conformal field theory [9-15]. The spinon in the Heisenberg chains could also show the non-abelian exchange statistics [16]. In a recent paper [15], Guruswamy and Schoutens had derived the occupation number distribution functions for the particle obeying the non-abelian statistics.
    [Show full text]
  • The Quantum Geometry of Spin and Statistics
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server The Quantum Geometry of Spin and Statistics Robert Oeckl∗ Centre de Physique Th´eorique, CNRS Luminy, 13288 Marseille, France and Department of Applied Mathematics and Mathematical Physics, University of Cambridge, Cambridge CB3 0WA, UK DAMTP-2000-81 27 January 2001 Abstract Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin- statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose-Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the un- derlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach. MSC: 81R50, 81S05, 81S40, 81T18 keywords: quantum groups, statistics, spin, quantum field theory, anyons, quons ∗email: [email protected] 1 1 Introduction While spin in quantum physics arises from the geometry of space-time, statistics is connected to the geometry of configuration space. Half-integer spin and Bose-Fermi statistics arise in 3 or higher dimensions, while in 2 dimensions more general fractional spin and anyonic statistics are possible (see [?] and references therein). In fact, both, spin and statistics are related to symmetries. In the case of spin this is plainly understood in terms of the symmetries of space-time.
    [Show full text]
  • Quantum Computing and Quantum Topology
    Topological Quantum Computation Zhenghan Wang Microsoft Station Q & UC Sana Barbara Texas, March 26, 2015 Classical Physics Turing Model Quantum field computing is the same as quantum computing. Quantum Mechanics Quantum Computing Quantum Field Theory ??? True for TQFTs String Theory ?????? (Freedman, Kitaev, Larsen, W.) Quantum Computation • There is a serious prospect for quantum physics to change the face of information science. • Theoretically, the story is quite compelling: – Shor’s factoring algorithm (1994) – Fault tolerance ~1996-1997 independently • P. Shor • A. M. Steane • A. Kitaev • But for the last twenty years the most interesting progress has been to build a quantum computer. Why Quantum More Powerful? • Superposition • Entanglement A (classical) bit is given by a Quantum states need not be physical system that can exist products. For example: in one of two distinct states: 0 or 1 A qubit is given by a physical system that can exist in a linear combination of two distinct quantum states: or This is the property that enables quantum state teleportation and Einstein’s “spooky action at a distance.” |휓 > ∈ 퐶푃1 ● Classical information source is modeled by a random variable X The bit---a random variable X {0,1} with equal probability. Physically, it is a switch I (p)= - n p log p , X i=1 i 2 i 1 ● A state of a quantum system is an information source The qubit---a quantum system whose states given by non-zero vectors in ℂퟐ up to non-zero scalars. Physically, it is a 2-level quantum system. Paradox: A qubit contains both more and less than 1 bit of information.
    [Show full text]
  • Arxiv:0808.0627V1 [Cond-Mat.Mes-Hall] 5 Aug 2008 Representations of the Symmetry Group
    Condensate induced transitions between topologically ordered phases F. A. Bais1, 2 and J.K.Slingerland3 1Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands 2Santa Fe Institute, Santa Fe, NM 87501, USA∗ 3Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Rd, Dublin, Ireland.y (Dated: May 28, 2018) We investigate transitions between topologically ordered phases in two spatial dimensions induced by the condensation of a bosonic quasiparticle. To this end, we formulate an extension of the theory of symmetry breaking phase transitions which applies to phases with topological excitations described by quantum groups or modular tensor categories. This enables us to deal with phases whose quasiparticles have non-integer quantum dimensions and obey braid statistics. Many examples of such phases can be constructed from two-dimensional rational conformal field theories and we find that there is a beautiful connection between quantum group sym- metry breaking and certain well-known constructions in conformal field theory, notably the coset construction, the construction of orbifold models and more general conformal extensions. Besides the general framework, many representative examples are worked out in detail. PACS numbers: 05.30.Pr,11.25.Hf. I. INTRODUCTION gously, traditional phases can be distinguished by the expecta- tion values of local order parameters, while different topolog- In both high energy and condensed matter physics, there is ical phases
    [Show full text]