Anyons: Field Theory and Applications
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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. -
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. -
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. -
Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation -
Arxiv:1709.02742V3
PIN GROUPS IN GENERAL RELATIVITY BAS JANSSENS Abstract. There are eight possible Pin groups that can be used to describe the transformation behaviour of fermions under parity and time reversal. We show that only two of these are compatible with general relativity, in the sense that the configuration space of fermions coupled to gravity transforms appropriately under the space-time diffeomorphism group. 1. Introduction For bosons, the space-time transformation behaviour is governed by the Lorentz group O(3, 1), which comprises four connected components. Rotations and boosts are contained in the connected component of unity, the proper orthochronous Lorentz group SO↑(3, 1). Parity (P ) and time reversal (T ) are encoded in the other three connected components of the Lorentz group, the translates of SO↑(3, 1) by P , T and PT . For fermions, the space-time transformation behaviour is governed by a double cover of O(3, 1). Rotations and boosts are described by the unique simply connected double cover of SO↑(3, 1), the spin group Spin↑(3, 1). However, in order to account for parity and time reversal, one needs to extend this cover from SO↑(3, 1) to the full Lorentz group O(3, 1). This extension is by no means unique. There are no less than eight distinct double covers of O(3, 1) that agree with Spin↑(3, 1) over SO↑(3, 1). They are the Pin groups abc Pin , characterised by the property that the elements ΛP and ΛT covering P and 2 2 2 T satisfy ΛP = −a, ΛT = b and (ΛP ΛT ) = −c, where a, b and c are either 1 or −1 (cf. -
Algebraic Topology
Algebraic Topology Len Evens Rob Thompson Northwestern University City University of New York Contents Chapter 1. Introduction 5 1. Introduction 5 2. Point Set Topology, Brief Review 7 Chapter 2. Homotopy and the Fundamental Group 11 1. Homotopy 11 2. The Fundamental Group 12 3. Homotopy Equivalence 18 4. Categories and Functors 20 5. The fundamental group of S1 22 6. Some Applications 25 Chapter 3. Quotient Spaces and Covering Spaces 33 1. The Quotient Topology 33 2. Covering Spaces 40 3. Action of the Fundamental Group on Covering Spaces 44 4. Existence of Coverings and the Covering Group 48 5. Covering Groups 56 Chapter 4. Group Theory and the Seifert{Van Kampen Theorem 59 1. Some Group Theory 59 2. The Seifert{Van Kampen Theorem 66 Chapter 5. Manifolds and Surfaces 73 1. Manifolds and Surfaces 73 2. Outline of the Proof of the Classification Theorem 80 3. Some Remarks about Higher Dimensional Manifolds 83 4. An Introduction to Knot Theory 84 Chapter 6. Singular Homology 91 1. Homology, Introduction 91 2. Singular Homology 94 3. Properties of Singular Homology 100 4. The Exact Homology Sequence{ the Jill Clayburgh Lemma 109 5. Excision and Applications 116 6. Proof of the Excision Axiom 120 3 4 CONTENTS 7. Relation between π1 and H1 126 8. The Mayer-Vietoris Sequence 128 9. Some Important Applications 131 Chapter 7. Simplicial Complexes 137 1. Simplicial Complexes 137 2. Abstract Simplicial Complexes 141 3. Homology of Simplicial Complexes 143 4. The Relation of Simplicial to Singular Homology 147 5. Some Algebra. The Tensor Product 152 6. -
Special Unitary Group - Wikipedia
Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group). -
Group Theory
Group theory March 7, 2016 Nearly all of the central symmetries of modern physics are group symmetries, for simple a reason. If we imagine a transformation of our fields or coordinates, we can look at linear versions of those transformations. Such linear transformations may be represented by matrices, and therefore (as we shall see) even finite transformations may be given a matrix representation. But matrix multiplication has an important property: associativity. We get a group if we couple this property with three further simple observations: (1) we expect two transformations to combine in such a way as to give another allowed transformation, (2) the identity may always be regarded as a null transformation, and (3) any transformation that we can do we can also undo. These four properties (associativity, closure, identity, and inverses) are the defining properties of a group. 1 Finite groups Define: A group is a pair G = fS; ◦} where S is a set and ◦ is an operation mapping pairs of elements in S to elements in S (i.e., ◦ : S × S ! S: This implies closure) and satisfying the following conditions: 1. Existence of an identity: 9 e 2 S such that e ◦ a = a ◦ e = a; 8a 2 S: 2. Existence of inverses: 8 a 2 S; 9 a−1 2 S such that a ◦ a−1 = a−1 ◦ a = e: 3. Associativity: 8 a; b; c 2 S; a ◦ (b ◦ c) = (a ◦ b) ◦ c = a ◦ b ◦ c We consider several examples of groups. 1. The simplest group is the familiar boolean one with two elements S = f0; 1g where the operation ◦ is addition modulo two. -
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. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Shapes of Finite Groups through Covering Properties and Cayley Graphs Permalink https://escholarship.org/uc/item/09b4347b Author Yang, Yilong Publication Date 2017 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Shapes of Finite Groups through Covering Properties and Cayley Graphs A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Yilong Yang 2017 c Copyright by Yilong Yang 2017 ABSTRACT OF THE DISSERTATION Shapes of Finite Groups through Covering Properties and Cayley Graphs by Yilong Yang Doctor of Philosophy in Mathematics University of California, Los Angeles, 2017 Professor Terence Chi-Shen Tao, Chair This thesis is concerned with some asymptotic and geometric properties of finite groups. We shall present two major works with some applications. We present the first major work in Chapter 3 and its application in Chapter 4. We shall explore the how the expansions of many conjugacy classes is related to the representations of a group, and then focus on using this to characterize quasirandom groups. Then in Chapter 4 we shall apply these results in ultraproducts of certain quasirandom groups and in the Bohr compactification of topological groups. This work is published in the Journal of Group Theory [Yan16]. We present the second major work in Chapter 5 and 6. We shall use tools from number theory, combinatorics and geometry over finite fields to obtain an improved diameter bounds of finite simple groups. We also record the implications on spectral gap and mixing time on the Cayley graphs of these groups. -
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].