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On exotic modular tensor categories
 Commun. Contemp. Math
"... Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a tot ..."
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Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 nontrivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 nontrivial prime UMTCs are given in Section 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Section 6. 1.
Nonabelian anyons and topological quantum computation
 Reviews of Modern Physics
"... Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are partic ..."
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Cited by 10 (0 self)
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Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as NonAbelian anyons, meaning that they obey nonAbelian braiding statistics. Quantum information is stored in states with multiple quasiparticles,
NonAbelian Anyons and Topological Quantum Computation
, 2007
"... Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles know ..."
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Cited by 6 (1 self)
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Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as NonAbelian anyons, meaning that they obey nonAbelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which
ARTICLES Fractional quantum Hall effect in a quantum point contact at filling fraction 5/2
, 2007
"... Recent theories suggest that the quasiparticles that populate certain quantum Hall states should exhibit exotic braiding statistics that could be used to build topological quantum gates. Confined systems that support such states at a filling fraction ν = 5/2 are of particular interest for testing th ..."
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Recent theories suggest that the quasiparticles that populate certain quantum Hall states should exhibit exotic braiding statistics that could be used to build topological quantum gates. Confined systems that support such states at a filling fraction ν = 5/2 are of particular interest for testing these predictions. Here, we report transport measurements of just such a system, which consists of a quantum point contact (QPC) in a twodimensional GaAs/AlGaAs electron gas that itself exhibits a welldeveloped fractional quantum Hall effect at a bulk filling fraction ν bulk = 5/2. We observe plateaulike features at an effective filling fraction of ν QPC = 5/2 for lithographic contact widths of 1.2 µm and 0.8 µm, but not 0.5 µm. Transport near ν QPC = 5/2 in the QPCs is consistent with a picture of chiral Luttingerliquid edge states with interedge tunnelling, suggesting that an incompressible state at ν QPC = 5/2 forms in this confined geometry. The discovery 1 of a fractional quantum Hall effect (FQHE) at the evendenominator filling fraction ν = 5/2 has sparked a series of experimental 2–6 and theoretical 7–9 studies, leading to a prevailing interpretation of the 5/2 state as comprising paired fermions condensed into a Bardeen–Cooper–Schriefferlike state 10–13. Within this picture, excitations of the 5/2 ground state possess nonabelian statistics 14–16 and associated topological properties. The possibility
Periodic table for topological insulators and superconductors
, 901
"... Abstract. Gapped phases of noninteracting fermions, with and without charge conservation and timereversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of re ..."
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Abstract. Gapped phases of noninteracting fermions, with and without charge conservation and timereversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of Khomology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the Ktheoretic classification is stable to interactions, but a counterexample is also given. Keywords: Topological phase, Ktheory, Khomology, Clifford algebra, Bott periodicity
Fermipoint scenario for emergent gravity”, Proceedings of From Quantum to Emergent Gravity: Theory and Phenomenology
"... Let us assume that gravity is an emergent lowenergy phenomenon arising from the topologically stable defect in momentum space – the Fermi point. What are the consequences? We discuss the natural values of fermion masses and cosmological constant; flatness of the Universe; bounds on the Lorentz viol ..."
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Let us assume that gravity is an emergent lowenergy phenomenon arising from the topologically stable defect in momentum space – the Fermi point. What are the consequences? We discuss the natural values of fermion masses and cosmological constant; flatness of the Universe; bounds on the Lorentz violation; etc.
Some nonbraided fusion categories of rank 3, arXiv: 0704.0208
"... Abstract. We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, c ..."
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Abstract. We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary reproof of the fact that the quadruple dual functor is naturally isomorphic to the identity. 1.
LOCALIZATION OF UNITARY BRAID GROUP REPRESENTATIONS
"... Abstract. Governed by locality, we explore a connection between unitary braid group representations associated to a unitary Rmatrix and to a simple object in a unitary braided fusion category. Unitary Rmatrices, namely unitary solutions to the YangBaxter equation, afford explicitly local unitary ..."
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Abstract. Governed by locality, we explore a connection between unitary braid group representations associated to a unitary Rmatrix and to a simple object in a unitary braided fusion category. Unitary Rmatrices, namely unitary solutions to the YangBaxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary Rmatrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 Rmatrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N> 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science. 1.
Skein Theory and Topological Quantum Registers: Braiding Matrices and Topological Entanglement Entropy of NonAbelian Quantum Hall states,” arXiv:0709.2409
"... ABSTRACT. We study topological properties of quasiparticle states in the nonAbelian quantum Hall states. We apply a skeintheoretic method to the Read–Rezayi state whose effective theory is the SU(2)K Chern–Simons theory. As a generalization of the Pfaffian (K = 2) and the Fibonacci (K = 3) anyon ..."
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ABSTRACT. We study topological properties of quasiparticle states in the nonAbelian quantum Hall states. We apply a skeintheoretic method to the Read–Rezayi state whose effective theory is the SU(2)K Chern–Simons theory. As a generalization of the Pfaffian (K = 2) and the Fibonacci (K = 3) anyon states, we compute the braiding matrices of quasiparticle states with arbitrary spins. Furthermore we propose a method to compute the entanglement entropy skeintheoretically. We find that the entanglement entropy has a nontrivial contribution called the topological entanglement entropy which depends on the quantum dimension of nonAbelian quasiparticle intertwining two subsystems. 1.