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23
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 30 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
The universal chiral partition function for exclusion statistics
, 1998
"... We demonstrate the equality between the universal chiral partition function, which was first found in the context of conformal field theory and RogersRamanujan identities, and the exclusion statistics introduced by Haldane in the study of the fractional quantum Hall effect. The phenomena of multipl ..."
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We demonstrate the equality between the universal chiral partition function, which was first found in the context of conformal field theory and RogersRamanujan identities, and the exclusion statistics introduced by Haldane in the study of the fractional quantum Hall effect. The phenomena of multiple representations of the same conformal field theory by different sets of exclusion statistics is discussed in the context of the û(1) theory of a compactified boson of radius R.
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 13 (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,
Rapidly rotating atomic gases
 Advances in Physics, 57:539–616
, 2008
"... This article reviews developments in the theory of rapidly rotating degenerate atomic gases. The main focus is on the equilibrium properties of a single component atomic Bose gas, which (at least at rest) forms a BoseEinstein condensate. Rotation leads to the formation of quantized vortices which o ..."
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This article reviews developments in the theory of rapidly rotating degenerate atomic gases. The main focus is on the equilibrium properties of a single component atomic Bose gas, which (at least at rest) forms a BoseEinstein condensate. Rotation leads to the formation of quantized vortices which order into a vortex array, in close analogy with the behaviour of superfluid helium. Under conditions of rapid rotation, when the vortex density becomes large, atomic Bose gases offer the possibility to explore the physics of quantized vortices in novel parameter regimes. First, there is an interesting regime in which the vortices become sufficiently dense that their cores – as set by the healing length – start to overlap. In this regime, the theoretical description simplifies, allowing a reduction to single particle states in the lowest Landau level. Second, one can envisage entering a regime of very high vortex density, when the number of vortices becomes comparable to the number of particles in the gas. In this regime, theory predicts the appearance of a series of strongly correlated phases, which can be viewed as bosonic versions of fractional quantum Hall states. This article describes the
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 7 (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
NonAbelian Anyons and Interferometry
 58 L. Zehnder, Ein neuer interferenzrefractor, Zeitschr. f. Instrkde
, 2007
"... To all my teachers, especially the three who have been with me from the very beginning: my parents, Mahrokh and Loren, and my sister, Roxana. iv Acknowledgments First and foremost, I would like to thank the members of my thesis defense committee: my advisor John Preskill for his guidance and support ..."
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To all my teachers, especially the three who have been with me from the very beginning: my parents, Mahrokh and Loren, and my sister, Roxana. iv Acknowledgments First and foremost, I would like to thank the members of my thesis defense committee: my advisor John Preskill for his guidance and support, and for giving me a chance and pointing me in the right direction when I was lost; Alexei Kitaev for providing inspiring and enlightening discussions; Kirill Shtengel for taking me under his wing and for all the help and advice he has given me; and NaiChang Yeh for her endless encouragement and enthusiasm. Also, I thank John Schwarz for his efforts and understanding during his time spent as my initial advisor at Caltech. I would like to recognize the hard work and affability of the Caltech staff, especially Donna Driscoll, Ann Harvey, and Carol Silberstein. I have had the pleasure and benefit of discussing physics, mathematics, and other interesting topics with Eddy Ardonne,
On a new universal class of phase transitions and quantum Hall effect
"... We study the possible phase transitions between (2+1)dimensional abelian ChernSimons theories. We show that they may be described by nonunitary rational conformal field theories with ceff = 1. As an example we choose the fractional quantum Hall effect and derive all its main features from the disc ..."
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We study the possible phase transitions between (2+1)dimensional abelian ChernSimons theories. We show that they may be described by nonunitary rational conformal field theories with ceff = 1. As an example we choose the fractional quantum Hall effect and derive all its main features from the discrete fractal structure of the moduli space of these nonunitary transition conformal field theories and some large scale principles. Rationality of these theories is intimately related to the modular group yielding severe conditions on the possible phase transitions. This gives a natural explanation for both, the values and the widths, of the observed fractional Hall plateaux.
Jacobson generators, Fock representations and statistics of sl(n
, 2000
"... The properties of Astatistics, related to the class of simple Lie algebras sl(n + 1), n ∈ Z+ (Palev, T.D.: Preprint JINR E1710550 (1977); hepth/9705032), are further investigated. The description of each sl(n + 1) is carried out via generators and their relations (see eq. (2.5)), first introduced ..."
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The properties of Astatistics, related to the class of simple Lie algebras sl(n + 1), n ∈ Z+ (Palev, T.D.: Preprint JINR E1710550 (1977); hepth/9705032), are further investigated. The description of each sl(n + 1) is carried out via generators and their relations (see eq. (2.5)), first introduced by Jacobson. The related Fock spaces Wp, p ∈ N, are finitedimensional irreducible sl(n + 1)modules. The Pauli principle of the underlying statistics is formulated. In addition the paper contains the following new results: (a) The Astatistics are interpreted as exclusion statistics; (b) Within each Wp operators B(p) ± 1,...,B(p) ± n, proportional to the Jacobson generators, are introduced. It is proved that in an appropriate topology (Definition 2) lim p→∞ B(p) ± i = B ± i, where B ± i are Bose creation and annihilation operators; (c) It is shown that the local statistics of the degenerated hardcore Bose models and of the related Heisenberg spin models is p = 1 Astatistics. 1
Statistical Phases and Momentum Spacings for One Dimensional Anyons
, 707
"... Anyons and fractional statistics1, 2 are by now well established in twodimensional systems. In one dimension, fractional statistics has been established so far only through Haldane’s fractional exclusion principle3, but not via a fractional phase the wave function acquires as particles are intercha ..."
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Anyons and fractional statistics1, 2 are by now well established in twodimensional systems. In one dimension, fractional statistics has been established so far only through Haldane’s fractional exclusion principle3, but not via a fractional phase the wave function acquires as particles are interchanged. At first sight, the topology of the configuration space appears to preclude such phases in one dimension. Here we argue that the crossings of onedimensional anyons are always unidirectional, which makes it possible to assign phases consistently and hence to introduce a statistical parameter θ. The fractional statistics then manifests itself in fractional spacings of the singleparticle momenta of the anyons when periodic boundary conditions are imposed. These spacings are given by ∆p = 2π�/L (θ/π + nonnegative integer) for a system of length L. This condition is the analogue of the quantisation of relative angular momenta according to lz = �(−θ/π + 2 · integer) for twodimensional anyons. The concept of fractional statistics, as introduced by Leinaas and Myrheim4 and Wilczek5, has generically been associated with identical particles in two space dimensions. It is intimately related to the topology of the configuration space, or the existence of fractional relative angular momentum. Angular momentum does not exist in one dimension (1D), and is quantised in units