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31
Noncommutative geometry of tilings and gap labelling
 Rev. Math. Phys
, 1995
"... To a given tiling a non commutative space and the corresponding C ∗algebra are constructed. This includes the definition of a topology on the groupoid induced by translations of the tiling. The algebra is also the algebra of observables for discrete models of one or many particle systems on the til ..."
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Cited by 43 (12 self)
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To a given tiling a non commutative space and the corresponding C ∗algebra are constructed. This includes the definition of a topology on the groupoid induced by translations of the tiling. The algebra is also the algebra of observables for discrete models of one or many particle systems on the tiling or its periodic identification. Its scaled ordered K0group furnishes the gap labelling of Schrödinger operators. The group is computed for one dimensional tilings and Cartesian products thereof. Its image under a state is investigated for tilings which are invariant under a substitution. Part of this image is given by an invariant measure on the hull of the tiling which is determined. The results from the Cartesian products of one dimensional tilings point out that the gap labelling by means of the values of the integrated density of states is already fully determined by this measure.
Discrete KaluzaKlein from scalar fluctuations in noncommutative geometry
 hepth/0104108, J. Math. Phys
, 2002
"... We compute the metric associated to noncommutative spaces described by a tensor product of spectral triples. Wellknown results of the twosheets model (distance on a sheet, distance between the sheets) are extended to any product of two spectral triples. The distance between different points on dif ..."
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Cited by 16 (10 self)
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We compute the metric associated to noncommutative spaces described by a tensor product of spectral triples. Wellknown results of the twosheets model (distance on a sheet, distance between the sheets) are extended to any product of two spectral triples. The distance between different points on different fibres is investigated. When one of the triples describes a manifold, one finds a Pythagorean theorem as soon as the direct sum of the internal states (viewed as projections) commutes with the internal Dirac operator. Scalar fluctuations yield a discrete KaluzaKlein model in which the extra component of the metric is given by the internal part of the geometry. In the standard model, this extra component comes from the Higgs field. 1
Dimensional regularization and renormalization of noncommutative quantum field theory
, 2008
"... Using the recently introduced parametric representation of noncommutative quantum field theory, we implement here the dimensional regularization and renormalization of the vulcanized Φ ⋆4 4 model on the Moyal space. ..."
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Cited by 16 (5 self)
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Using the recently introduced parametric representation of noncommutative quantum field theory, we implement here the dimensional regularization and renormalization of the vulcanized Φ ⋆4 4 model on the Moyal space.
Algebras of Random Operators Associated to Delone Dynamical Systems
, 2003
"... We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain C∗subalgebra to discu ..."
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Cited by 8 (7 self)
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We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain C∗subalgebra to discuss a Shubin trace formula.
Geometry of Quantum Principal Bundles I
"... Abstract: A noncommutativegeometric generalization of the theory of principal bundles is sketched. A differential calculus over corresponding quantum principal bundles is analysed. The formalism of connections is presented. In particular, operators of covariant derivative and horizontal projection ..."
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Cited by 8 (0 self)
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Abstract: A noncommutativegeometric generalization of the theory of principal bundles is sketched. A differential calculus over corresponding quantum principal bundles is analysed. The formalism of connections is presented. In particular, operators of covariant derivative and horizontal projection are described and analysed. Quantum counterparts for the Bianchi identity and the Weil’s homomorphism are found. Illustrative examples are considered.
A WALK IN THE NONCOMMUTATIVE GARDEN
"... 2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9 ..."
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Cited by 7 (0 self)
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2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9
A Spectral Sequence for the Ktheory of Tiling Spaces
"... Let T be an aperiodic and repetitive tiling of R^d with finite local complexity. We present a spectral sequence that converges to the Ktheory of T with page2 isomorphic to the Pimsner cohomology of T. It is a generalization of the Serre spectral sequence to a class of spaces which are not fibered. ..."
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Cited by 4 (2 self)
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Let T be an aperiodic and repetitive tiling of R^d with finite local complexity. We present a spectral sequence that converges to the Ktheory of T with page2 isomorphic to the Pimsner cohomology of T. It is a generalization of the Serre spectral sequence to a class of spaces which are not fibered. The Pimsner cohomology of T generalizes the cohomology of the base space of a fibration with local coefficients in the Ktheory of its fiber. We prove that it is isomorphic to the Čech cohomology of the hull of T (the closure for an appropriate topology of the family of its translates).
Magnetic Fingerprints of Fractal Spectra and Duality of Hofstadter Models
, 2008
"... We study the deHaas van Alphen oscillations in the magnetization of the Hofstadter model. Near a split band the magnetization is a rapidly oscillating function of the Fermi energy with lip shaped envelopes. For generic magnetic fields this structure appears on all scales and provides a thermodynami ..."
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Cited by 4 (1 self)
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We study the deHaas van Alphen oscillations in the magnetization of the Hofstadter model. Near a split band the magnetization is a rapidly oscillating function of the Fermi energy with lip shaped envelopes. For generic magnetic fields this structure appears on all scales and provides a thermodynamic fingerprint of the fractal properties of the model. The analysis applies equally well to the two dual interpretations of the Hofstadter model and the nature of the duality transformation is elucidated. The Hofstadter model [1] describes noninteracting (spinless) electrons moving in the plane under the combined action of magnetic field and a periodic potential. The model is a paradigm for quantum systems with singular continuous spectra [2, 3]; Chern numbers in the integer quantum Hall effect [4]; Fractal quantum phase diagrams [5]; Quantum integrable models [6] and Noncommutative geometry [7]. It has been realized experimentally in [8] and measurements of the Hall conductance phase diagram are in good agreement with theoretical predictions of [4]. Previous studies of the thermodynamics of the Hofstadter model have focused on minima of the ground state energy [9]. Our aim is to examine the possibility that the de Haasvan Alphen oscillations, which are a basic tool in the study of the geometry of Fermi surfaces of metals [10], also provide a tool for studying its noncommutative analog. We show that when the magnetic flux per unit cell is close to rational multiple of the flux quantum, the magnetization oscillates as a function of the chemical potential in the vicinity of the bands of the rational flux, as in the de Hassvan Alphen effect. As the deviation from rational flux becomes smaller, the frequency of the oscillations increases, so that in the limit of the deviation tending to zero the oscillations fill an area bounded by a limiting envelope. The envelopes have the following universal features: They are smooth functions of the chemical potential except for logarithmic pinching at a single point, associated with a logarithmic divergence of