The Noncommutative Geometry of Aperiodic Solids

Bellissard, Jean

doi:10.1142/9789812705068_0002

Cited by 41 publications

(43 citation statements)

References 99 publications

(134 reference statements)

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“…An unbounded, selfadjoint, covariant operator, such as H ω , is said to be affiliated to this algebra if all its spectral projections for bounded Borel subsets of R belong to this algebra. There is a canonical trace on this algebra given by [5,9] …”

Section: The Models Hypotheses and The Main Resultsmentioning

confidence: 99%

Smoothness of Correlations in the Anderson Model at Strong Disorder

Bellissard

Hislop

2006

Ann. Henri Poincaré

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We study the higher-order correlation functions of covariant families of observables associated with random Schrödinger operators on the lattice in the strong disorder regime. We prove that if the distribution of the random variables has a density analytic in a strip about the real axis, then these correlation functions are analytic functions of the energy outside of the planes corresponding to coincident energies. In particular, this implies the analyticity of the density of states, and of the current-current correlation function outside of the diagonal. Consequently, this proves that the current-current correlation function has an analytic density outside of the diagonal at strong disorder.

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Section: The Models Hypotheses and The Main Resultsmentioning

confidence: 99%

Smoothness of Correlations in the Anderson Model at Strong Disorder

Bellissard

Hislop

2006

Ann. Henri Poincaré

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“…The resulting non-commutative space is called the non-commutative Brillouin torus. The notion of noncommutative Brillouin torus (not to be confused with the non-commutative torus) was introduced in the inspiring work of Jean Bellissard [20], who developed an entire noncommutative geometry program for aperiodic solids [21]. The non-commutative Brillouin torus is a true gift to the condensed matter physics, since it enables one to de ne an equivalent Bloch-Floquet calculus for homogeneous aperiodic systems.…”

Section: The Non-commutative Brillouin Torusmentioning

confidence: 99%

“…The construction of the non-commutative Brillouin torus for the homogenous models discussed above is standard and is described next [21].…”

Section: Homogeneous Aperiodic Lattice Modelsmentioning

confidence: 99%

The Non-Commutative Geometry of the Complex Classes of Topological Insulators

Prodan¹

2014

Topological Quantum Matter

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Alain Connes' Non-Commutative Geometry program [1] has been recently carried out [2,3] for the entire A-and AIII-symmetry classes of topological insulators, in the regime of strong disorder where the insulating gap is completely lled with dense localized spectrum. This is a short overview of these results, whose goal is to highlight the methods of Non-Commutative Geometry involved in these studies. The exposition proceeds gradually through the cyclic cohomology, quantized calculus with Fredholm-modules, local formulas for the odd and even Chern characters and index theorems for the odd and even Chern numbers. The characterization of the A-and AIIIsymmetry classes in the presence of strong disorder and magnetic elds emerges as a natural application of these tools.

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“…Ces deux observations peuvent être illustrées plus précisément, par exemple par les théorèmes de Ghys [20] et CantwellConlon [5] sur la topologie des feuilles d'une feuilletage de dimension 2. Il est également intéressant de noter, réciproquement, qu'un espace métrique possédant certaines propriétés de quasi-périodicité peut parfois être plongé dans une lamination minimale, ou ergodique, de façon interessante ; pour un exemple concret d'une telle construction, nous renvoyons le lecteur à l'étude des quasi-cristaux telle qu'exposée dans [4] par exemple.…”

Section: Annales De L'institut Fourierunclassified

Espaces mesurés singuliers fortement ergodiques (Étude métrique–mesurée)

Pichot¹

2007

Ann. inst. Fourier

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The Noncommutative Geometry of Aperiodic Solids

Cited by 41 publications

References 99 publications

Smoothness of Correlations in the Anderson Model at Strong Disorder

Smoothness of Correlations in the Anderson Model at Strong Disorder

The Non-Commutative Geometry of the Complex Classes of Topological Insulators

Espaces mesurés singuliers fortement ergodiques (Étude métrique–mesurée)

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