Papers by Jean Bellissard
Journal of Mathematical Physics, Dec 1, 2020
This material is based upon work supported by the National Science Foundation
Annals of Physics, 2016
The current-current correlation function is a useful concept in the theory of electron transport ... more The current-current correlation function is a useful concept in the theory of electron transport in homogeneous solids. The finite-temperature conductivity tensor as well as Anderson's localization length can be computed entirely from this correlation function. Based on the critical behavior of these two physical quantities near the plateau-insulator or plateau-plateau transitions in the integer quantum Hall effect, we derive an asymptotic formula for the current-current correlation function, which enables us to make several theoretical predictions about its generic behavior. For the disordered Hofstadter model, we employ numerical simulations to map the current-current correlation function, obtain its asymptotic form near a critical point and confirm the theoretical predictions.
Journal of Mathematical Physics, 1994
We give an overview of the Integer Quantum Hall Effect. We propose a mathematical framework using... more We give an overview of the Integer Quantum Hall Effect. We propose a mathematical framework using Non-Commutative Geometry as defined by A. Connes. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states.
Abstract. Let (C, d) be an ultrametric Cantor set. Then it admits an isometric embedding into an ... more Abstract. Let (C, d) be an ultrametric Cantor set. Then it admits an isometric embedding into an infinite dimensional Euclidean space [30]. Associated with it is a weighted rooted tree, the reduced Michon graph T [28]. It will be called f-embeddable if there is a bi-Lipshitz map from (C, d) into a finite dimensional Euclidean space. The main result establishes that (C, d) is f-embeddable if and only if it can be represented by a weighted Michon tree such that (i) the number of children per vertex is uniformly bounded, (ii) if κ denotes the weight, there are constants c> 0 and 0 < δ < 1 such that κ(v)/κ(u) ≤ c δd(u,v) where v is a descendant of u and where d(u, v) denotes the graph distance between the vertices u, v. Several examples are provided: (a) the tiling space of a linear repetitive sequence is f-embeddable, (b) the tiling space of a Sturmian sequence is f-embeddable if and only if the irrational number characterizing it has bounded type, (c) the boundary of a Galton...
arXiv: Operator Algebras, 2010
Let (A,H,D) be a spectral triple, namely: A is a C^*-algebra, H is a Hilbert space on which A act... more Let (A,H,D) be a spectral triple, namely: A is a C^*-algebra, H is a Hilbert space on which A acts and D is a selfadjoint operator with compact resolvent such that the set of elements of A having a bounded commutator with D is dense. A spectral metric space, the noncommutative analog of a complete metric space, is a spectral triple (A,H,D) with additional properties which guaranty that the Connes metric induces the weak^*-topology on the state space of A. A ^*-automorphism respecting the metric defined a dynamical system. This article gives various answers to the question: is there a canonical spectral triple based upon the crossed product algebra Ax_αZ, characterizing the metric properties of the dynamical system ? If α is the noncommutative analog of an isometry the answer is yes. Otherwise, the metric bundle construction of Connes and Moscovici is used to replace (A,α) by an equivalent dynamical system acting isometrically. The difficulties relating to the non compactness of this...
Given a compact metrizable space X a Cantorizarion is a continuous surjective map from the Cantor... more Given a compact metrizable space X a Cantorizarion is a continuous surjective map from the Cantor set onto X. If such a Cantorization is non degenerate, it gives rise to the concept of reconnection cohomology. Such a cohomology is defined and is shown to be isomorphic to the Čech cohomology of X.
arXiv: Quantum Physics, 2017
This comment about the article "Hamiltonian for the Zeros of the Riemann Zeta Function"... more This comment about the article "Hamiltonian for the Zeros of the Riemann Zeta Function", by C. M. Bender, D. C. Brody, and M. P. M\"uller, published recently in Phys. Rev. Lett. (Phys. Rev. Lett., 118, 130201, (2017)) gives arguments showing that the strategy proposed by the authors to prove the Riemann Hypothesis, does not actually work.
arXiv: Combinatorics, 2007
Given a square matrix with elements in the group-ring of a group, one can consider the sequence f... more Given a square matrix with elements in the group-ring of a group, one can consider the sequence formed by the trace (in the sense of the group-ring) of its powers. We prove that the corresponding generating series is an algebraic G-function (in the sense of Siegel) when the group is free of finite rank. Consequently, it follows that the norm of such elements is an exactly computable algebraic number, and their Green function is algebraic. Our proof uses the notion of rational and algebraic power series in non-commuting variables and is an easy application of a theorem of Haiman. Haiman’s theorem uses results of linguistics regarding regular and context-free language. On the other hand, when the group is free abelian of finite rank, then the corresponding generating series is a G-function. We ask whether the latter holds for general hyperbolic groups.
Reminder: an aperiodic solid can be described as follows: let G C Rd be the discrete set of equil... more Reminder: an aperiodic solid can be described as follows: let G C Rd be the discrete set of equilibrium positions of its atoms in the real space in which the solid is setting. The Hull of G is obtained as the closure of the family IL + a; a e Rd } of translated of the solid. The topology used here is the following: a subsequence of discrete subsets converges if and only if their intersections with any bounded open set of R d converges for the Hausdorff distance. If the initial set of atoms is uniformly discrete, the Hull is a compact space Q endowed with an action of R d by homeomorphisms. Each point in the Hull is itself a discrete subset of R d . This dynamical system has a canonical transversal made of points such that the corresponding discrete subset contains the origin in R d . This transversal is called the atomic surface by physicist studying quasicrystals. Such a Hull can be described through various points of view: as a dynamical system as previously, as a groupoid (of the...
arXiv: Spectral Theory, 2018
The existence and construction of periodic approximations with convergent spectra is crucial in s... more The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schr\"odinger operators. In a forthcoming work [9] (arXiv:1709.00975) this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.
1.1.1. Periodic Media. In Condensed Matter Physics, the basic tool to describe models and to perf... more 1.1.1. Periodic Media. In Condensed Matter Physics, the basic tool to describe models and to perform calculations is Bloch’s Theory [13]. The main ingredient is to use the invariance of the Hamiltonian under the action of the translation group and to diagonalize simultaneously the Hamiltonian and the unitary group representing the translations. Since the translation group (either Rd or Zd, with d = 1, 2, 3 in practice) is both Abelian and locally compact, the diagonalization of the unitaries representing it can be done through its group of characters (Pontryagin dual) known as the Brillouin zone [16] in Condensed Matter theory and it will be denoted here by B. If the focus of attention is put on the one-electron motion, then for each quasi-momentum k ∈ B, there is a self-adjoint Hamiltonian H(k) = H(k)† which is mostly a matrix, either finite dimensional, when the energy is restricted to a neighborhood of the Fermi level (in the so called tight-binding representation), or infinite d...
A simplistic model, based on the concept of anankeon, is proposed to predict the value of the vis... more A simplistic model, based on the concept of anankeon, is proposed to predict the value of the viscosity of a material in its liquid phase. As a result, within the simplifications and hypothesis made to define the model, it is possible to predict (a) the existence of a difference between strong and fragile liquids, (b) a fast variation of the viscosity near the glass transition temperature for fragile liquids. The model is a mechanical analog of the Drude model for the electric transport in metal.
An ultrametric Cantor set can be seen as the boundary of a rooted weighted tree called the Michon... more An ultrametric Cantor set can be seen as the boundary of a rooted weighted tree called the Michon tree. The notion of Assouad dimension is re-interpreted as seen on the Michon tree. The Assouad dimension of an ultrametric Cantor set is finite if and only if the space is bi-Lipschitz embeddable in a finite dimensional Euclidean space. This result, due to Assouad and refined by Luukkainen--Movahedi-Lankarani is re-proved in the Michon tree formalism. It is applied to answer the embedding question for some spaces which can be seen naturally as boundary of trees: linearly repetitive subshifts, Sturmian subshifts, and the boundary of Galton--Watson trees with random weights. Some of these give examples of nonembeddable spaces with finite Hausdorff dimension.
Journal of Functional Analysis
A correct statement of Theorem 4 in [1] is provided. The change does not affect the main results.... more A correct statement of Theorem 4 in [1] is provided. The change does not affect the main results. As stated [1, Theorem 4] is not yet proved (nor disproved). The correct version is: Theorem 4.([6]) If a second countable locally compact Hausdorff groupoid admits a leftcontinuous Haar system, then it is open, i.e., the range map is an open map. The proof was provided in [1, Theorem 4], which is due to Westman [6]. The converse direction is a conjecture [3, Conjecture 3.3.]: Conjecture.([3]) If a second countable locally compact Hausdorff groupoid is open then it admits a left-continuous Haar system. A left-continuous Haar system is a family of Borel measures (µ x) x∈Γ (0) on the groupoid satisfying the conditions (H1), (H2) and (H3), see [1, Definition 5]. Blanchard [2] proved that if a second countable locally compact Hausdorff groupoid Γ is open, then it admits a family of Borel measures (µ x) x∈Γ (0) satisfying (H1) and (H2), see also [7, Lemma 2.3]. However, it remains an open question to prove (or disprove) that also (H3) holds. It is worth pointing out that all the results contained in [1] remain valid provided that [1, Definition 4] is replaced with: Definition 4. A topological groupoid will be called handy whenever: (i) it is locally compact, Hausdorff, second countable, (ii) its unit set is compact, (iii) it admits a left-continuous Haar system.
Mathematics of Aperiodic Order, 2015
These notes are proposing a program liable to provide physicists working in material science, esp... more These notes are proposing a program liable to provide physicists working in material science, especially metallic liquids and glasses, the mathematical tools they need to build an atomic scale theory of Continuous Mechanics including plasticity, fluidity and, hopefully, fractures. Using the long list of datas and numerical simulations accumulated during the last forty years, physicists have identified a new class of degrees of freedom, besides the elastic ones, which will be called anankeons here [7]. They are dominant in the liquid phase and they explain the properties related to plastic deformations of the solid phase. It is advocated that Delone sets provide a natural frame within which such a theory can be expressed. The use of Voronoi tiling and its dual construction, called the Delaunay triangulation, gives a discretization of the data. The concept of Pachner move or Delaunay flips permits to describe very precisely what the anankeons are. A partition of the configuration space into contiguity domains leads to a graph on which a Markov process can be built to describe the anakeon dynamics. At last, a speculative Section is giving an attempt to describe the Continuous Mechanics of a condensed material in terms of a Noncommutative Geometry of the configuration space.
Annales Henri Poincaré
We study the spectral location of a strongly pattern equivariant Hamiltonians arising through con... more We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.
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Papers by Jean Bellissard