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# Cantor spectra of magnetic chain graphs

**Article**

*in*Journal of Physics A Mathematical and Theoretical 50(16) · November 2016

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Abstract

We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with $\delta$ coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.

- Article
- Apr 2018
- J PHYS A-MATH THEOR

The paper discusses quantum graphs with the vertex coupling which interpolates between the common one of the $\delta$ type and a coupling introduced recently by two of the authors which exhibits a preferred orientation. Describing the interpolation family in terms of circulant matrices, we analyze the spectral and scattering property of such vertices, and investigate the band spectrum of the corresponding square lattice graph. - We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux $\Phi/2\pi$ through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero.

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S.C. Power: Simplicity of C * -algebras of minimal dynamical systems, J. London Math. Soc., 3 (1978), 534–538. - Article
- Oct 2015
- COMMUN MATH PHYS

We study the almost Mathieu operator at critical coupling. We prove that there exists a dense $G_\delta$ set of frequencies for which the spectrum is of zero Hausdorff dimension. - Article
- Dec 2014

We analyze spectral properties of a quantum graph in the form of a ring chain with a $\delta$ coupling in the vertices exposed to a homogeneous magnetic field perpendicular to the graph plane. We find the band spectrum in the case when the chain exhibits a translational symmetry and study the discrete spectrum in the gaps resulting from changing a finite number of vertex coupling constants. In particular, we discuss in details some examples such as perturbations of one or two vertices, weak perturbation asymptotics, and a pair of distant perturbations. - Article
- Dec 1982

We review the recent rigorous literature on the one dimensional Schördinger equation, \(H = - \frac{{d^2 }}{{dx^2 }} + V(x)\)with V(x) al most periodic and the discrete (= tight binding) analogy, i. e. the doubly infinite Jacobi matrix, hij = σi,j+1 + σi,j−1 + viσi,j with vi almost periodic on the integers. Two themes dominate. The first is that the gaps in the spectrum tend to be dense so that the spectrum is a Cantor set. We describe intuitions for this from the point of view of where gaps open and from the point of view of anamalous long time behaviour. We give a theorem of Avron-Simn, Chulasvsky and Moser that for a generic sequence with Σ|an| < ∞, the continuum operator with V(x) = Σ an cos(x/2n) has a Cantor spectrum. The second theme involves unusual spectral types that tend to occur. We describe recurrent absolutely continuous spectrum and show it occurs in some examples of the type just discussed. We give an intuition for dense point spectrum to occur and some theorems on the occurende of point spectrum. We sketch the proof of Avron-Simon that for the discrete case with Vn = λcos(2παn + θ) if λ > 2 and α is a Lionville number, then for a.e. θ, h has purely singular continuous spectrum. - Azbel: Energy spectrum of a conduction electron in a magnetic field
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M.Ya. Azbel: Energy spectrum of a conduction electron in a magnetic field, J. Exp. Theor. Phys 19 (1964), 634–645. - Superlattices have attracted great interest because their use may make it possible to modify the spectra of two-dimensional electron systems and, ultimately, create materials with tailored electronic properties. In previous studies (see, for example, refs 1, 2, 3, 4, 5, 6, 7, 8), it proved difficult to realize superlattices with short periodicities and weak disorder, and most of their observed features could be explained in terms of cyclotron orbits commensurate with the superlattice. Evidence for the formation of superlattice minibands (forming a fractal spectrum known as Hofstadter's butterfly) has been limited to the observation of new low-field oscillations and an internal structure within Landau levels. Here we report transport properties of graphene placed on a boron nitride substrate and accurately aligned along its crystallographic directions. The substrate's moiré potential acts as a superlattice and leads to profound changes in the graphene's electronic spectrum. Second-generation Dirac points appear as pronounced peaks in resistivity, accompanied by reversal of the Hall effect. The latter indicates that the effective sign of the charge carriers changes within graphene's conduction and valence bands. Strong magnetic fields lead to Zak-type cloning of the third generation of Dirac points, which are observed as numerous neutrality points in fields where a unit fraction of the flux quantum pierces the superlattice unit cell. Graphene superlattices such as this one provide a way of studying the rich physics expected in incommensurable quantum systems and illustrate the possibility of controllably modifying the electronic spectra of two-dimensional atomic crystals by varying their crystallographic alignment within van der Waals heterostuctures.
- Electrons moving through a spatially periodic lattice potential develop a quantized energy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving through a magnetic field also develop a quantized energy spectrum, consisting of highly degenerate Landau energy levels. When subject to both a magnetic field and a periodic electrostatic potential, two-dimensional systems of electrons exhibit a self-similar recursive energy spectrum. Known as Hofstadter's butterfly, this complex spectrum results from an interplay between the characteristic lengths associated with the two quantizing fields, and is one of the first quantum fractals discovered in physics. In the decades since its prediction, experimental attempts to study this effect have been limited by difficulties in reconciling the two length scales. Typical atomic lattices (with periodicities of less than one nanometre) require unfeasibly large magnetic fields to reach the commensurability condition, and in artificially engineered structures (with periodicities greater than about 100 nanometres) the corresponding fields are too small to overcome disorder completely. Here we demonstrate that moiré superlattices arising in bilayer graphene coupled to hexagonal boron nitride provide a periodic modulation with ideal length scales of the order of ten nanometres, enabling unprecedented experimental access to the fractal spectrum. We confirm that quantum Hall features associated with the fractal gaps are described by two integer topological quantum numbers, and report evidence of their recursive structure. Observation of a Hofstadter spectrum in bilayer graphene means that it is possible to investigate emergent behaviour within a fractal energy landscape in a system with tunable internal degrees of freedom.
- Article
- Sep 1997
- MONATSH MATH

We study the spectrum of the continuous Laplacian Δ on a countable connected locally finite graph Γ without self-loops, whose edges have suitable positive conductances and are identified with copies of segments [0, 1], with the condition that the sum of the weighted normal exterior derivatives is 0 at every node (Kirchhoff-type condition). In particular, we analyse the relation-between the spectrum of the operator Δ and the spectrum of the discrete Laplacian (I - P) defined on the vertices of Γ. - Chapter
- May 2005

This review discusses some of the central developments in the spectral theory of Sturm-Liouville operators on infinite intervals over the last thirty years or so. We discuss some of the natural questions that occur in this framework and some of the main models that have been studied. - Newton's binomial theorem is extended to an interesting noncommutative setting as follows: If, in a ring,ba=γab with γ commuting witha andb, then the (generalized) binomial coefficient\(\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)_r \) arising in the expansion$$\left( {a + b} \right)^n = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} _\gamma a^{n - k} b^k $$ (resulting from these relations) is equal to the value at γ of the Gaussian polynomial$$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right] = \frac{{\left[ n \right]}}{{\left[ k \right]\left[ {n - k} \right]}}$$ where [m]=(1-x m )(1-x m−1)...(1-x). (This is of course known in the case γ=1.) From this it is deduced that in the (universal)C *-algebraA gq generated by unitariesu andv such thatvu=e 2πiθuv, the spectrum of the self-adjoint element (u+v)+(u+v)* has all the gaps that have been predicted to exist-provided that either θ is rational, or θ is a Liouville number. (In the latter case, the gaps are labelled in the natural way-viaK-theory-by the set of all non-zero integers, and the spectrum is a Cantor set.)
- In the present article magnetic Laplacians on a graph are analyzed. We provide a complete description of the set of all operators which can be obtained from a given self-adjoint Laplacian by perturbing it by magnetic fields. In particular, it is shown that generically this set is isomorphic to a torus. We also describe the conditions under which the operator is unambiguously (up to unitary equivalence) defined by prescribing the magnetic fluxes through all loops of the graph.
- We consider a class of self-adjoint extensions using the boundary triple technique. Assuming that the associated Weyl function has the special form $M(z)=\big(m(z)\Id-T\big) n(z)^{-1}$ with a bounded self-adjoint operator $T$ and scalar functions $m,n$ we show that there exists a class of boundary conditions such that the spectral problem for the associated self-adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for $T$. As a motivating example we consider differential operators on equilateral metric graphs, and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete laplacians.
- Article
- Jan 1990

Introducción a los fundamentos matemáticos y las aplicaciones de la geometría de fractales. - Article
- Nov 2006
- ANN MATH

We show that for almost every frequency ¿¿ ¿¿ R\Q, for every C¿Ö potential v : R/Z ¿¿ R, and for almost every energy E the corresponding quasiperiodic Schr¿Nodinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schr¿Nodinger operator, and allows us to complete the proof of the Aubry-Andr¿Le conjecture on the measure of the spectrum of the Almost Mathieu Operator. - Article
- Dec 1999
- ANN MATH

We prove that for Diophantine ù and almost every è, the almost Mathieu operator, (Hù,ë,è )(n) = (n+1)+ (n.1)+ë cos 2ð(ùn+è) (n), exhibits localization for ë > 2 and purely absolutely continuous spectrum for ë < 2. This completes the proof of (a correct version of) the Aubry-Andr/e conjecture. - Article
- May 1997

this paper is to show that the same duality can be established for a wide class of Schrodinger operators on graphs, including the case of a nonempty boundary. In general, the resulting Jacobi matrices exhibit a varying "mass". 1 II Preliminaries - We prove the conjecture (known as the ``Ten Martini Problem'' after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all non-zero values of the coupling and all irrational frequencies.
- We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is given. Applications include quantum graphs, point interactions, hybrid spaces, singular perturbations.
- We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.