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Asymptotic Analysis of the Spectrum of the DiscreteHamiltonian with Period Doubling Potential
Meg Fields, Tara Hudson, Maria Markovich
Cornell University Department of Mathematics Summer Math Institute
Introduction
Quasi-crystals are materials that have some properties ofcrystals, specifically x-ray diffraction patterns, but haveaperiodic lattices.
• We model one-dimensional quasi-crystals using Hamil-tonian operators with aperiodic potentials [1] [2], [5].
• Fractal properties of spectra affect quantum diffusionpatterns of electrons [7], [4], [8].
Background
Period Doubling Sequence: We say S∞(A) is the perioddoubling sequence
S : A 7→ AB S : B 7→ AAS1(A) = A BS2(A) = A B A AS3(A) = A B A A A B A B
The Hamiltonian Operator The Hamiltonian operatorwith period doubling potential, HV, is:
HV = −∆ + V̂Where,• ∆ is the discrete Laplacian of Z
• V̂ is a multiplication operator which inputs the perioddoubling sequence into the main diagonal.
The Hamiltonian operator acts on a sequence in `2(Z)[2].
Trace Map: The trace map is recursively defined:xn+1 = xnyn− 2 and yn+1 = x2
n− 2where x0 = E−V and y0 = E + V for E, V ∈ R.• A point (x, y) is unstable if and only if there exists n such
that (xn, yn) ∈ A = {(x, y) | y > 2 , |x| > 2} [2].
Theorem 1 (Bellissard, Bovier, Ghez) E is in the spectrumof HV if and only if x = E + V and y = E − V are suchthat (x, y) ∈ UC, where U is the set of all unstable points.
Theorem 2 (Bellisard, Bovier, Ghez) For a fixed couplingconstant V > 0, the spectrum of the Hamiltonian is a Can-tor set of Lebesgue measure zero.
Two methods will be used to approximate the spec-trum.• Iterate the trace map and identify not unstable points• Truncate the operator and find the eigenvaluesEmploy fractal dimension analysis (box counting di-mension, thickness and Hausdorff dimension).
Visualization of the Spectrum
−3 −2 −1 0 1 2 3−3
−2
−1
0
1
2
3
y
x
−2 −1.5 −1 −0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
V
E
Student Version of MATLAB
Top: A numerical approximation of UC.Bottom: An approximation of the spectrum for a givencoupling constant between 0 and 1.
Box-Counting Dimension
Definition 2 Box-counting dimension of a Cantor set, K,is:
dimB K := limδ→0
log Nδ(K)− log δ
,
where Nδ(K) is the number of delta covers needed tocover the set, if the limit exists [6].
As V approaches 0, dimB K approaches 1.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−3
0.85
0.9
0.95
1
Coupling Constant
Box
Cou
ntin
g D
imen
sion
Box Counting Dimension as Coupling Constant Approaches Zero
Student Version of MATLAB
In similar models, the box-counting dimension and theHausdorff dimension coincide [5].
Thickness
Definition 3 [9]. Thickness of a set K at a boundary pointu is:
τ(K, u) =`(C)`(U)
where `(U) is the length of the gap and `(C) is the lengthof the bridge from u to the next gap U′ satisfying `(U′) ≥`(U).Overall thickness is defined as:
τ(K) = inf{τ(K, ui)}
As coupling constant approaches zero, thicknessapproaches infinity. This implies that the Hausdorffdimension is bounded below by one [9].
ThicknessV n = 1000 n = 10000
0.03 0.0106 0.01160.003 0.0833 0.0125
0.0003 3.0 0.09090.00003 383 1.0
Issue:As the number of iterations increase, the thicknesstrend is weaker.
Hausdorff Dimension
In the formal definition of Hausdorff dimension, dimH K,when s < dimH K the Hausdorff measure is ∞; whens > dimH K, the Hausdorff measure is 0. This trend is rep-resented in the following numerical estimation method[3]. Consider the quantity n∆s, where ∆ = 2−i ands ∈ [0, 1].
Suppose the value of n∆s1 is increasing while n∆s2 is de-creasing. Then the Hausdorff dimension is given by,
s1 < dimH(K) < s2
The Hausdorff dimension has the following bound,0.92 < dimH(K) < 1.00.
si 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.001 4.757 4.724 4.691 4.659 4.627 4.595 4.563 4.531 4.5002 4.748 4.683 4.619 4.555 4.492 4.430 4.369 4.309 4.2503 4.872 4.771 4.673 4.577 4.483 4.391 4.300 4.212 4.1254 5.071 4.933 4.798 4.667 4.539 4.415 4.294 4.177 4.0635 5.319 5.138 4.963 4.794 4.631 4.473 4.321 4.173 4.0316 5.601 5.373 5.154 4.944 4.742 4.549 4.364 4.186 4.016
Table 1: Values of n∆s,V=0.001
Future Work
Further work for this problem includes refining the ap-proximations of the two fractal dimensions, refining theapproach to the thickness computations, and analyzingthe physical implications of the results.
Acknowledgments
We would like to thank our advisor, May Mei, and ourproject TA, Drew Zemke. Thanks also to Ravi Ramakr-ishna, Summer Math Institute program director, and themath department at Cornell University.This work wassupported by NSF grant DMS-0739338.
Contact Information• Meg Fields, University of North Carolina at [email protected]
• Tara Hudson, University at [email protected]
• Maria Markovich, Shippensburg [email protected]
References[1] Michael Baake, Uwe Grimm, and Robert V. Moody. What is aperiodic
order? Spektrum der Wissenschaft, pages 64–74, 2002. The translatedversion was used for our research.
[2] Jean Bellissard, Anton Bovier, and Jean-Michel Ghez. Spectral prop-erties of a tight binding hamiltonian with period doubling potential.Communications in Mathematical Physics, 135(2):379–399, 1991.
[3] Alexandre Joel Chorin. Numerical estimates of hausdorff dimension.Journal of Computational Physics, 46(3):390 – 396, 1982.
[4] Jean-Michel Combes. Connections between quantum dynamics andspectral properties of time-evolution operators. In E.M. HarrellW.F. Ames and J.V. Herod, editors, Differential Equations with Appli-cations to Mathematical Physics, volume 192 of Mathematics in Scienceand Engineering, pages 59 – 68. Elsevier, 1993.
[5] Daminik, Embree, and Gorodetski. Spectral properties of schrdingeroperators arising in the study of quasicrystalsl. 1210.5753, October2012.
[6] Kenneth Falconer. Hausdorff Measure and Dimension, pages 27–38. JohnWiley & Sons, Ltd, 2005.
[7] I. Guarneri. Spectral properties of quantum diffusion on discrete lat-tices. EPL, 10(2):95–100, 1989.
[8] Yoram Last. Quantum dynamics and decompositions of singular con-tinuous spectra. Journal of Functional Analysis, 142(2):406 – 445, 1996.
[9] Jacob Palis and Floris Takens. Hyperbolicity and sensitive chaotic dy-namics at homoclinic bifurcations, volume 35 of Cambridge Studies in Ad-vanced Mathematics. Cambridge University Press, Cambridge, 1993.Fractal dimensions and infinitely many attractors.
Advisor: May Mei, Denison University, [email protected]
