Diamond Fractal. March 19, University of Connecticut REU Magnetic Spectral Decimation on the. Diamond Fractal

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1 University of Connecticut REU 2015 March 19, 2016

2 Motivation In quantum physics, energy levels of a particle are the spectrum (in our case eigenvalues) of an operator. Our work gives the spectrum of a magnetic operator on the. This gives the energy levels of a quantum particle confined to the fractal and under the influence of a magnetic field.

3 The Each edge in the n th level approximating graph becomes a diamond in the (n + 1) th level. We can represent this operation using an iterated function system of four contraction maps f i. The diamond fractal is the unique, non-empty, compact set invariant under the IFS: K = f i (K).

4 Laplacian Matrix The graph Laplacian at level n is an operator on functions given by n u(x) = y x ( u(x) u(y) ) This can be represented as the difference of the graph s degree matrix and adjacency matrix. At level zero we have: Degree Matrix Adjacency Matrix Laplacian Matrix [ ] [ ] [ ] We normalize by replacing the a ij entry with a ij aii a jj.

5 Laplacian Matrix At any level n the graph Laplacian is a block matrix: [ ] A B B t D The A block corresponds to vertices from the (n 1) th level. The D block corresponds to new vertices introduced at the n th level. B and B t correspond to connections between levels n 1 and n. 4 n n converges to an operator that is the correct replacement for the usual Euclidean Laplacian.

6 Spectral Laplacian It is known that we can relate the spectrum of n to n 1 via the Schur complement (S λ ). [ ] [ ] [ ] u A λ B u ( n λ) = v B t = D λ v If λ is not an eigenvalue of D then [ ] 0 0 S λ = A λ B(D λ) 1 B t = 0.

7 Spectal Laplacian Theorem For the there exist rational functions ϕ 0 (λ) and ϕ 1 (λ) such that S λ = ϕ 0 (λ) n 1 ϕ 1 (λ)i. Corollary If λ is not an eigenvalue of D and ϕ 0 (λ) 0 then λ is an eigenvalue of n if and only if R(λ) = ϕ 1(λ) ϕ 0 (λ) is an eigenvalue of n 1.

8 Magnetic Laplacian Approximating graph becomes a weighted, directed graph. Edges are weighted by e iθ in the direction of the edge e iθ in the opposite direction. M n u(x) = ( y x u(x) e iθ xy u(y) )

9 Spectral Decimation on Magnetic Laplacian A variant of Spectral Decimation still works for M n on the. Instead of fixed functions ϕ 0 and ϕ 1 we have functions that depend on the magnetic field strength. Theorem For the there exist rational functions ϕ 0 (λ, γ) and ϕ 1 (λ, γ) such that S λ,γ = ϕ 0 (λ, γ)m n 1 ϕ 1 (λ, γ)i.

10 Spectral Decimation on Magnetic Laplacian Example on level 1: [ ] A B M 1 = B D A = D = (1 λ)i B = S λ = = [ e iγ 2 e iγ 2 e iγ 2 e iγ 2 [ 1 λ 1 ] 2(1 λ) cos(2γ) cos(2γ) 2(1 λ) 2(1 λ) 1 λ 1 2(1 λ) ( 2λ 2 + 4λ 1 + cos (2γ) cos (2γ) 2(1 λ) M 0 ] 2(1 λ) ) I

11 Spectral Decimation on Magnetic Laplacian It does not immediately follow that this method can be used to reduce M n to M n 1 Gluing M n together in the right way results in the spectral decimation operators The additional feature we need is that the operator on each piece may be gauge-transformed.

12 Gauge Transforms from Level to Level The spectral decimation relations for the magnetic operators of the Diamond fractal can be understood through gauge transforms. The gauge transforms can be found through analyzing the magnetic field strength through each cell. The magnetic field through an approximating graph can be written as an equivalent sequence of magnetic fields. The equivalent magnetic field strengths determine the gauge transforms.

13 Gauge Transforms-Level 2 Example [ ] [ ] µ1 = µ 2 [ δ1 δ 2 ]

14 Gauge Transforms from Level to Level In General: µ 1 δ µ 2 δ µ µ 4 = δ 3 δ µ k δ k We can invert the left most matrix in order to solve for the vector of µ values, if we know the δ field strengths from level to level.

15 Gauge Transforms from Level to Level Lemma Fix a level n. Given a magnetic field in which the field strength through a hole depends only on the level of the hole, let δ j denote the field strength through each j th level hole, and define µ j = δ j + n i=j+1 2 2i (2j+1) δ i. Define a field of strength µ j on each j-level cell, and extend it to act as a gauge transform on all smaller cells. Then the original field is the sum of the µ j fields.

16 Spectral Decimation with Magnetic Field Theorem Suppose {λ n } is a sequence such that for each n, λ n 1 and µ n π 2 Z. Then for each n, λ n is an eigenvalue of M n if and only if R(λ n, µ n ) is an eigenvalue of M n 1, where R(λ, µ) = 4λ 2λ2 1 cos (µ/2) + 1.

17 Spectral Decimation with Magnetic Field The exceptional case λ n = 1 does correspond to eigenvalues, and we can calculate their multiplicity. Another exceptional case occurs when ϕ 0 (λ, µ) = 0. This does not immediately lead to eigenvalues.

18 Spectral Decimation with Magnetic Field This theorem gives an algorithm for finding eigenvalues of the Laplacian. For a given magnetic field defined by the sequence {δ i } n i=1, compute {µ i} n i=1. Then for each i: 1 For every eigenvalue λ (i 1) k of M i 1, find its two preimages under R(, µ i ). 2 Incorporate 4i 4 3 copies of the exceptional eigenvalue λ = 1. 3 Re-scale all eigenvalues by 4 n in order to maintain compatible energy levels between operators. 4 Take the limit as n.

19 Spectral Decimation with Magnetic Field δ j (the strength of the magnetic field through the level k hole) could be any sequence of fields we want. As a special case we consider δ j to be proportional to area of the j th level cell. For a suitable embedding of the fractal, the area occupied by the j th level cells may be taken to be geometrically decreasing. δ j = 4 1 j A j for some A < 1. µ j = 2 1 2j( A n+1 + A j+1 A j).

20 Predicting Eigenvalues Example R 1 ± (λ, µ) = 1 ± cos( µ 2 ) λcos( µ 2 )+1 2 Recall the Laplacian for Level 0: [ Eigenvalues = {0, 1, 1, 2} R 1 ({0,1,1,2}, µ 1 ) gives the eigenvalues of the Laplacian of the Level 1 graph approximation ]

21 Analysis of Eigenvalues To analyze the eigenvalues, define a counting function.. N(x, δ k ) = Number of Eigenvalues < x

22 Spectrum of Eigenvalues

23 Spectrum of Eigenvalues

24 Spectrum of Eigenvalues

25 Spectrum of Eigenvalues

26 References and Acknowledgements References: L. Malozemov and A. Teplyaev, Self-similarity, Operators and Dynamics. Mathematical Physics, Analysis and Geometry 6 (2003), J. Bellissard, Renormalization Group Analysis and Quasicrystals. Ideas and Methods in Analysis, Stochastics, and Applications (1992), Acknowledgements: We would like to acknowledge our collaborator Aubrey Coffey, thank the 2015 UConn Math REU, and our mentors Dr. Luke Rogers, Dr. Alexander Teplyaev, Dr. Joe Chen, and Antoni Brzoska. This work was made possible by the generous support of the National Science Foundation under the grant DMS