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Spectral Decimation of the Magnetic Laplacian on the Sierpinski Gasket: Solving the Hofstadter–Sierpinski Butterfly

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Abstract

The magnetic Laplacian (also called the line bundle Laplacian) on a connected weighted graph is a self-adjoint operator wherein the real-valued adjacency weights are replaced by unit complex-valued weights \(\{\omega _{xy}\}_{xy\in E}\), satisfying the condition that \(\omega _{xy}=\overline{\omega _{yx}}\) for every directed edge xy. When properly interpreted, these complex weights give rise to magnetic fluxes through cycles in the graph. In this paper we establish the spectrum of the magnetic Laplacian, as a set of real numbers with multiplicities, on the Sierpinski gasket graph (SG) where the magnetic fluxes equal \(\alpha \) through the upright triangles, and \(\beta \) through the downright triangles. This is achieved upon showing the spectral self-similarity of the magnetic Laplacian via a 3-parameter map \({\mathcal {U}}\) involving non-rational functions, which takes into account \(\alpha \), \(\beta \), and the spectral parameter \(\lambda \). In doing so we provide a quantitative answer to a question of Bellissard [Renormalization Group Analysis and Quasicrystals (1992)] on the relationship between the dynamical spectrum and the actual magnetic spectrum. Our main theorems lead to two applications. In the case \(\alpha =\beta \), we demonstrate the approximation of the magnetic spectrum by the filled Julia set of \({\mathcal {U}}\), the Sierpinski gasket counterpart to Hofstadter’s butterfly. Meanwhile, in the case \(\alpha ,\beta \in \{0,\frac{1}{2}\}\), we can compute the determinant of the magnetic Laplacian and the corresponding asymptotic complexity.

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Notes

  1. See also [4, Figure 2] for what appears to be a higher-resolution picture of [21, Figure 2], though it is the authors’ opinion that the two pictures have major differences.

  2. For the graph Laplacian \({\mathcal {L}}_\infty \) on an infinite, locally finite, connected graph with geometric self-similarity, it is expected that \(\sigma ({\mathcal {L}}_\infty ) = {\mathcal {J}} \cup {\mathcal {D}}\), where the set \({\mathcal {D}}\) depends on the self-similar structure of the graph under study. See [38] for illustrating examples. An example where \({\mathcal {D}}=\emptyset \) appears in a one-parameter family of self-similar “pq-Laplacians” on \({{\mathbb {Z}}}_+\) [10].

  3. The answer is yes for 1-parameter rational functions on \(\hat{{{\mathbb {C}}}}\) of degree \(\ge 2\) [40, Corollary 4.13]. This provides an algorithm for numerically generating pictures of the Julia set of a rational function.

  4. The semiconductance of an undirected edge with end vertices x and y is defined as \(\frac{1}{2}(\mathbf{c}_{xy}+\mathbf{c}_{yx})\).

  5. If \(G_\infty \) has bounded degree, the proof in [35, Theorem 4.1] suffices. If \(G_\infty \) has unbounded degree, then the proof proceeds according to [36, Theorem 3.1], which is based on von Neumann algebras.

  6. The limit of USTs on an infinite connected graph is a spanning forest. On \({{\mathbb {Z}}}^d\) the limit is a tree iff \(d\le 4\) [41].

  7. In (5.8) the weights associated to the logarithmic factors are probability weights. This is merely coincidental: for the graphical \((d-1)\)-dimensional Sierpinski simplex, the tree entropy equals \(\frac{d-2}{d}\log 2+\frac{d-2}{d-1}\log d + \frac{d-2}{d(d-1)}\log (d+2)\) [11, Corollary 4.1].

    More generally, the tree entropy of a unimodular random infinite connected weighted graph can take values in \([-\infty , \infty )\). For an example of a unimodular random graph with tree entropy equal to \(-\infty \), see [36, pp. 308-309].

  8. A power law modulated by log-periodic oscillations was proved for the growth of deterministic single-source abelian sandpile on SG [9].

  9. See [39, Chapter 5] for the proof of sandpile height distributions on the Hanoi tower graphs, a variant of SG. A nice exposition of the connection between sandpile density and the looping rate on periodic planar graphs is [29].

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Acknowledgements

We thank Alexander Teplyaev and Richard Kenyon for useful conversations during the initial stage of this work; Quan Vu for his early numerical contributions to cycle-rooted spanning forests on the Sierpinski gasket; and the anonymous referee for critical comments which helped us improve the paper.

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  1. Ruoyu Guo

    Present address: Department of Mathematics, Brandeis University, Waltham, MA, 02453, USA

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  1. Department of Mathematics, Colgate University, Hamilton, NY, 13346, USA

    Joe P. Chen & Ruoyu Guo

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Communicated by S. Chatterjee.

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This work is an outgrowth of the High Honors Bachelor’s thesis by the second-named author at Colgate University, advised by the first-named author. We acknowledge partial financial support from the Research Council of Colgate University (Grant No. 826568), the Simons Foundation (Collaboration Grant for Mathematicians #523544), and the National Science Foundation (DMS-1855604).

Appendix A: Numerical Approximation of the Filled Julia Set in Fig. 6

Appendix A: Numerical Approximation of the Filled Julia Set in Fig. 6

To numerically generate the filled Julia set of the map \({\mathcal {U}}\), we initialize with a uniform sample of points \(w=(\alpha ,\lambda )\) in the rectangle \([0,1]\times [0,2]\). We then discard points w for which \(|{\mathcal {U}}^k(w)|\) exceeds a threshold (10) after \(k(=20)\) iterations, and keep those points which remain bounded within. Below is a working MATLAB code.

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Chen, J.P., Guo, R. Spectral Decimation of the Magnetic Laplacian on the Sierpinski Gasket: Solving the Hofstadter–Sierpinski Butterfly. Commun. Math. Phys. 380, 187–243 (2020). https://doi.org/10.1007/s00220-020-03850-w

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