Spectral decimation of the magnetic Laplacian on the Sierpinski gasket: Solving the Hofstadter-Sierpinski butterfly

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 determinant and the corresponding asymptotic complexity.

arXiv comment46 pages, 10 figures. Final version, to appear in Comm. Math. Phys

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