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.