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

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 { ω xy } x y ∈ E , satisfying the condition that ω xy = ω 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 α through the upright triangles, and β through the downright triangles. This is achieved upon showing the spectral self-similarity of the magnetic Laplacian via a 3-parameter map U involving non-rational functions, which takes into account α , β , and the spectral parameter λ . 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 α = β , we demonstrate the approximation of the magnetic spectrum by the filled Julia set of U , the Sierpinski gasket counterpart to Hofstadter’s butterfly. Meanwhile, in the case α , β ∈ { 0 , 1 2 } , we can compute the determinant of the magnetic Laplacian and the corresponding asymptotic complexity.