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The authors adapt Bellissard’s noncommutative algebraic framework to bridge atomistic electronic-structure calculations of twisted quasiperiodic bilayers with macroscopic topological gap labels and integrated densities of states. Resolving this connection is essential for moiré systems where finite-size spectral weight redistribution often obscures whether observed gaps carry genuine bulk topological invariants. Using an exact second-moment identity on a model honeycomb bilayer, they isolate interlayer contributions to spectral variance and establish the precise basis requirements—including nonorthogonality and relative sliding coordinates—needed to reliably compute bulk topological response from microscopic simulations.
Spectral weight redistribution in twisted moiré bilayers can mimic bulk gap formation without generating genuine topological invariants, revealing why standard electronic-structure diagnostics fail without Bellissard's noncommutative algebraic machinery.
Bellissard's framework uses covariant observables to describe aperiodic solids. We examine how butterfly-like spectra in twisted bilayers can be related to bulk gap labels and topological response. A bulk gap permits a spectral projector and an integrated density of states. A topological response requires an additional pairing of that projector. In an illustrative finite honeycomb bilayer, an exact second-moment identity isolates the interlayer contribution to spectral variance. Angle-resolved spectra, finite state counts, an uncoupled reference, and size and broadening checks show how coupling redistributes spectral weight without assigning a bulk gap label. We then examine the requirements for atomistic electronic-structure calculations, including nonorthogonal orbitals, projected densities of states, and basis functions that move during relative sliding. The resulting framework specifies the steps needed to connect atomistic spectra with bulk state counting and topological response.
