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  • Open Access

Contamination-Free Resolution of Cantor Spectra and Magic-Angle Localization in Moiré Quasicrystals and Aperiodic Media

James Chok1,*, Matthew J. Colbrook2,†, Mark Embree3,‡, and Jake Fillman4,§

  • 1School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh
  • 2Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
  • 3Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
  • 4Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA

  • *Contact author: james.chok@ed.ac.uk
  • †Contact author: mjc249@cam.ac.uk
  • ‡Contact author: embree@vt.edu
  • §Contact author: fillman@tamu.edu
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Phys. Rev. Lett. 137, 176403 – Published 21 October, 2026

DOI: https://doi.org/10.1103/7931-lsqk

Abstract

Quasicrystalline Hamiltonians host Cantor-like spectra, gaps, and critical states, yet conventional finite-patch diagonalization is often compromised by spectral pollution: spurious eigenvalues absent from the infinite system. We introduce a certified spectral-computation framework for aperiodic media, computing infinite-volume spectra of general Hermitian Hamiltonians with rigorous a posteriori error bounds and without imposing artificial boundaries or artificial periodicity. The method resolves fine spectral features and fractal dimensions universally across aperiodic systems. For twisted Penrose bilayers, angle-resolved certified spectra identify a magic angle at θ≈27.228°, where ring-worm interference produces twist-controlled confinement. Across one- and two-dimensional quasicrystals, the results show how fractal spectral geometry organizes localization: lower box-counting dimension correlates with larger inverse participation ratio in substitution chains, while dodecagonal tilings exhibit compactly supported and algebraically confined critical modes. The framework also scales to three dimensions, delivering contamination-free, high-resolution spectra for a quasicrystal arising in acoustic metamaterials. Further examples involving quantum walks, continuum superlattices, longer-range hopping, and inhomogeneous magnetic fields demonstrate the broad reach of certified spectral computation as a route to predictive spectral design in aperiodic media.

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Phys. Rev. Lett. 137, 176403– Published 21 October, 2026
  • Received 11 October 2025
  • Accepted 24 September 2026
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