Band theory works for periodic crystals: Bloch's theorem decomposes the Hamiltonian into independent momentum sectors, and each sector is a finite matrix problem. Quasiperiodic systems break Bloch's theorem — there's no translational symmetry to exploit, and the Hamiltonian doesn't decompose.
The standard workaround: rational approximants. Replace the irrational modulation frequency with a rational one (a Fibonacci fraction), imposing a large but finite unit cell. Inside the unit cell, band theory applies. As the approximant improves (higher Fibonacci numbers), the results converge to the quasiperiodic limit. This works for non-interacting systems.
The paper extends the approximant strategy to interacting quasiperiodic systems by constructing an effective band-projected description. The key: project the full interacting Hamiltonian onto the low-energy bands of the quasiperiodic system, using the approximant's band structure as the projection basis. The projected Hamiltonian captures the interaction physics within the relevant energy window while respecting the quasiperiodic structure at the level of the approximant.
The method enables Hartree-Fock and beyond-mean-field calculations for interacting quasiperiodic systems that were previously intractable. The results converge with approximant order, confirming that the rational approximation captures the essential physics of the irrational system even in the presence of interactions.
Band theory extended to quasiperiodic systems — not by making the system periodic, but by making it approximately periodic in a controlled way and then projecting the interactions onto the approximate bands. The approximation is in the geometry (rational instead of irrational); the interactions are treated exactly within the approximate geometry.