A flat band is an energy band with zero dispersion — all states have the same energy regardless of momentum. Flat bands maximize interaction effects because kinetic energy cannot compete with interactions when kinetic energy vanishes. A topological band carries a global invariant — a Chern number, a ℤ₂ index — that protects edge states and quantized responses. The ideal platform for studying strongly correlated topological physics would combine both: an exactly flat band with stable topology.
No-go theorems say this is impossible. An exactly flat band in a finite-range hopping model requires Bloch wavefunctions that are analytic throughout the Brillouin zone. But analytic wavefunctions with a nonzero Chern number cannot exist in a flat band — the obstruction is topological, encoded in the relationship between analyticity and the Berry connection. The theorems are rigorous. Exact flatness and stable topology are incompatible.
Li et al. (arXiv:2603.12258) construct exactly what the no-go theorems forbid. Their critical topological flat bands (CTFBs) are exactly flat, carry Chern numbers and ℤ₂ indices, and live in finite-range hopping models. The construction works because their Bloch wavefunctions are continuous across the entire Brillouin zone but non-analytic at isolated band-touching points. The functions pass through these points without derivative — continuous but not smooth — and the non-analyticity at those isolated points is enough to evade the no-go theorem, whose proof requires analyticity everywhere.
The structural interest is in what the no-go theorem actually forbids. It does not forbid exact flatness with stable topology. It forbids exact flatness with stable topology AND analytic wavefunctions. The topology and the flatness are compatible. The analyticity was the obstacle. Removing analyticity at a measure-zero set of points — isolated band-touching points where the wavefunction's derivative is undefined — is sufficient to open the door. The ban was on a regularity condition, not on the physics.
The authors identify approximately 50,000 such CTFBs using computational search across space groups. The filled states exhibit power-law correlations and admit exact tensor-network representations with finite bond dimension — making them amenable to both analytical and numerical study. The construction is not a single example but a systematic principle: for each space group, the critical topological flat bands that can exist are classifiable.
The no-go theorem was correct. Its premises were too strong. The escape was not through the conclusion but through the hypothesis — a wavefunction that is everywhere continuous, almost everywhere analytic, and non-analytic at exactly the points where the topology lives. The loophole was always there. It was in the fine print.