Luttinger's theorem states that the volume enclosed by a metal's Fermi surface is fixed by the electron density alone, regardless of interaction strength. It is one of the most robust results in many-body physics — surviving perturbation theory, renormalization, and decades of increasingly exotic materials. The electron count determines the Fermi volume. Period.
Fractional Chern insulators violate it.
Using exact diagonalization of the Harper-Hofstadter-Hubbard model in a fractional Chern insulating phase, the authors directly measure the Luttinger count — the integral over the Brillouin zone that should equal the particle number. It doesn't. The violation is not a numerical artifact or a finite-size effect. It is a consequence of topological order: the ground state has fractionalized excitations that carry quantum numbers not attributable to any individual electron.
The violation has structure. The fractional nature of the many-body Chern number — the topological invariant that makes the phase “fractional” — is encoded in the Středa response of the Luttinger integral. The integer part comes from the Luttinger count; the fractional part comes from its response to flux insertion. The theorem doesn't simply fail — it fails in a way that reveals the topological order responsible for the failure.
The authors also identify experimentally measurable signatures: local density-of-states measurements that can detect the Luttinger violation in fractional quantum Hall systems. The theorem's breakdown is not just a theoretical curiosity but a diagnostic tool — the deviation from Luttinger's prediction is a probe of fractionalization.
When a fundamental theorem fails, the pattern of its failure is often more informative than the theorem itself. Luttinger's theorem doesn't hold here, and the way it breaks encodes exactly the exotic physics responsible for breaking it.