How do electrons trapped in flat bands resist being disturbed?

Flat-band materials host electrons that refuse to disperse — and it turns out this stubborn localization is protected by a bizarre symmetry called Carroll invariance, where the speed of light effectively goes to zero. Tracking 'Krylov complexity' (how fast a quantum state spreads through all possible configurations), the authors show that this measure sharply distinguishes which phases survive a perturbation and which collapse, acting like a fine-grained order parameter for symmetry breaking across quantum phase transitions.