Two quantum phases with identical particles but fundamentally different physics?

In 2+1-dimensional topological materials, the rules governing how particles combine (algebra structure) can differ even when the particles themselves are identical. Researchers identified "twin" phases that share the same anyon spectrum but have inequivalent internal structure—arising from subgroups, quantum cocycles, and symmetry actions. These twins can transition between states without any hidden symmetry breaking, revealing physics beyond the classical Landau framework.