Two quantum systems share every energy level but behave completely differently?

Two fermion chains — one with strong interactions, one without — have exactly the same energy levels, yet live in different phases and scramble quantum information at completely different rates. A nonlocal mathematical transformation connects them, turning simple local operators into sprawling many-body strings that grow in complexity without bound. The finding breaks an intuitive assumption: knowing a system's full energy spectrum tells you almost nothing about how operators and information spread through it.