The paper shows that finding the usual signature of equilibrium in correlation functions is not enough to prove a quantum system has truly thermalized. In a controlled example based on the Sachdev‑Ye‑Kitaev (SYK) model, the authors demonstrate that all fermion correlators evaluated after a sudden change (a quench) match those of a thermal state. Still, a second quench back to the original Hamiltonian reveals a persistent memory of the initial state. This hidden memory fades in time at a rate set by the Lyapunov exponent, a number normally associated with scrambling and chaos.

To make this concrete the researchers study a return‑quench protocol in a large, strongly interacting SYK model built from Majorana fermions with random q‑body interactions. They change the interaction strengths suddenly, wait a controlled time, and then switch back. Working in the thermodynamic limit of large system size and large interaction order, they solve the full real‑time evolution exactly at leading order. The calculation uses standard real‑time tools (Kadanoff‑Baym equations) so the result is an explicit analytic solution for the nonequilibrium dynamics under this protocol.

What looks thermal at first glance is the Kubo‑Martin‑Schwinger (KMS) property — a technical periodicity condition of imaginary‑time correlation functions that signals equilibrium. All post‑quench two‑point correlators satisfy this KMS relation, so any observable built solely from them also looks thermal. However, the second quench probes correlators that connect to times before the first quench. Those mixed correlators keep a dependence on the waiting time. The authors identify two concrete nonequilibrium witnesses: the expectation value of the original Hamiltonian after the return quench, and an extensive mismatch in thermodynamic entropy between the actual state and its thermal reference. Both encode memory of the initial state.