Single-eigenstate test of eigenstate thermalization hypothesis via perturbed eigenstate quench

How can we efficiently determine whether a specific energy eigenstate of a quantum many-body system satisfies the Eigenstate Thermalization Hypothesis (ETH) without the computationally expensive process of constructing explicit thermal ensembles?

The authors propose a 'perturbed eigenstate quench' protocol. By adding a weak random perturbation to an energy eigenstate, they break its stationarity, allowing the state to evolve under the original Hamiltonian. They then track the 'subsystem evolution speed'—the rate at which the reduced density matrix of a subsystem changes over time—as a function of the ratio between the subsystem size and the total system size.

The study demonstrates that the qualitative shape of the evolution speed curve serves as a robust signature of thermalization:

• ETH-satisfying eigenstates: These exhibit an S-shaped curve with a clear inflection point at half the system size. This reflects the distinct entanglement structure of thermal states, where subsystems smaller than half the total size behave thermally.
• ETH-violating eigenstates: These display a convex J-shaped profile across the entire range of subsystem sizes, indicating a lack of the thermal-like entanglement structure found in ergodic systems.

The authors successfully benchmarked this method across several paradigmatic models, including chaotic and integrable Ising chains, disordered XXZ chains (covering both ergodic and Many-Body Localized regimes), and spin-1 XX chains containing quantum many-body scar states. In all cases, the diagnostic correctly distinguished between thermal and non-thermal eigenstates.

This method provides a computationally efficient and experimentally feasible way to probe ergodicity breaking at the level of individual eigenstates. By avoiding the need to construct thermal ensembles, it simplifies the study of complex quantum phases, such as many-body localization and quantum scars, which are central to modern condensed matter physics.