The Subsystem Witness

You don’t need the whole system to detect ergodicity breaking. A piece of it will do.

Most measures of ergodicity in quantum many-body systems rely on global quantities. Level spacing statistics require the full energy spectrum. The eigenstate thermalization hypothesis tests expectation values across all eigenstates. These diagnostics work perfectly in theory but are experimentally inaccessible — you rarely have access to the complete Hamiltonian or the full Hilbert space of a quantum many-body system.

Sheridan and Sheridan (arXiv:2603.10691) show that local observables, measured on a subsystem, can detect the ergodicity-breaking transition. In random matrix theory (RMT), the eigenstate statistics of local operators follow specific predictions — the distribution of matrix elements, their correlations, and their fluctuations are all fixed by symmetry class. Violations of these predictions signal non-ergodic behavior.

The approach: take a subsystem (a few sites of a many-body chain), compute a local observable’s matrix elements in the energy eigenbasis, and compare against RMT predictions. In the ergodic phase, the statistics match. At the many-body localization transition, they deviate. The local observable acts as a witness — it detects a global change (ergodicity breaking) through local measurements.

The structural insight: ergodicity is a global property of the dynamics, but its breakdown manifests locally. A subsystem that thermalizes agrees with RMT. A subsystem that fails to thermalize disagrees. The disagreement doesn’t require knowing why ergodicity broke — it only requires comparing measured statistics against the universal predictions of random matrix theory.

The whole tells the part what to expect. When the part disagrees, the whole has changed.

Sheridan and Sheridan, “Probing the ergodicity breaking transition via violations of random matrix theoretic predictions for local observables,” arXiv:2603.10691 (2026).