Two models of how quantum systems break ergodicity: the Rosenzweig-Porter (RP) model, where off-diagonal elements are suppressed by a power of system size, and the ultrametric model, where the matrix structure mirrors a hierarchical tree. They look different. The RP model has a fractal phase — eigenstates occupy a fraction of Hilbert space that shrinks with system size. The ultrametric model has a fading-ergodicity regime — eigenstates spread but their thermalization becomes increasingly incomplete. Swietek et al. (arXiv: 2603.23616) show these are the same thing.
By aligning parameters using Thouless times — the timescale at which quantum dynamics first explores the available Hilbert space — the authors demonstrate that local observables in both models show comparable statistical behavior. Quantum-quench dynamics, temporal fluctuations, power spectra, survival probabilities: all match when the Thouless times are matched. The fractal phase of RP is the fading-ergodicity regime of the ultrametric model, viewed through a different lens.
A key finding: local observables thermalize within the fading-ergodicity regime on timescales shorter than the Heisenberg time. The system appears thermal if you don't wait too long. Full ergodicity eventually fails — the infinite-time averages disagree with thermal predictions — but on intermediate timescales, the system is indistinguishable from a thermalizing one. Ergodicity doesn't break sharply; it fades.
The through-claim: ergodicity breaking is a phenomenon, not a model. Two formally distinct random matrix constructions — one parametric, one hierarchical — produce identical ergodicity-breaking signatures when matched on the right observable. The universality isn't in the Hamiltonian structure but in the timescale structure. How a system loses ergodicity is determined by its Thouless time, not by the microscopic details that produce it.
Swietek, Kliczkowski, Hopjan & Vidmar, 2603.23616. Quantum dynamics / random matrices / ergodicity breaking / Rosenzweig-Porter / ultrametric model.