Emergent Replica Clock Unifies Many-Body Localization and Thermalization

Abstract

Many-body localization (MBL) and eigenstate thermalization (ETH) are traditionally distinguished by a collection of separate diagnostics, not by a single order parameter. We show that replication and folding of isolated unitary dynamics generate a complex statistical mechanics of forward--backward history pairings and, in a closed cyclic infrared sector, an emergent clock $r_x\in\mathbb Z_t$. Its correlator has three distinct asymptotics: exponential decay in MBL, scale-free decay at criticality, and long-range locking in the thermal phase. A controlled $l$-bit reduction makes the clock action quasi-local, while its inverse correlation length obeys $\xi_{\rm pair}^{-1}=\kappa_{\rm el}=\ln|\rho_0/\rho_q|$, unifying history coherence, the defect line tension, and the transfer spectrum. Replica order supplies a second, discrete coordinate---the first order at which a hidden dynamical invariant becomes visible---and can distinguish localized dynamics that share the same spatial correlation length.