Quantum birthmarks: Ergodicity breaking beyond scarring
A hallmark of classical ergodicity is the complete loss of memory of the initial conditions due to eventual uniform covering of {\it a priori} available phase space. In quantum counterparts of such systems, however, this classical ergodic ideal is fundamentally limited: Here, we introduce the concept of a \emph{quantum birthmark}, a permanent signature left by the initial state and its early-time evolution in a general quantum system, which gives rise to non-ergodic behavior persisting even in the infinite-time limit. We present a birthmark framework outlining a ubiquitous memory effect for an arbitrary, non-stationary state composed of two factors conspiring together: the universal and the revival-enhancement. The former sets the minimal amplification carried by the time evolution of a quantum state based on global symmetries, whereas the latter incorporates the further enhancement stemming from the early dynamics, particularly prominent in the presence of recurrences that occur before the Heisenberg time. As a concrete example, we identify quantum birthmarks in the venerable stadium billiard, where they can be significantly enhanced by quantum scars. Finally, we discuss the broader implications of quantum birthmarks, including their role as a natural extension of all types of scarring theories to generic non-stationary quantum systems and prospects for experimental observation. Generally, our work opens an unexplored avenue for understanding the elusive quantum nature of ergodicity.