Translation symmetry restoration in integrable systems: the noninteracting case

The study of symmetry restoration has recently emerged as a fruitful means to extract high-level information on the relaxation of quantum many-body systems. However, while the restoration of internal symmetries has been investigated intensively, that of spatial symmetries has hitherto only been considered in the context of random unitary circuits. Here we present a complementary study of translation symmetry restoration in integrable systems. In particular, we consider a one-dimensional chain of spinless, non-interacting fermions quenched from a $ν>1$ shift invariant state, and follow the local restoration of one-site shift invariance using the Frobenius distance $ΔF_A$ between the state on a subsystem and its symmetrised counterpart. Distinct from the case of random unitary circuits, where symmetry restoration occurs abruptly for times proportional to the subsystem size, we find that symmetry here is restored smoothly and over timescales of the order of the subsystem size squared. We also find that the so-called `quantum Mpemba effect' is readily observed. Most importantly, we show that - in contrast to the case of continuous internal symmetries - this discrete symmetry restoration is not qualitatively described by a quasiparticle picture for $ΔF_A$, and therefore goes beyond the hydrodynamic description. Our results can be directly extended to higher dimensions.