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We characterize the long-term state of the 1D Dirac vacuum stirred by animpenetrable object, modeled as the ground state of a finite free-fermionicchain dynamically perturbed by a moving classical obstacle which suppresses thelocal hopping amplitudes. We find two different regimes, depending on thevelocity of the obstacle. For a slow motion, the effective Floquet Hamiltonianpresents features which are typical of the Gaussian orthogonal ensemble, andthe occupation of the Floquet modes becomes roughly homogeneous. Moreover, thelong term entanglement entropy of a contiguous block follows a Gaussiananalogue of Page's law, i.e. a volumetric behavior. Indeed, the statisticalproperties of the reduced density matrices correspond to those of a randomSlater determinant, which can be described using the Jacobi ensemble fromrandom matrix theory. On the other hand, if the obstacle moves fast enough, theeffective Floquet Hamiltonian presents a Poissonian behavior. The nature of thetransition is clarified by the entanglement links, which determine theeffective geometry underlying the entanglement structure, showing that theone-dimensionality of the physical Hamiltonian dissolves into a randomadjacency matrix as we slow down the obstacle motion.

Long term behavior of the stirred vacuum on a Dirac chain: geometry blur and the random Slater ensemble