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See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.130.250402 for details on the tube state construction (A), the multidimensional generalization of Heller's criterion (B), quantum scarring when approaching low (Equation presented) (C), the emergence of a uniform time scale (D), the comparison with regular tunnelling oscillations in the self-trapping regime (E), and quantum scarring in larger chains (F).
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In principle, minor chaotic layers in the mixed regime, e.g., for (Equation presented) in Figs. 2(a) and 2(b), may also be able to host quantum scars for sufficiently large total particle number (Equation presented). For all investigated parameters, however, a clear characterization of tube states as strongly embedded within locally ergodic patches by means of Eq. (10) was only possible for the steadily chaotic regime (Equation presented).
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