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In two dimensional isotropic scale invariant theories, the time scaling of the entanglement entropy of a segment is fixed via the conformal symmetry. We consider scale invariance in a more general sense and show that in integrable theories that the scale invariance is anisotropic between time and space, parametrized by z, most of the entanglement is carried by the slow modes for z > 1. At early times entanglement grows linearly due to the contribution of the fast modes, before smoothly entering a slow mode regime where it grows forever with t^1 /(1âz) . The slow mode regime admits a logarithmic enhancement in bosonic theories. We check our analytical results against numerical simulations in corresponding fermionic and bosonic lattice models finding extremely good agreement. We show that in these non-relativistic theories that the slow modes are dominant, local quantum information is universally scrambled in a stronger way compared to their relativistic counterparts.
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