Time Scaling of Entanglement in Integrable Scale-Invariant Theories

In two-dimensional conformal field theories (isotropic scale-invariant theories), the time scaling of the entanglement entropy of a segment is fixed via conformal symmetry. I will talk about anisotropic scale-invariance between time and space. The anisotropy is parametrized by $z$, the so-called dynamical critical exponent. I will show that in anisotropic integrable theories with $z>1$, most of the entanglement is carried by slow modes. 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 $t^{1/(1－z)}$. I present numerical results for corresponding scalar and fermion lattice models which show extremely good agreement with our analytical results. Due to the dominance of the slow modes in these non-relativistic theories, local quantum information is "weakly" scrambled independently of the dynamical exponent. This "weak" scrambling is stronger than its counterpart in integrable relativistic theories.