On the Time Scaling of Entanglement in Integrable Scale-Invariant Theories

TL;DR

This paper investigates how entanglement entropy evolves over time in integrable, scale-invariant theories with anisotropic scaling, revealing a universal slow growth regime dominated by slow modes, supported by analytical and numerical evidence.

Contribution

It introduces a detailed analysis of entanglement dynamics in anisotropic scale-invariant integrable theories, highlighting the dominance of slow modes and their impact on information scrambling.

Findings

Entanglement growth transitions from linear to power-law with exponent 1/(1-z).

Logarithmic enhancement of entanglement in bosonic theories during slow mode regime.

Numerical simulations confirm analytical predictions with high accuracy.

Abstract

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 $t^{\frac{1}{1-z}}$. 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.