Short-time versus long-time dynamics of entanglement in quantum lattice models

We study the short-time evolution of the bipartite entanglement in quantum lattice systems with local interactions in terms of the purity of the reduced density matrix. A lower bound for the purity is derived in terms of the eigenvalue spread of the interaction Hamiltonian between the partitions. Starting from an initially separable state the purity decreases as 1-(t/t)(2) (i. e., quadratically in time, with a characteristic timescale tau that is inversely proportional to the boundary size of the subsystem, that is, as an area law). For larger times an exponential lower bound is derived corresponding to the well-known linear-in-time bound of the entanglement entropy. The validity of the derived lower bound is illustrated by comparison to the exact dynamics of a one-dimensional spin lattice system as well as a pair of coupled spin ladders obtained from numerical simulations.