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 $1-(t/\tau ){}^2$ (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.

Figure 1

Time evolution of the purity in the $80+80$ site $\mathit{XX}$ chain compared to the bounds [Eqs. (22) and (34)] (see text). The inset shows a closeup for short times. Note that $t$ has no units since we have chosen the dimensionless Hamiltonian Eq. (35).

Figure 2

Time evolution of the purity in the $80+80$ site $\mathit{XXZ}$ chain for $\Delta =\frac{1}{2}$ compared to the bounds [Eqs. (22) and (34)] in analogy to Fig. 1. The inset shows a closeup for short times. The dotted lines show the purity in the $\mathit{XX}$ model (Fig. 1) for comparison.

Figure 3

Short-time scaling of the purity with boundary size for a pair of coupled spin chains with Ising interaction. Exact diagonalization was done for $t=0.001$, such that the quadratic order in Eqs. (22) and (28) suffices to describe the result. Although the GHZ-like initial states show the fastest increase of entanglement [$\tau ~N$; Eq. (22), straight line], and the product-like ones stick to the moderate $\tau ~\sqrt{N}$ [Eq. (28), dashed line], $\mathit{W}$-type states show an intermediate behavior.