Entanglement production and information scrambling in a noisy spin system

We study theoretically entanglement and operator growth in a spin system coupled to an environment, which is modeled with classical dephasing noise. Using exact numerical simulations we show that the entanglement growth and its fluctuations are described by the Kardar-Parisi-Zhang equation. Moreover, we find that the wavefront in the out-of-time ordered correlator (OTOC), which is a measure for the operator growth, propagates linearly with the butterfly velocity, and broadens diffusively with a diffusion constant that is larger than the one of spin transport. The obtained entanglement velocity is smaller than the butterfly velocity for finite noise strength, yet both of them are strongly suppressed by the noise. We calculate perturbatively how the effective timescales depend on the noise strength, both for uncorrelated Markovian and for correlated non-Markovian noise.

Figure 1

Entanglement production in a noisy spin chain. We compute the von Neumann entanglement entropy for a Heisenberg chain with $N$ spins starting with initial unentangled product states. In the due course of the dynamics the spins are subjected to dephasing noise of amplitude $\mathrm{\Lambda }=3\sqrt{J}$. The entanglement cut is chosen to separate the system into two equal halves. (a) The von Neumann entanglement entropy obeys a KPZ scaling with a strong sublinear power-law correction $S_{\text{vN}}=s_{\text{eq}}{v}_{e}t+B{t}^{1/3}$, dashed line. Inset: The entanglement growth for cuts at different positions $x$ (later times are indicated by more intense colors). (b) As a consequence of the KPZ scaling, the entanglement fluctuations also scale as a power law with exponent $1/3$. At late times the entanglement starts to saturate and hence the power-law scaling in the fluctuations breaks down as well.

Figure 2

Operator scrambling. We calculate the out-of-time ordered correlator (OTOC) $C(x_i-x_j,t)$ for noise amplitude $\mathrm{\Lambda }=3\sqrt{J}$. (a) A contour plot of the OTOC is shown in addition to the contour line at which the OTOC takes half (red curve) and one percent (white curve) of its saturation value. Error bars indicate the standard error of the mean. The slope of the linear red curve defines the inverse butterfly velocity $v_b$. (b) Scaling collapse over five orders of magnitude of the diffusively broadening right wavefront of the OTOC, which moves with the butterfly velocity $+v_b$. The inset shows the spin diffusion constant $D_{\text{S}}$, red line, and the diffusion constant of the wavefront $D_O$. Both diffusion constants scale as $1/{\mathrm{\Lambda }}^2$. (c) The velocity dependent Lyapunov exponent $\lambda \left(v\right)$, blue dots, becomes negative at the butterfly velocity $v_b$.

Figure 3

Comparison of the entanglement velocity and the butterfly velocity. The entanglement velocity $v_e$ (blue squares) is strictly smaller than the butterfly velocity $v_b$ (red dots) for all values of the noise $\mathrm{\Lambda }$, even though they approach each other with increasing noise. In the strong noise limit, the velocities are governed by the inverse timescale ${\tau }^{-1}\sim J^2/{\mathrm{\Lambda }}^2$, which is shown as a black dashed line.

Figure 4

von Neumann, Rényi, and purity entanglement entropies. (a) Entanglement growth for different values of the noise amplitude $\mathrm{\Lambda }$ in a spin chain of length $N=20$. Inset: The ratio between the subleading contribution to the entanglement growth and the entanglement growth rate increases with noise. (b) The Rényi entropies $S_n$ are shown for noise amplitude $\mathrm{\Lambda }=2\sqrt{J}$ along with the von Neumann entropy and the purity entropy. The purity entropy, diamonds, is very close the second Rényi entropy. The subleading corrections $B$ to the entanglement entropies decreases with the index $n$ of the Rényi entropy. Inset: The saturation value of the entropy decreases with increases Rényi index $n$.

Figure 5

Comparison of the entanglement dynamics in interacting and noninteracting systems. We compare the entanglement dynamics of the interacting spin chain studied in the main body of this work with the one of free fermions for noise amplitude $\mathrm{\Lambda }=2\sqrt{J}$. At short time $tJ\lesssim 1$ both results agree well, indicating that interactions are not relevant for the initial build up of the entanglement. At late time, the free fermion entanglement crosses over to a $\sqrt{t}$ behavior, whereas interactions in the spin chain lead to the KPZ entanglement growth. The short time free fermion evolution, and the late time KPZ evolution of the entanglement are separated by a crossover regime.