Entanglement Growth after a Global Quench in Free Scalar Field Theory

TL;DR

The paper demonstrates that entanglement growth after global quenches in free massless scalar field theories can be accurately captured by a simple quasiparticle picture based on EPR pairs, across 1D intervals, strips, and spheres in up to 3 spatial dimensions. Using a Gaussian/correlation-matrix formalism, it computes entanglement and Rényi entropies in a double scaling limit and shows quantitative agreement with the quasiparticle model for both boundary-state and mass quenches. A robust linear growth with entangling-surface area and a shape-dependent saturation are established, with a universal entanglement velocity v_E, and subleading logarithmic growth from the zero mode identified as a key caveat. The results provide a high-precision benchmark for entanglement dynamics in free theories and offer a baseline for comparing strongly coupled or holographic systems where entanglement spreads faster than free streaming.

Abstract

We compute the entanglement and Rényi entropy growth after a global quench in various dimensions in free scalar field theory. We study two types of quenches: a boundary state quench and a global mass quench. Both of these quenches are investigated for a strip geometry in 1, 2, and 3 spatial dimensions, and for a spherical geometry in 2 and 3 spatial dimensions. We compare the numerical results for massless free scalars in these geometries with the predictions of the analytical quasiparticle model based on EPR pairs, and find excellent agreement in the limit of large region sizes. At subleading order in the region size, we observe an anomalous logarithmic growth of entanglement coming from the zero mode of the scalar.

Figures (9)

• Figure 1: The two types of geometries we examine in this work. Regions $\mathcal{A}$ and $\bar{\mathcal{A}}$ partition the system into two distinct regions. Starting with a pure state, we trace out region $\bar{\mathcal{A}}$ to obtain a reduced density matrix $\rho_{\mathcal{A}}$, from which we compute the entanglement and Rényi entropies. Left: The strip geometry with two sides separated by a distance $2R$. Right: A spherical geometry of radius $R$.
• Figure 2: Quenches for strips (intervals) in 1+1 dimensions. The top figure is linear time and the bottom figure is logarithmic time. We have taken intervals of length $2R/a = 100, 200, 300$, where $a$ is the lattice unit, both for boundary state quenches with $\beta = 10 \, a$ and for mass quenches. For the mass quench we have chosen $m ={4 \pi \over 3(\pi-2) \beta}$ such that the resulting entropy density matches that of the boundary state quench, see \ref{['RenyiEntropyDensityBoundaryStateQuench']}, \ref{['EntropyDensityBoundaryStateQuench']}. Curves from lower to upper represent $2R/a = 100$ (black), $2R/a = 200$ (blue), and $2R/a = 300$ (red). Curves with circle-markers represent boundary state quenches; curves with star-markers represent mass quenches (almost indistinguishable from the boundary state quenches). In the top figure, the lines with no markers represent the quasiparticle model predictions \ref{['qpPrediction']}, using the entropy density corresponding to the quench. Note that the linear ramp of the quasiparticle model is indistinguishable from the numerical results. The lines in the bottom figure with no markers show the fit \ref{['FitCurve']} to the linear asymptotic behavior $\sim \frac{1}{2} \log(t)$.
• Figure 3: Modified mass quench for intervals of length $2R/a = 100, 200, 300$, with $m^2(k)$ a smoothed step function as shown in the inset graph. $s$ in the quasiparticle formula \ref{['qpPrediction']} is adjusted to match the numerical data points.
• Figure 4: Time evolution of the entropy of a finite harmonic chain with Dirichlet boundary conditions on both ends. The region at the end of the chain is of size $2R/a=300$, and the full chain is chosen to be $2L/a=1050$ long in order to avoid any special ratio $L/R$. We follow the time evolution for a long time and find exact revivals. Note that because the region is at the end of the chain, the slope of the curves is half of that in \ref{['qpPrediction']}.
• Figure 5: Time evolution for a boundary state quench with $\beta = 10 \, a$ of entanglement entropy (blue) and the Rényi entropies for $q = 2, 3, 4$ (in red, $q$ increasing from top to bottom), for a strip of width $2R = 300 \, a$. The numerical data points (circles) are shifted to match the quasiparticle curves (solid lines) at the single point $t = R$. Top figure is for $2+1$ dimensions; bottom figure is for $3+1$ dimensions.