Numerical Computations of Entanglement Measures in Curved Space

TL;DR

This work develops a covariant, lattice-like numerical framework to compute entanglement entropy and logarithmic negativity for scalar and abelian gauge fields in curved backgrounds, extending Srednicki's flat-space results to AdS$_4$, RT surfaces in AdS$_3$, and dS$_4$. The method discretizes space via proper distance, builds quadratic Hamiltonians, and diagonalizes reduced-density matrices across angular modes to obtain EE and E_N. It demonstrates a clear area-law scaling $S\sim \kappa R^2$, with the slope controlled by the AdS radius $L$ and reconcilable with flat-space limits when $L\to\infty$, and it verifies the universal term predictions using heat-kernel coefficients. The study also reveals divergences in the negativity in 3+1d and above, while showing convergent behavior in 2+1d, providing a bridge between numerical entanglement measures in curved spacetimes and analytic heat-kernel analyses.

Abstract

We numerically compute the entanglement entropy and negativity for scalar fields and abelian gauge fields in a variety of situations. These extend computations of Srednicki to situations involving curved space. We discretize space in a covariant way. Finally, we compare some of our results with those obtained via the heat kernel coefficients.

Figures (16)

• Figure 1: Foliation of a constant $t$ slice using constant $\eta$ and $x$ surfaces. $L$ is taken to be 1 here. Constant $\eta$ surfaces (red) are RT surfaces. Both $\eta$ and $x$ can take values from $-\infty$ to $\infty$.
• Figure 2: Entanglement entropy of scalar field in a sphere in flat space time vs. radius. Entanglement entropy increases linearly with $R^2$.
• Figure 3: Entanglement entropy vs. $R^2$ for different masses. Addition of mass reduces the entanglement entropy.
• Figure 4: Fitting the dependence of entanglement entropy on mass (i) $1/(1+p\hat{m}^b)$ with $p=0.6339, \ b=1.6649$, (ii)$ce^{-\hat{m}d}$ with $c=0.9943,\ d= 0.5013$.
• Figure 5: Entanglement entropy vs. $R^2$ calculated from working with discretizing proper distance $u$ in radial direction for different values of AdS radius. We see the area law, ie. entanglement entropy $\propto R^2$. We also observe the slope is a function of the AdS radius $L$.