Numerical Computations of Entanglement Measures in Curved Space

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.

💡 Research Summary

The paper presents a comprehensive numerical study of entanglement entropy (EE) and logarithmic negativity for both scalar fields and abelian gauge fields in a variety of curved spacetimes, extending the classic lattice approach pioneered by Srednicki to non‑flat backgrounds. The authors first review the original flat‑space calculations, emphasizing the area law—EE proportional to the area of the entangling surface—and the relevance of this result to black‑hole thermodynamics and the AdS/CFT correspondence.

To treat curved spaces, they introduce a covariant discretization scheme: instead of discretizing the coordinate radius directly, they discretize the proper geodesic distance along the radial direction. In global AdS₄ this proper distance is (u = L \sinh^{-1}(r/L)); in AdS₃ Rindler‑type coordinates the analogous variables are (\eta) and (x). By using the proper distance as the lattice spacing, the UV cutoff becomes a geometric invariant, avoiding coordinate‑dependent artifacts.

The numerical method follows Srednicki’s recipe: the field is expanded in spherical harmonics, the radial (or proper‑distance) direction is discretized, and the resulting Hamiltonian is cast as a chain of coupled harmonic oscillators. The matrix (K) of couplings is diagonalized to obtain (\Omega = K^{1/2}); the ground‑state wavefunction is Gaussian, and tracing out interior degrees of freedom yields a reduced density matrix whose eigenvalues (\xi_n) are analytically known. EE is then computed as (S = -\sum_n p_n \log p_n) with (p_n = (1-\xi)\xi^n). The method is straightforwardly generalized to massive scalars by adding a diagonal mass term, which reduces EE as the mass increases, in line with intuition that larger diagonal entries suppress entanglement.

For gauge fields, the authors construct vector spherical harmonics adapted to AdS₄ and decompose the abelian field into transverse electric and magnetic modes, each behaving like an independent scalar. The same lattice procedure is applied, confirming that EE again obeys an area law, with a slope that depends on the AdS radius (L). As (L\to\infty) the slope approaches the flat‑space value, providing a non‑trivial check of the method.

The paper also investigates entanglement across Rindler‑type (RT) surfaces in AdS₃. Using the (\eta)–(x) foliation, the authors discretize the two‑dimensional slice and compute EE for both scalars and gauge fields confined within the RT region. In 2+1 dimensions the logarithmic negativity can be obtained directly from the Renyi entropy of order (1/2), because for pure states the negativity equals this Renyi entropy. The numerical results again display an area law, while in higher dimensions the angular integration leads to divergences that prevent a straightforward negativity calculation.

A separate section treats scalar fields in de Sitter space (dS₄), showing that the area law persists even for positively curved backgrounds.

To validate the numerical findings, the authors perform an analytic heat‑kernel expansion of the effective action on the same geometries. The heat‑kernel coefficients generate a series of UV‑divergent terms; among them, a universal logarithmic term appears in even dimensions. By comparing the coefficient of this universal term obtained from the heat‑kernel with the numerical slope, they confirm quantitative agreement. In the limit (L\to\infty) the universal coefficient reduces to the known flat‑space value, providing a stringent cross‑check.

The paper concludes with several key insights: (1) covariant proper‑distance discretization is an effective and coordinate‑independent way to extend lattice EE calculations to curved spacetimes; (2) both scalar and abelian gauge fields obey the area law in AdS₄, AdS₃ RT regions, and dS₄, with the EE slope being a smooth function of the curvature scale (L); (3) the universal logarithmic term extracted from heat‑kernel methods matches the numerical data, reinforcing the reliability of both approaches; (4) logarithmic negativity in 2+1 dimensions follows the same area scaling, offering a simple probe of quantum correlations in curved backgrounds. These results provide valuable quantitative input for bulk‑entanglement contributions in the FLM formula, quantum extremal surface calculations, and recent island proposals, thereby strengthening the bridge between numerical quantum field theory and holographic quantum gravity.