Entanglement Entropy: A Perturbative Calculation

TL;DR

The paper develops a perturbative Euclidean path-integral framework to compute entanglement entropy in quantum field theories, focusing on small geometric deformations of the entangling surface and mild relevant perturbations. By foliating spacetime near Σ and mapping to a thermal setup on hyperbolic space, entanglement entropy is expressed through stress-tensor correlators, enabling explicit leading-order corrections from geometric perturbations for planar and spherical entangling regions. For a planar surface in 4D, the leading universal logarithmic term matches the known 4D Solodukhin structure, while for a perturbed sphere in 4D, the analysis reproduces Solodukhin’s universal term and clarifies the roles of intrinsic and extrinsic geometry. Overall, the framework provides a controllable, QFT-based method to compute universal entanglement terms beyond traditional replica/holographic approaches and supports conformal-invariance arguments across geometries.

Abstract

We provide a framework for a perturbative evaluation of the reduced density matrix. The method is based on a path integral in the analytically continued spacetime. It suggests an alternative to the holographic and `standard' replica trick calculations of entanglement entropy. We implement this method within solvable field theory examples to evaluate leading order corrections induced by small perturbations in the geometry of the background and entangling surface. Our findings are in accord with Solodukhin's formula for the universal term of entanglement entropy for four dimensional CFTs.

Figures (5)

• Figure 1: Abstract sketch of the two dimensional transverse space to the entangling surface $\Sigma$. $\mathcal{C}_\pm$ are the two sides of the cut $\mathcal{C}$ where the values $\phi_\pm$ of the field $\phi$ are imposed.
• Figure 2: A sketch of a slightly deformed entangling surface (curved line) in three dimensions. ($x_1,x_2$) span the transverse space to $\Sigma$ , while $y$ parametrizes $\Sigma$. The foliation (\ref{['ans']}) is designed to capture the geometry of the neighborhood of a given entangling surface $\Sigma$.
• Figure 3: Transverse space to the entangling surface in the analytically continued spacetime. $\Sigma$ is located at the origin. The reduced density matrix is given by a path integral (\ref{['reden']}) with fixed boundary conditions $\phi_+$ ($\phi_-$) on the upper (lower) dashed blue lines.
• Figure 4: We conformally transform between $\mathcal{H}$ (left) and $R^{d}$ (right). We first map from the $\sigma \equiv u +i \tau$ coordinates of $\mathcal{H}$ to $e^{-\sigma}$ (middle); here the origin is $u = \infty$ and the boundary circle is $u =0$. We then map via (\ref{['magic']}) to $R^{d}$. Dashed lines on the left represent $\tau_{\textrm{\tiny E}}=0^+,\beta^-$ slices of $\mathcal{H}$ that are mapped through an intermediate step onto $t=0^\pm$ sides of the cut throughout the interior of the sphere $r=R$ on the right
• Figure 5: We show the constant $\tau_{\textrm{\tiny E}}$ slices (blue) and constant $u$ slices (red) in the $(r,t_{\textrm{\tiny E}})$ plane (\ref{['magic2']}). The sphere is located at $r/R=1$, $t_{\textrm{\tiny E}}=0$ and corresponds to $u\rightarrow \infty$. The vertical line ($r=0$) corresponds to $u=0$.