Entanglement entropy, planar surfaces, and spectral functions

TL;DR

This work presents a non-perturbative expression for the universal part of entanglement entropy across a planar surface in flat space in terms of a spectral function, and develops both perturbative and non-perturbative analyses of how EE changes under deformations of couplings and background geometry. Central to the approach is the exact knowledge of the planar modular (Rindler) Hamiltonian, enabling a flow equation $\frac{\partial S}{\partial \lambda} = - \int d^d x\, \langle K_{\lambda}\, \mathcal{O}(x) \rangle_{\lambda}$ and its spectral decomposition via $c^{(0)}(\mu)$ and $c^{(2)}(\mu)$. The paper computes explicit universal area-like terms for free massive fields in even dimensions, showing, for example, $S = (-)^{d/2} \frac{d-2}{6 (2\pi)^{(d-2)/2} \Gamma(d/2)} m^{d-2} \mathcal{A}_\Sigma \log(m\delta)$ for Dirac fermions and $S = (-)^{d/2} \frac{1}{6 (4\pi)^{(d-2)/2} \Gamma(d/2)} m^{d-2} \mathcal{A}_\Sigma \log(m\delta)$ for minimally coupled scalars. Extending to deformed geometries, the authors derive new universal terms that mix curvature with couplings, expressed in terms of curvature invariants and the same spectral data, including explicit results in four and higher dimensions and for both fermions and scalars. Overall, the spectral-function framework provides a powerful, broadly applicable method to characterize entanglement entropies across planes and in curved backgrounds, with potential applications to interacting theories where spectral data are accessible perturbatively.

Abstract

We consider the universal part of entanglement entropy across a plane in flat space for a QFT, giving a non-perturbative expression in terms of a spectral function. We study the change in entanglement entropy under a deformation by a relevant operator, providing a pertrubative expansion where the terms are correlation functions in the undeformed theory. The entanglement entropy for free massive fermions and scalars easily follows. Finally, we study entanglement entropy across a plane in a background geometry that is a deformation of flat space, finding new universal terms arising from mixing of geometry and couplings of the QFT.

Figures (1)

• Figure 1: (a) An entangling surface that is a plane. We use coordinates $x_{\mu} = (x_a,y_i)$, with $x_a$ transverse to the plane and $y_i$ along the plane. (b) The transverse space to the plane.