Entanglement spectra from holography

TL;DR

This paper computes the entanglement spectrum in holographic conformal field theories, generalizing the Cardy formula to higher dimensions, analyzing shape deformations, and revealing universal high-energy behavior.

Contribution

It provides a holographic computation of the entanglement spectrum in higher-dimensional CFTs, extending the Cardy formula and analyzing shape deformation effects.

Findings

High-energy entanglement spectra follow a generalized Cardy formula.

Shape deformations do not affect the high-energy entropy scaling.

Universal high-energy behavior is independent of entangling surface shape.

Abstract

The entanglement spectrum of a bipartite quantum system is given by the distribution of eigenvalues of the modular Hamiltonian. In this work, we compute the entanglement spectrum in the vacuum state for a subregion of a $d$-dimensional conformal field theory (CFT) admitting a holographic dual. In the case of a spherical (or planar) entangling surface, we recover known results in two dimensions, including the Cardy formula in the high energy regime. In higher dimensions $d>2$, we analytically determine a generalization of the Cardy formula valid at large energies and consistent with previous studies of CFT spectra in the literature. We also investigate numerically the spectrum at energy levels far above the modular ground state energy. We extend our analysis to the supersymmetric point of Einstein-Maxwell gravity, providing exact results when $d=2,3$, and a generalization of the Cardy formula at high energies in generic dimension $d$. We consider small shape deformations of a spherical entangling surface, for both the non-supersymmetric and the supersymmetric cases. In all cases we find that the high-energy scaling of the microcanonical entropy with the modular energy is unaffected by the shape deformation. This result suggests that the high-energy regime of the entanglement spectra carries universal information, independent of the shape of the entangling surface.