Holographic entanglement entropy inequalities beyond strong subadditivity

Abstract

The vacuum entanglement entropy in quantum field theory provides nonperturbative information about renormalization group flows. Most studies so far have focused on the universal terms, related to the Weyl anomaly in even space-time dimensions, and the sphere free energy $F$ in odd dimensions. In this work we study the entanglement entropy on a sphere of radius $R$ in a large radius limit, for field theories with gravity duals. At large radius the entropy admits a geometric expansion in powers of $R$; the leading term is the well-known area term, and there are subleading contributions. These terms can be physical, they contain information about the full renormalization group flow, and they reproduce known monotonicity theorems in particular cases. We set up an efficient method for calculating them using the Hamilton-Jacobi equation for the holographic entanglement entropy. We first reproduce the known result for the area term, the coefficient multiplying $R^{d-2}$ in the entanglement entropy. We then obtain the holographic result for the $R^{d-4}$ term and establish its irreversibility. Finally, we derive the $R^{d-6}$ coefficient for holographic theories, and also establish its irreversibility. This result goes beyond what has been proved in quantum field theory based on strong subadditivity, and hints towards new methods for analyzing the monotonicity of the renormalization group in space-time dimensions bigger than four.