Corrections to holographic entanglement plateau

Abstract

We investigate the robustness of the Araki-Lieb inequality in a two-dimensional (2D) conformal field theory (CFT) on torus. The inequality requires that $\Delta S=S(L)-|S(L-\ell)-S(\ell)|$ is nonnegative, where $S(L)$ is the thermal entropy and $S(L-\ell)$, $S(\ell)$ are the entanglement entropies. Holographically there is an entanglement plateau in the BTZ black hole background, which means that there exists a critical length such that when $\ell \leq \ell_c$ the inequality saturates $\Delta S=0$. In thermal AdS background, the holographic entanglement entropy leads to $\Delta S=0$ for arbitrary $\ell$. We compute the next-to-leading order contributions to $\Delta S$ in the large central charge CFT at both high and low temperatures. In both cases we show that $\Delta S$ is strictly positive except for $\ell = 0$ or $\ell = L$. This turns out to be true for any 2D CFT. In calculating the single interval entanglement entropy in a thermal state, we develop new techniques to simplify the computation. At a high temperature, we ignore the finite size correction such that the problem is related to the entanglement entropy of double intervals on a complex plane. As a result, we show that the leading contribution from a primary module takes a universal form. At a low temperature, we show that the leading thermal correction to the entanglement entropy from a primary module does not take a universal form, depending on the details of the theory.