Holographic entanglement beyond classical gravity

Abstract

The Renyi entropies and entanglement entropy of 1+1 CFTs with gravity duals can be computed by explicit construction of the bulk spacetimes dual to branched covers of the boundary geometry. At the classical level in the bulk this has recently been shown to reproduce the conjectured Ryu-Takayanagi formula for the holographic entanglement entropy. We study the one-loop bulk corrections to this formula. The functional determinants in the bulk geometries are given by a sum over certain words of generators of the Schottky group of the branched cover. For the case of two disjoint intervals on a line we obtain analytic answers for the one-loop entanglement entropy in an expansion in small cross-ratio. These reproduce and go beyond anticipated universal terms that are not visible classically in the bulk. We also consider the case of a single interval on a circle at finite temperature. At high temperatures we show that the one-loop contributions introduce expected finite size corrections to the entanglement entropy that are not present classically. At low temperatures, the one-loop corrections capture the mixed nature of the density matrix, also not visible classically below the Hawking-Page temperature.Comment: 1+41 pages. Three figures. v2 References adde

Comments on Rényi entropy in AdS3/CFT2

“…Because the higher spin currents do not contribute at O(c) as described earlier, this is in fact their leading contribution. Following [18], we compute these 1-loop determinants in the small interval expansion using known formulas for handlebodies [21] and spin-s gauge fields [22], and find complete agreement with CFT. That is, if S (s) n is the contribution to the CFT Rényi entropy from the pair of spin-s currents, and S (s) n (M) is the holographic contribution to the Rényi entropy obtained from linearized spin-s gauge 2 The analog of (1.2) was only proven in [2,3] to hold for a noncompact CFT in its ground state and, implicitly, for all states related by conformal transformations.…”

“…The same is true of our proof of (1.2). As in [18], we take the perspective that it holds more generally, e.g. for a CFT on the torus.…”

“…One can always write any Riemann surface as a quotient of the complex plane, Σ = C/Γ, where Γ is an order g discrete subgroup of PSL(2,C), the Möbius group. Γ is called the Schottky group, and the procedure of writing Σ as such a quotient is known as Schottky uniformization; more details in this context can be found in [2,3,18].…”

“…The minimization condition that determines the Rényi entropy follows from the usual Hawking-Page-type transition among competing saddles of fixed genus. Building on [2,3,18] generalized this construction to the case of a single interval in a CFT of finite size and at finite temperature T , under the assumption that the singleinterval version of (2.9) still holds on the torus. Among other results, this showed that finite size effects are invisible at O(c).…”

“…Unless M is the solid torus, this expression is generally not 1-loop exact. This technology was put to use in [18] for the graviton, with physically interesting results. First, [18] provided a systematic algorithm for computing the eigenvalues q γ for the case of two intervals in the ground state, in an expansion in short intervals.…”