Abstract
Recent works have related the bulk first law of black hole mechanics to the first law of entanglement in a dual CFT. These are first order relations, and receive corrections for finite changes. In particular, the latter is naively expected to be accurate only for small changes in the quantum state. But when Newton’s constant is small relative to the AdS scale, the former holds to good approximation even for classical perturbations that contain many quanta. This suggests that — for appropriate states — corrections to the first law of entanglement are suppressed by powers of N in CFTs whose correlators satisfy ’t Hooft large-N power counting. We take first steps toward verifying that this is so by studying the large-N structure of the entropy of spatial regions for a class of CFT states motivate dby those created from the vacuum by acting with real-time single-trace sources. We show that 1/N counting matches bulk predictions, though we require the effect of the source on the modular hamiltonian to be non-singular. The magnitude of our sources is ϵN with ϵ fixed-but-small as N → ∞. Our results also provide a perturbative derivation — without relying on the replica trick — of the subleading Faulkner-Lewkowycz-Maldacena correction to the Ryu-Takayagi and Hubeny-Rangamani-Takayanagi conjectures at all orders in 1/N .
Highlights
Much of the recent discussion has centered on the so-called first law of entanglement δSA = δ HA
We show that 1/N counting matches bulk predictions, though we require the effect of the source on the modular hamiltonian to be non-singular
A is a subregion of some Cauchy surface for the CFT, SA := − Tr ρA log(ρA) is the von Neumann entropy of the associated reduced density matrix ρA, HA := − log(ρA) is the modular Hamiltonian, and δ denotes the first variation with respect to the state when the operator HA is held fixed on the right hand side
Summary
We wish to study excitations of the vacuum |0 of a large N CFT in d spacetime dimensions on R × Sd−1. Light operators are those whose scaling dimension ∆i of Oi(x) remains finite as we take N → ∞ and the sparse spectrum condition requires that for any fixed ∆ the number of such operators with ∆i < ∆ remains finite at large N. We introduce a basis for the space of light local gauge-invariant single trace operators {OAi (x)} ({OAi c(x)}) on the domain of dependence D(A) (D(Ac)) that again satisfy (2.1) and (2.2). We will consider ball-shaped regions in a constant-time Cauchy surface Σ = Sd−1 so that the modular Hamiltonians HA, HAc can be expressed as integrals of the stress tensor over A, Ac in any CFT [16]
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