Abstract
Recent results suggest that new corrections to holographic entanglement entropy should arise near phase transitions of the associated Ryu-Takayanagi (RT) surface. We study such corrections by decomposing the bulk state into fixed-area states and conjecturing that a certain ‘diagonal approximation’ will hold. In terms of the bulk Newton constant G, this yields a correction of order O(G−1/2) near such transitions, which is in particular larger than generic corrections from the entanglement of bulk quantum fields. However, the correction becomes exponentially suppressed away from the transition. The net effect is to make the entanglement a smooth function of all parameters, turning the RT ‘phase transition’ into a crossover already at this level of analysis.We illustrate this effect with explicit calculations (again assuming our diagonal approximation) for boundary regions given by a pair of disconnected intervals on the boundary of the AdS3 vacuum and for a single interval on the boundary of the BTZ black hole. In a natural large-volume limit where our diagonal approximation clearly holds, this second example verifies that our results agree with general predictions made by Murthy and Srednicki in the context of chaotic many-body systems. As a further check on our conjectured diagonal approximation, we show that it also reproduces the O(G−1/2) correction found Penington et al. for an analogous quantum RT transition. Our explicit computations also illustrate the cutoff-dependence of fluctuations in RT-areas.
Highlights
The net effect is to make the entanglement a smooth function of all parameters, turning the RT ‘phase transition’ into a crossover already at this level of analysis. We illustrate this effect with explicit calculations for boundary regions given by a pair of disconnected intervals on the boundary of the AdS3 vacuum and for a single interval on the boundary of the BTZ black hole
Using a decomposition into fixedarea states we found that, when a so-called diagonal approximation holds, the result can be written in the form (3.14)
At the phase transition where the mean value A1 − A2 vanishes, we find a correction of order G−1/2 controlled by the width σ− = G1/2σ− of the fluctuations in (A1 − A2)/2
Summary
After defining the fixed-area states, we will review their connection with the probability distribution P (A∗) for a holographic state to have RT-area A∗, features of the semiclassical approximation for such states, and the simple form of their Renyi entropies. One expects the RT area AγR to define superselection sectors of the quantum error correcting code associated with CFT reconstruction of the bulk entanglement wedges of R and R [19]. When this is the case, the density matrix on R of a CFT state |ψ is.
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