Holographic RG flows, entanglement entropy and the sum rule

Abstract

Abstract:We calculate the two-point function of the trace of the stress tensor in holographic renormalization group flows between pairs of conformal field theories. We show that the term proportional to the momentum squared in this correlator gives the change of the central charge between fixed points in d = 2 and in d > 2 it gives the holographic entanglement entropy for a planar region. This can also be seen as a holographic realization of the Adler-Zee formula for the renormalization of Newton's constant. Holographic regularization is found to provide a perfect match of the finite and divergent terms of the sum rule, and it is analogous to the regularization of the entropy in terms of mutual information. Finally, we provide a general proof of reflection positivity in terms of stability of the dual bulk action, and discuss the relation between unitarity constraints, the null energy condition and regularity in the interior of the gravity solution.

Relative entropy and the RG flow

“…The situation is analogous to what happened in d = 2, where positivity of the stress-tensor two-point function leads to the c-theorem [36], while our proof relied on positivity of the relative entropy. In fact, the derivation based on the relative entropy emphasizes the common origin between the c-theorem and the area theorem, something that was also seen in the holographic context in [23]. Furthermore, our approach identifies ∆µ with a well-defined continuum quantity, and suggests further connections between quantum corrections to gravity and relative entropy.…”

“…This is shown to be always decreasing between fixed points, but there is a restricted window of conformal dimensions ∆ < (d + 2)/2 where the change is finite. This is parallel to studies of the renormalization of the Newton constant [20][21][22][23].…”

“…The idea that part of the black hole entropy is due to entanglement entropy, suggests that the universal area term in the EE should agree with the renormalization of Newton's constant. This was made more precise in [22,23,34,35], who related the Adler-Zee formula [20,21],…”

Holographic torus entanglement and its renormalization group flow

“…As proven by Casini and Huerta [27] -see also [28] -a renormalized version of this quantity [29] is monotonously decreasing under the entire RG flow connecting two fixed points, and it coincides with F at fixed points. This "F-theorem" is one of the most celebrated applications of EE to QFT, and generalizes the earlier EE-based proof of the two-dimensional "ctheorem" [30,31]. Extensions of these monotonicity theorems to CFTs defined on R 1,d≥3 relying on the EE of smooth surfaces -typically spheres -have been also proposed, see e.g., [32][33][34][35].…”

Holographic entanglement entropy of anisotropic minimal surfaces in LLM geometries

“…This should be manifested in the holographic calculation of REE for LLM geometries. See also [12][13][14][15][16][17][18] for the behavior of EE under relevant perturbations from the UV fixed point.…”