Renormalization of entanglement entropy and the gravitational effective action

Abstract

The entanglement entropy of quantum fields across a spatial boundary is UV divergent, its leading contribution proportional to the area of this boundary. We demonstrate that the Callan-Wilczek formula provides a renormalized geometrical definition of this entanglement entropy for a class of quantum states defined by a path integral over quantum fields propagating on a curved background spacetime. In particular, UV divergences localized on the spatial boundary do not contribute to the entanglement entropy, the leading contribution to the renormalized entanglement entropy is given by the Bekenstein-Hawking formula, and subleading UV-sensitive contributions are given in terms of renormalized couplings of the gravitational effective action. These results hold even if the UV-divergent contribution to the entanglement entropy is negative, for example, in theories with non-minimal scalar couplings to gravity. We show that subleading UV-sensitive contributions to the renormalized entanglement entropy depend nontrivially on the quantum state. We compute new subleading UV-sensitive contributions to the renormalized entanglement entropy, finding agreement with the Wald entropy formula in all cases. We speculate that the entanglement entropy of an arbitrary spatial boundary may be a welldefined observable in quantum gravity.

Entanglement entropy: a perturbative calculation

“…If, however, the unperturbed background is such that the undeformed entangling surface exhibits rotational symmetry in the transverse space, then this symmetry will be inherent in the path integral representation ofρ 0 . In particular, as shown in [23] (see also [37][38][39][40][41]) in this caseK 0 is identical to the generator of angular evolution around Σ andÛ takes the form…”

Holographic RG flows, entanglement entropy and the sum rule

“…Susskind and Uglum proposed that entanglement entropy should renormalize in the same way as (4G N ) −1 [17]. The subject was revisited several times in the past [18][19][20][21][22][23][24][25].…”

“…Here K is the trace of the extrinsic curvature of the boundary metric (discussed in more detail below), and 20) where all quantities are evaluated at the boundary. The first two counterterms were found in [40] by requiring a finite energy-momentum tensor; the one proportional to φ 2 cancels the boundary term generated when integrating by parts to evaluate the scalar field action on-shell.…”

Conformal tightness of holographic scaling in black hole thermodynamics

“…The leading near-horizon gravitational potential arises from the first term in the second line of equation (3.8), and takes the form 14) which involves the effective coupling 15) and power-law exponent (3.13). By comparison with the radial derivative terms of the normal-reduced equation (2.7), the scale-invariant case defining CQM occurs if and only if 16) in which case the 'extra term' carries the critical coupling 1/4 that guarantees the relevant physics in the strong-coupling regime [27] (and is absorbed in equation (3.15)).…”

“…In addition and most importantly, a complete understanding of the predominant role of CQM in the determination of the thermodynamic properties of black holes has not yet been fully achieved, especially vis-à-vis the physical and geometric meaning of the underlying conformal symmetry. It should be noticed that semiclassical methods, based on quantum fields in the black hole background, include both the brick wall approach or thermal atmosphere proposal, and the entanglement entropy approach [14,15]. Moreover, the brick wall and entanglement entropies are essentially equivalent §.…”