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arXiv:2202.06391 · Physical review. D/Physical review. D.

Checking volume-law entropy with Hubeny-Rangamani-Takayanagi surfaces

Tyler Guglielmo, Phuc Nguyen

Published Jun 1, 2022• 0 citations
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Abstract

We check formally that the Hubeny-Rangamani-Takayanagi prescription for holographic entanglement entropy---when applied to a static black brane spacetime and to a wide class of subregions that do not lie on a constant time slice---gives rise to volume-law entropy in the limit of large subregion. By volume-law entropy, we mean that the entanglement entropy scales with the volume of the projection of the boundary subregion onto a static time slice with respect to the boundary thermal state (with the same coefficient as in the volume law on the static time slice). Our result applies to subregions that have reflection symmetry as well as strips, and we also present field-theoretic arguments in support of our holographic findings.

DOI
10.1103/physrevd.105.126004

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We check formally that the Hubeny-Rangamani-Takayanagi prescription for holographic entanglement entropy---when applied to a static black brane spacetime and to a wide class of subregions that do not lie on a constant time slice---gives rise to volume-law entropy in the limit of large subregion. By volume-law entropy, we mean that the entanglement entropy scales with the volume of the projection of the boundary subregion onto a static time slice with respect to the boundary thermal state (with the same coefficient as in the volume law on the static time slice). Our result applies to subregions that have reflection symmetry as well as strips, and we also present field-theoretic arguments in support of our holographic findings.
Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.
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Abstract: \nTensor networks,ToverlineT T\\overline{T} ToverlineTTT¯, and broader notions of a holographic principle all motivate the idea that some notion of gravitational holography should persist in the presence of a radial cutoff. But in the absence of time-reflection symmetry, the areas of Hubeny-Rangamani-Takayanagi surfaces anchored to the radial cutoff generally violate strong sub-additivity, even when the associated boundary regions are spacelike separated as defined by both bulk and boundary notions of causality. We thus propose an alternate definition of cutoff-holographic entropy using a restricted maximin prescription anchored to a codimension 2 cutoff surface. For bulk solutions that respect the null energy condition, we show that the resulting areas satisfy SSA, entanglement wedge nesting, and monogamy of mutual information in parallel with cutoff free results in AdS. These results hold even when the cutoff surface fails to be convex.
We compute holographic entanglement entropy in two strongly coupled nonlocal field theories: the dipole and the noncommutative deformations of SYM theory. We find that entanglement entropy in the dipole theory follows a volume law for regions smaller than the length scale of nonlocality and has a smooth cross-over to an area law for larger regions. In contrast, in the noncommutative theory the entanglement entropy follows a volume law for up to a critical length scale at which a phase transition to an area law occurs. The critical length scale increases as the UV cutoff is raised, which is indicative of UV/IR mixing and implies that entanglement entropy in the noncommutative theory follows a volume law for arbitrary large regions when the size of the region is fixed as the UV cutoff is removed to infinity. Comparison of behaviour between these two theories allows us to explain the origin of the volume law. Since our holographic duals are not asymptotically AdS, minimal area surfaces used to compute holographic entanglement entropy have novel behaviours near the boundary of the dual spacetime. We discuss implications of our results on the scrambling (thermalization) behaviour of these nonlocal field theories. ar X iv
We show that Renyi entropies satisfy an area law in quantum field theories with a holographic dual description in terms of a bulk gravitational theory. A derivative of the Renyi entropy SnS_nSn​ with respect to its index nnn is given by the area in Planck units of a bulk cosmic brane homologous to the entangling region. This geometric prescription for holographic Renyi entropies is a one-parameter generalization of the Ryu-Takayanagi prescription for entanglement entropy. As an example of applying this prescription, we calculate the mutual Renyi information InI_nIn​ between two disks to linear order in n−1n-1n−1 in holographic conformal field theories of arbitrary dimension.
When a spacetime has boundaries, the entangling surface does not have to be necessarily compact and it may have boundaries as well. Then there appear a new, boundary, contribution to the entanglement entropy due to the intersection of the entangling surface with the boundary of the spacetime. We study the boundary contribution to the logarithmic term in the entanglement entropy in dimensions d=3d=3d=3 and d=4d=4d=4 when the entangling surface is orthogonal to the boundary. In particular, we compute a boundary term in the entropy of N=4\mathcal{N}=4N=4 supergauge multiplet at weak coupling. For gauge fields we use a prescription which is consistent with the positive area law. The boundary term is compared with the holographic calculation of the entropy based on the Ryu-Takayanagi proposal adapted appropriately to the present situation. We find a complete agreement between these two calculations provided the boundary conditions imposed on the gauge multiplet preserve 1/21/21/2 of the supersymmetry and the extension of the boundary into the anti--de Sitter bulk is a minimal hypersurface.