We study aspects of entanglement and extremal surfaces in various families of spacetimes exhibiting cosmological, Big-Crunch, singularities, in particular isotropic $AdS$ Kasner. The classical extremal surface dips into the bulk radial and time directions. Explicitly analysing the extremization equations in the semiclassical region far from the singularity, we find the surface bends in the direction away from the singularity. In the 2-dim cosmologies obtained by dimensional reduction of these and other singularities, we have studied quantum extremal surfaces by extremizing the generalized entropy. The resulting extremization shows the quantum extremal surfaces to always be driven to the semiclassical region far from the singularity. We give some comments and speculations on our analysis.
Following arXiv:2012.07351 [hep-th], we study quantum extremal surfaces in various families of cosmologies with Big-Crunch singularities, by extremizing the generalized entropy in 2-dimensional backgrounds which can be thought of as arising from dimensional reduction. Focussing first on the isotropic $AdS$ Kasner case, introducing a spatial regulator enables relating the locations in time of the quantum extremal surface and the observer. This shows that the quantum extremal surface lags behind the observer location. A potential island-like region, upon analysing more closely near the island bo…
We have obtained the equation of the extremal hypersurface by considering the Jacobson-Myers functional and computed the entanglement entropy. In this context, we show that the higher derivative corrected extremal surfaces can not penetrate the horizon. Also, we have studied the entanglement temperature and entanglement entropy for low excited states for such higher derivative theories when the entangling region is of the strip type.
In this note, we use entanglement entropy as a tool to explore the universal properties of CFTs dual to extremal BTZ black holes. We demonstrate that the entanglement entropies computed in the CFTs at the boundary of the extremal BTZ and the boundary of the near-horizon limit of the extremal BTZ are in perfect agreement and have the form appropriate to a two-dimensional CFT with only the chiral part excited and the anti-chiral in the ground state. Furthermore, we analyze the universal limits of the entanglement entropy and recover the correct value of the thermal entropy for large entangling i…
I compute the entanglement entropy of a strongly coupled 2+1d quantum field theory containing fermions at finite density using gauge/gravity duality. The dual geometry is an extremal black hole in 3+1d Einstein-Maxwell theory. This system was recently shown to exhibit non-Fermi liquid behavior, but the leading geometrical contribution to the entanglement entropy does not produce an expected violation of the boundary law. I discuss this negative result in the context of attempts to find highly entangled states of quantum matter.
We develop further the codim-2 future-past extremal surfaces stretching between the future and past boundaries in de Sitter space, discussed in previous work. We first make more elaborate the construction of such surfaces anchored at more general subregions of the future boundary, and stretching to equivalent subregions at the past boundary. These top-bottom symmetric future-past extremal surfaces cannot penetrate beyond a certain limiting surface in the Northern/Southern diamond regions: the boundary subregions become the whole boundary for this limiting surface. For multiple disjoint subregi…
Recent progress in our understanding of the black hole information paradox has lead to a new prescription for calculating entanglement entropies, which involves special subsystems in regions where gravity is dynamical, called \textit{quantum extremal islands}. We present a simple holographic framework where the emergence of quantum extremal islands can be understood in terms of the standard Ryu-Takayanagi prescription, used for calculating entanglement entropies in the boundary theory. Our setup describes a $d$-dimensional boundary CFT coupled to a ($d$-1)-dimensional defect, which are dual to…
The island rule for the entanglement entropy is applied to an eternal Reissner-Nordström black hole. The key ingredient is that the black hole is assumed to be in thermal equilibrium with a heat bath of an arbitrary temperature and so the generalized entropy is treated as being off-shell. Taking the on-shell condition to the off-shell generalized entropy, we find the generalized entropy and then obtain the entanglement entropy following the island rule. For the non-extremal black hole, the entanglement entropy grows linearly in time and can be saturated after the Page time as expected. The ent…
We point out that the entropy of (near) extremal black holes can be interpreted as the entanglement entropy of dual conformal quantum mechanics via AdS2/CFT1. As an explicit example, we study near extremal BTZ black holes and derive this claim from AdS3/CFT2. We also analytically compute the entanglement entropy in the two dimensional CFT of a free Dirac fermion compactified on a circle at finite temperature. From this result, we clarify the relation between the thermal entropy and entanglement entropy, which is essential for the entanglement interpretation of black hole entropy.
In arXiv:1501.03019 [hep-th], the areas of certain complex extremal surfaces in de Sitter space were found to have resemblance with entanglement entropy in appropriate dual Euclidean non-unitary CFTs, with the area being real and negative in $dS_4$. In this paper, we study some toy models of 2-dim ghost conformal field theories with negative central charge with a view to exploring this further from the CFT point of view. In particular we consider $bc$-ghost systems with central charge $c=-2$ and study the replica formulation for entanglement entropy for a single interval, and associated issues…
We propose that holographic entanglement entropy can be calculated at arbitrary orders in the bulk Planck constant using the concept of a "quantum extremal surface": a surface which extremizes the generalized entropy, i.e. the sum of area and bulk entanglement entropy. At leading order in bulk quantum corrections, our proposal agrees with the formula of Faulkner, Lewkowycz, and Maldacena, which was derived only at this order; beyond leading order corrections, the two conjectures diverge. Quantum extremal surfaces lie outside the causal domain of influence of the boundary region as well as its …
We test whether static charged dilaton black holes in 2+1 dimensions can be turned into naked singularities, by sending in test particles from infinity. We derive that overcharging is possible and generic for both extremal and nearly extremal black holes. Our analysis also imply that nearly extremal charged dilaton black holes can be continuously driven to extremality and beyond, unlike nearly extremal Baunados-Teitelboim-Zanelli, Kerr and Reissner-Nordström black holes which are overspun or overcharged by a discrete jump. Thus the weak form of the cosmic censorship conjecture and the third la…
Using a solution-generating method, we derive an exact solution of the Einstein's field equations in five dimensions describing multi-black hole configurations. More specifically, this solution describes systems of non-extremal static black holes with Kaluza-Klein asymptotics. As expected, we find that, in general, there are conical singularities in-between the Kaluza-Klein black holes that cannot be completely eliminated. Notwithstanding the presence of these conical singularities, such solutions still exhibit interesting thermodynamical properties. By choosing an appropriate set of thermodyn…
Motivated by the holographic prescriptions for computing entanglement entropy and complexity, we study the properties of volumes/areas of bulk surfaces. We obtain a simple formula for the shape dependence of holographic entanglement entropy in terms of a certain integral over the entangling surface. This easily generalizes to any bulk codimension-$p$ extremal surface. We study additional properties of bulk codimension-$p$ extremal surfaces corresponding to strip/plane "entangling surfaces" in various geometries. We compute universal terms for codim-one volumes (conjectured to be dual to hologr…
We elaborate on the role of extremal surfaces probing the internal space in AdS/CFT. Extremal surfaces in AdS quantify the "geometric" entanglement between different regions in physical space for the dual CFT. This, however, is just one of many ways to split a given system into subsectors, and extremal surfaces in the internal space should similarly quantify entanglement between subsectors of the theory. For the case of AdS$_5\times$S$^5$, their area was interpreted as entanglement entropy between U(n) and U(m) subsectors of U(n+m) N=4 SYM. Making this proposal precise is subtle for a number o…
We study the evolution of holographic complexity in various AdS/CFT models containing cosmological crunch singularities. We find that a notion of complexity measured by extremal bulk volumes tends to decrease as the singularity is approached in CFT time, suggesting that the corresponding quantum states have simpler entanglement structure at the singularity.
We investigate the behavior of the entanglement entropy of a confining gauge theory near cosmological singularities using gauge/gravity duality. As expected, the coefficients of the UV divergent terms are given by simple geometric properties of the entangling surface in the time-dependent background. The finite (universal) part of the entanglement entropy either grows without bound or remains bounded depending on the nature of the singularity and entangling region. We also discuss a confinement/deconfinement phase transition as signaled by the entanglement entropy.
We study the canonical purification (with respect to one of the parties) of pure, bi-partite states obtained by turning on sources in the Euclidean path integral. In holographic conformal field theories, the Lorentzian bulk dual of the canonical purification consists of the corresponding entanglement wedge glued to its CPT image at the quantum extremal surface. However, the mismatch in the classical expansions at the QES due to quantum corrections needs to be supported by a shock in the bulk matter stress tensor in order for the bulk to satisfy Einstein's equations. Working perturbatively to f…
Extremal black holes are studied in a two dimensional model motivated by a dimensional reduction from four dimensions. Their quantum corrected geometry is calculated semiclassically and a mild singularity is shown to appear at the horizon. Extensions of the geometry past the horizon are not unique but there are continuations free from malevolent singularities. A few comments are made about the relevance of these results to four dimensions and to the study of black hole entropy and information loss.
The Ryu-Takayanagi conjecture predicts a holographic dual of the entanglement entropy of a CFT. It proposes that the entanglement entropy is given by the area of the minimal surface in the dual spacetime. In the semi-classical limit, this conjecture is supported by the saddle point approximation. If there are multiple classical solutions, it is assumed that only the minimal action contributes to the entanglement entropy. However, we will point out that these saddles equally contribute to the entanglement entropy in some cases. Therefore, the derivation of the conjecture is incomplete if there …
We study a class of extremal transitions between topological distinct Calabi--Yau manifolds which have an interpretation in terms of the special massless states of a type II string compactification. In those cases where a dual heterotic description exists the exceptional massless states are due to genuine strong (string-) coupling effects. A new feature is the appearance of enhanced non-abelian gauge symmetries in the exact nonperturbative theory.
We present a natural generalization of holographic entanglement entropy proposals beyond the scope of AdS/CFT by anchoring extremal surfaces to holographic screens. Holographic screens are a natural extension of the AdS boundary to arbitrary spacetimes and are preferred codimension 1 surfaces from the viewpoint of the covariant entropy bound. A broad class of screens have a unique preferred foliation into codimension 2 surfaces called leaves. Our proposal is to find the areas of extremal surfaces achored to the boundaries of regions in leaves. We show that the properties of holographic screens…
We study the entanglement entropy between a strip region with width $2R$ and its complement in strongly coupled large-$N$ conformal field theory (CFT) on $\mathbb{R}^{1,n}$ with chemical potential and angular momentum in an thermal equilibrium state, which dual to the cylindrical Kerr-Newman black hole. Using Ryu-Takanayagi conjecture, we are able to explore the entanglement entropy through calculating the area of the extremal surface which is anchored on the entangled surface. Because we consider the rotating charged black hole for the gravitational dual, the entropy can be characterized by t…
We study extremal surfaces in a traversable wormhole geometry that connects two locally AdS$_5$ asymptotic regions. In the context of the AdS/CFT correspondence, we use these to compute the holographic entanglement entropy for different configurations: First, we consider an extremal surface anchored at the boundary on a spatial $2$-sphere of radius $R$. The other scenario is a slab configuration which extends in two of the boundary spacelike directions while having a finite size $L$ in the third one. We show that in both cases the divergent and the finite pieces of the holographic entanglement…
We study diffraction catastrophes of wave functions in diffeomorphism invariant quantum theories, for which $\hat HΨ=0$. These wave functions can be represented in terms of integrations over cycles in a complexified lapse variable $N$. The integrand $\exp(i{\mathbb S}(N))$ may have multiple essential singularities at finite values of $N$ and at infinity. A basis set for Greens functions and solutions of the wave equation is represented by Lefschetz thimbles connecting these singularities. The finite $N$ singularities are shown to be directly related to $A_{n\ge 3}$ caustics. We give an example…
We classify all fundamental electrically charged thin shells in general relativity, i.e., static spherically symmetric perfect fluid thin shells with a Minkowski spacetime interior and a Reissner-Nordström spacetime exterior, characterized by the spacetime mass and electric charge. The fundamental shell can exist in three states, nonextremal, extremal, and overcharged. The nonextremal state allows the shell to be located such that its radius can be outside its own gravitational radius, or can be inside its own Cauchy radius. The extremal state allows the shell to be located such that its radiu…
We show that the bulk region reconstructable from a given boundary subregion --- which we term the reconstruction wedge --- can be much smaller than the entanglement wedge even when backreaction is small. We find arbitrarily large separations between the reconstruction and entanglement wedges in near-vacuum states for regions close to an entanglement phase transition, and for more general regions in states with large energy (but very low energy density). Our examples also illustrate situations for which the quantum extremal surface is macroscopically different from the Ryu-Takayanagi surface.
Probing the bubble interior with entanglement entropy and bulk-cone singularities
Università Cattolica del Sacro Cuore（天主教撒克雷圣心大学）；INFN Sezione di Perugia（意大利国家核物理研究所佩鲁加分部）；Institute for Theoretical Physics, KU Leuven（荷语鲁汶大学理论物理研究所）；TIFPA - INFN, c/o Dipartimento di Fisica, Università di Trento（特伦托大学物理系意大利国家核物理研究所理论与应用物理中心）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
We study extremal surfaces in de Sitter space in the Poincare slicing in the upper patch, anchored on spatial subregions at the future boundary ${\cal I}^+$, restricted to constant boundary Euclidean time slices (focussing on strip subregions). We find real extremal surfaces of minimal area as the boundaries of past lightcone wedges of the subregions in question: these are null surfaces with vanishing area. We also find complex extremal surfaces as complex extrema of the area functional, and the area is not always real-valued. In $dS_4$ the area is real. The area has structural resemblance wit…
Extremal black holes tend to expel magnetic and electric fields. Fields are unable to reach the horizon because the length of the black hole throat blows up in the extremal limit. The length of the throat is related to the amount of entanglement between modes on either side of the horizon. So it is natural to try to relate the black hole Meissner effect to entanglement. We derive the black hole Meissner effect directly from the low temperature limit of two-point functions in the Hartle-Hawking vacuum. Then we discuss several new examples of the black hole Meissner effect, its applications to a…
The covariant holographic entropy conjecture of AdS/CFT relates the entropy of a boundary region R to the area of an extremal surface in the bulk spacetime. This extremal surface can be obtained by a maximin construction, allowing many new results to be proven. On manifolds obeying the null curvature condition, these extremal surfaces: i) always lie outside the causal wedge of R, ii) have less area than the bifurcation surface of the causal wedge, iii) move away from the boundary as R grows, and iv) obey strong subadditivity and monogamy of mutual information. These results suggest that the in…
The Ryu-Takayanagi and covariant Hubeny-Rangamani-Takayanagi proposals relate entanglement entropy in CFTs with holographic duals to the areas of minimal or extremal surfaces in the bulk geometry. We show how, in three dimensional pure gravity, the relevant regulated geodesic lengths can be obtained by writing a spacetime as a quotient of AdS3, with the problem reduced to a simple purely algebraic calculation. We explain how this works in both Lorentzian and Euclidean formalisms, before illustrating its use to obtain novel results in a number of examples, including rotating BTZ, the RP2 geon, …
Bulk quantum fields are often said to contribute to the generalized entropy $\frac{A}{4G_N} +S_{\mathrm{bulk}}$ only at $O(1)$. Nonetheless, in the context of evaporating black holes, $O(1/G_N)$ gradients in $S_{\mathrm{bulk}}$ can arise due to large boosts, introducing a quantum extremal surface far from any classical extremal surface. We examine the effect of such bulk quantum effects on quantum extremal surfaces (QESs) and the resulting entanglement wedge in a simple two-boundary $2d$ bulk system defined by Jackiw-Teitelboim gravity coupled to a 1+1 CFT. Turning on a coupling between one bo…
The Ryu-Takayanagi formula relates entanglement entropy in a field theory to the area of extremal surfaces anchored to the boundary of a dual AdS space. It is interesting to ask if there is also an information theoretic interpretation of the areas of non-extremal surfaces that are not necessarily boundary-anchored. In general, the physics outside such surfaces is associated to observers restricted to a time-strip in the dual boundary field theory. When the latter is two-dimensional, it is known that the differential entropy associated to the strip computes the length of the dual bulk curve, an…
