The Schwarzschild singularity is known to be classically unstable. We demonstrate a simple holographic consequence of this fact, focusing on a perturbation that is uniform in boundary space and time. Deformation of the thermal state of the dual CFT by a relevant operator triggers a nonzero temperature holographic renormalization group flow in the bulk. This flow continues smoothly through the horizon and, at late interior time, deforms the Schwarzschild singularity into a more general Kasner universe. We show that the deformed near-singularity, trans-horizon Kasner exponents determine specific non-analytic corrections to the thermal correlation functions of heavy operators in the dual CFT, in the analytically continued `near-singularity' regime.
Considering a thermal state of the dual CFT with a uniform deformation by a scalar operator, we study a holographic renormalization group flow at nonzero temperature in the bulk described by the Einstein-scalar field theory with the self-interaction term $λϕ^4$ in asymptotic anti-de Sitter spacetime. We show that the holographic flow with the self-interaction term could run smoothly through the event horizon of a black hole and deform the Schwarzschild singularity to a Kasner universe at late times. Furthermore, we also study the effect of the scalar self-interaction on the deformed near-singu…
The recent study of holographic superconductors has shown the emergence of a Kasner universe behind the event horizon. This paper serves to add to the discussion by introducing two modifications to the holographic superconductor model: an axion field term and an Einstein-Maxwell-scalar (EMS) coupling term. We first discuss the effect the modification parameters have on the condensate then explore the black hole interior dynamics. Features previously identified in the interior are found in the model presented, including the collapse of the Einstein-Rosen bridge, Josephson oscillations and Kasne…
Charged black holes in anti-de Sitter space become unstable to forming charged scalar hair at low temperatures $T < T_\text{c}$. This phenomenon is a holographic realization of superconductivity. We look inside the horizon of these holographic superconductors and find intricate dynamical behavior. The spacetime ends at a spacelike Kasner singularity, and there is no Cauchy horizon. Before reaching the singularity, there are several intermediate regimes which we study both analytically and numerically. These include strong Josephson oscillations in the condensate and possible 'Kasner inversions…
We explore properties of path integral complexity in field theories on time dependent backgrounds using its dual description in terms of Hartle-Hawking wavefunctions. In particular, we consider boundary theories with time dependent couplings which are dual to Kasner-AdS metrics in the bulk with a time dependent dilaton. We show that holographic path integral complexity decreases as we approach the singularity, consistent with earlier results from holographic complexity conjectures. Furthermore, we find examples where the complexity becomes universal i.e., independent of the Kasner exponents, b…
We consider a holographic dual of hydrodynamics of N=4 SYM plasma that undergoes non-isotropic three-dimensional expansion relevant to RHIC fireball. Our model is a natural extension of the Bjorken's one-dimensional expansion, and it describes an elliptic flow whose v2 and eccentricity are given in terms of anisotropy parameters. Holographic renormalization shows that absence of conformal anomaly in the SYM theory constrains our local rest frame to be a Kasner spacetime. We show that the Kasner spcetime provides a simple description of the anisotropically expanding fluid within a well-controle…
We study the interior dynamics of a top-down holographic superconductor from M-theory. The condense of the charged scalar hair necessarily removes the inner Cauchy horizon and the spacetime ends at a spacelike singularity. Although there is a smooth superconducting phase transition at the critical temperature, the onset of superconductivity is accompanied by intricate interior dynamics, including the collapse of the Einstein-Rosen bridge, the Josephson oscillations of the condensate, and the final Kasner singularity. We obtain analytically the transformation rule for the alternation of differe…
Exact solutions to Einstein's equations for holographic models are presented and studied. The IR geometry has a timelike cousin of the Kasner singularity, which is the less generic case of the BKL (Belinski-Khalatnikov-Lifshitz) singularity, and the UV is asymptotically AdS. This solution describes a holographic RG flow between them. The solution's appearance is an interpolation between the planar AdS black hole and the AdS soliton. The causality constraint is always satisfied. The entanglement entropy and Wilson loops are discussed. The boundary condition for the current-current correlation f…
We study the black hole interiors in holographic Weyl semimetals and holographic nodal line semimetals. We find that the black hole singularities are of Kasner form. In the topologically nontrivial phase at low temperature, both the Kasner exponents of the metric fields and the proper time from the horizon to the singularity are almost constant, likely reflecting the topological nature of the topological semimetals. We also find some specific behaviors inside the horizon in each holographic semimetal model.
Black hole interiors of homogeneous holographic solids under shear strain
Dalian University of Technology（大连理工大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院，中国科学院大学）
Probing the Singularity of Scalar-Haired Black Holes with Holographic Complexity
Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris（巴黎高等师范学院物理实验室，ENS，PSL大学，CNRS，索邦大学，巴黎大学）；Departement Physik, ETH Zürich（苏黎世联邦理工学院物理系）；Département de Physique, École Polytechnique, IP Paris（巴黎综合理工学院物理系）
We consider Kasner space-time describing anisotropic three dimensional expansion of RHIC and LHC fireball and study the generalization of Bjorken's one dimensional expansion by taking into account second order relativistic viscous hydrodynamics. Using time dependent AdS/CFT correspondence, we study the late time behaviour of the Bjorken flow. From the conditions of conformal invariance and energy-momentum conservation, we obtain the explicit expression for the energy density as a function of proper time in terms of Kasner parameters. The proper time dependence of the temperature and entropy ha…
School of Mathematics and Hamilton Mathematics Institute, Trinity College（三一学院数学与哈密顿数学研究所）；School of Mathematics & Statistics, University College Dublin（都柏林大学学院数学与统计学院）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）
In the context of massive gravity theories, we study holographic flows driven by a relevant scalar operator and interpolating between a UV 3-dimensional CFT and a trans-IR Kasner universe. For a large class of scalar potentials, the Cauchy horizon never forms in presence of a non-trivial scalar hair, although, in absence of it, the black hole solution has an inner horizon due to the finite graviton mass. We show that the instability of the Cauchy horizon triggered by the scalar field is associated to a rapid collapse of the Einstein-Rosen bridge. The corresponding flows run smoothly through th…
We compare the behavior of the vacuum free energy (i.e. the Casimir energy) of various $(2+1)$-dimensional CFTs on an ultrastatic spacetime as a function of the spatial geometry. The CFTs we consider are a free Dirac fermion, the conformally-coupled scalar, and a holographic CFT, and we take the spatial geometry to be an axisymmetric deformation of the round sphere. The free energies of the fermion and of the scalar are computed numerically using heat kernel methods; the free energy of the holographic CFT is computed numerically from a static, asymptotically AdS dual geometry using a novel app…
Using AdS/CFT prescription, we compute two point Yang-Mills correlator on a constant time slice for the Kasner background. Pushing the surface close to the initial singularity, we find, in some cases, the correlator does not develop pole. This is consistent with the observations made in a recent work. We further numerically calculate similar correlator where the bulk is a Kasner AdS soliton. We find that the qualitative behaviour of the correlator remains unchanged.
We argue that gravity theories in AdS4 are holographically dual to either of two three-dimensional CFT's: the usual Dirichlet CFT1 where the fixed graviton acts as a source for the stress-energy tensor, and a dual CFT2 with a fixed dual graviton which acts as a source for a dual stress-energy tensor. The dual stress-energy tensor is shown to be the Cotton tensor of the Dirichlet CFT. The two CFT's are related by a Legendre transformation generated by a gravitational Chern-Simons coupling. This duality is a gravitational version of electric-magnetic duality valid at any radius r, where the reno…
Considering a doubly holographic model, we study the evolution of holographic subregion complexity corresponding to deformations of bath state by a relevant scalar operator, which corresponds to a renormalization group flow from the AdS-Schwarzschild to the Kasner universe in the bulk. The subregion complexity shows a discontinuous jump at Page time at a fixed perturbation, where the discontinuity depends solely on the system's parameters. We show that the amount of discontinuity decreases with the perturbation as well as with the scaling dimension of the relevant scalar operator.
In this note we study the symmetry-breaking phases of 3D gravity coupled to matter. In particular, we consider black holes with scalar hair as a model of symmetry-breaking phases of a strongly coupled 1+1 dimensional CFT. In the case of a discrete symmetry, we show that these theories admit metastable phases of broken symmetry and study the thermodynamics of these phases. We also demonstrate that the 3D Einstein-Maxwell theory shows continuous symmetry breaking at low temperature. The apparent contradiction with the Coleman-Mermin-Wagner theorem is discussed.
In the AdS$_3$/CFT$_2$ correspondence, we find some conformal field theory (CFT) states that have no bulk description by the Bañados geometry. We elaborate the constraints for a CFT state to be geometric, i.e., having a dual Bañados metric, by comparing the order of central charge of the entanglement/Rényi entropy obtained respectively from the holographic method and the replica trick in CFT. We find that the geometric CFT states fulfill Bohr's correspondence principle by reducing the quantum KdV hierarchy to its classical counterpart. We call the CFT states that satisfy the geometric constrai…
We study a non-relativistic limit of physics in AdS which retains the curvature through a harmonic Newtonian potential. This limit appears in a CFT dual through the spectrum of operators of large dimension and correlation functions of those operators with appropriate kinematics. In an additional flat spacetime limit, the spectrum is determined by scattering phase shifts (proportional to anomalous dimensions), and a CFT correlation function is proportional to the S-matrix. In particular, we describe the effect of resonances on the spectrum and correlation functions. As an example, we discuss th…
We extend our topological renormalization scheme for Entanglement Entropy to holographic CFTs of arbitrary odd dimensions in the context of the AdS/CFT correspondence. The procedure consists in adding the Chern form as a boundary term to the area functional of the Ryu-Takayanagi minimal surface. The renormalized Entanglement Entropy thus obtained can be rewritten in terms of the Euler characteristic and the AdS curvature of the minimal surface. This prescription considers the use of the Replica Trick to express the renormalized Entanglement Entropy in terms of the renormalized gravitational ac…
We propose a new non-holographic formulation of AdS/CFT correspondence, according to which quantum gravity on AdS and its dual non-gravitational field theory both live in the same number D of dimensions. The field theory, however, appears (D-1)-dimensional because the interactions do not propagate in one of the dimensions. The D-dimensional action for the field theory can be identified with the sum over (D-1)-dimensional actions with all possible values $Λ$ of the UV cutoff, so that the extra hidden dimension can be identified with $Λ$. Since there are no interactions in the extra dimension, m…
We consider a (2+1)-dimensional holographic CFT on a static spacetime with globally timelike Killing vector. Taking the spatial geometry to be closed but otherwise general we expect a non-trivial vacuum energy at zero temperature due to the Casimir effect. We assume a thermal state has an AdS/CFT dual description as a static smooth solution to gravity with a negative cosmological constant, which ends only on the conformal boundary or horizons. A bulk geometric argument then provides an upper bound on the ratio of CFT free energy to temperature. Considering the zero temperature limit of this bo…
In this note, we establish the 2-D Quantum Inequalities - first proved by Flanagan - for all CFTs with a causal holographic dual. Following the treatment of Kelly \& Wall, we establish that the Boundary Causality Condition in an asymptotic AdS spacetime implies the Quantum Inequalities on the boundary. Our results extend easily to curved spacetime and are stable under deformations of the CFT by relevant operators. We discuss higher dimensional generalizations and possible connections to recent bounds on $a/c$ in 4-D CFTs.
In this work, we derive an upper bound on energetic quantities, namely vacuum energy and free energy, for static solutions of Einstein-Scalar theory in four dimensional asymptotically locally Anti-de Sitter(AlAdS) spacetime with a nontrivial scalar potential where the scalar field mass parameter($m^2$) is equal to 0 or -2. This system is the holographic dual of strongly coupled conformal field theory(CFT) in three dimensions being deformed by a relevant or marginal scalar operator of conformal dimension $Δ=1, 2, 3$. The bound is derived from a purely gravitational perspective regardless of the…
We further explore the quantum dynamics near past cosmological singularities in anisotropic Kasner-AdS solutions using gauge/gravity duality. The dual description of the bulk evolution involves N=4 super Yang-Mills on the contracting branch of an anisotropic de Sitter space and is well defined. We compute two-point correlators of Yang-Mills operators of large dimensions using spacelike geodesics anchored on the boundary. The correlator between two points separated in a direction with negative Kasner exponent p always exhibits a pole at horizon scales, in any dimension, which we interpret as a …
Holographic Renormalization Group (RG) in nine dimensions is considered. The d8 holographic conformal anomaly is found. It should correspond to d8 CFT in AdS_9/CFT_8 correspondence. The comparison of holographic and QFT anomalies in d8 de Sitter space is done. It may give the indication for rigorous AdS_9/CFT_8 correspondence proposal.
In this work, we formulate a Non-Commutative (NC) extension of AdS-CFT correspondence that is manifested in the modification of behavior of a holographic superconductor. The noncommutativity is introduced in the model through the NC corrected AdS charged black hole metric developed by Nicolini, Smailagic and Spallucci. First of all we discuss thermodynamic properties of this black hole in Euclidean formalism. In particular, we compute trace of the boundary energy-momentum tensor which, as expected, is non-zero due to the NC scale introduced in the model. Our findings indicate that the non-comm…
We present a general procedure for constructing tensor networks for geometric states in the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. Given a state in a large-$N$ CFT with a static, semiclassical gravitational dual, our procedure produces a tensor network for the boundary state whose internal geometry matches (a discretization of) the bulk spacetime geometry. By invoking the "holographic entanglement of purification" conjecture, our construction can be made to capture the structure of the bulk spacetime at sub-AdS scales.
The final (close to the singularity) dynamical behavior of the metric inside black holes with massive charged scalar or vector hair is analyzed for general anisotropic and inhomogeneous initial conditions. These solutions are relevant to a holographic realization of superconductivity. It is shown that the dynamics falls within the scope of the "cosmological billiard" description and that in both cases, the corresponding hyperbolic billiard region has infinite volume so that the system ultimately settles down to a final Kasner regime. For massive vector hair, the conclusion holds because the lo…
In the presence of a scalar hair perturbation, the Cauchy horizon of a Reissner-Nordström black hole disappears and is replaced by the rapid collapse of the Einstein-Rosen bridge, which leads to a Kasner singularity [1,2]. We study the time-dependence of holographic complexity, both for the volume and for the action proposals, in a class of models with hairy black holes. Volume complexity can only probe a portion of the black hole interior that remains far away from the Kasner singularity. We provide numerical evidence that the Lloyd bound is satisfied by the volume complexity rate in all the …
The aim of the present letter is to find the holographic entanglement entropy (HEE) in 2D holographic superconductors (HSC). Indeed, it is possible to compute the exact form of this entropy due to an advantage of approximate solutions inside normal and superconducting phases with backreactions. By making the UV and IR limits applied to the integrals, an approximate expression for HEE is obtained. In case the software cannot calculate minimal surface integrals analytically, it offers the possibility to proceed with a numerical evaluation of the corresponding terms. We'll understand how the area…
In this article, we explore the holographic Q-picture description for the charged rotating black holes in the five-dimensional minimal supergravity. The central charge in the Q-picture depends only on black hole charge, therefore can be computed from the near horizon geometry of the extremal and non-rotating counterpart. Moreover, the CFT temperatures can be identified by studying the hidden conformal symmetry, and the related gravity-CFT dictionary can be translated via thermodynamics analysis. The entropy and absorption cross section computed from both gravity and CFT sides properly agree wi…
Entanglement entropies are notoriously difficult to compute. Large-N strongly-coupled holographic CFTs are an important exception, where the AdS/CFT dictionary gives the entanglement entropy of a CFT region in terms of the area of an extremal bulk surface anchored to the AdS boundary. Using this prescription, we show -- for quite general states of (2+1)-dimensional such CFTs -- that the renormalized entanglement entropy of any region of the CFT is bounded from above by a weighted local energy density. The key ingredient in this construction is the inverse mean curvature (IMC) flow, which we su…
Using a holographic model, we study quantum field theories with a layer of one CFT surrounded by another CFT, on either a periodic or an infinite direction. We study the vacuum energy density in each CFT as a function of the central charges, the thickness of the layer(s), and the properties of the interfaces between the CFTs. The dual spacetimes in the holographic model include two regions separated by a dynamical interface with some tension. For two or more spatial dimensions, we find that a layer of CFT with more degrees of freedom than the surrounding one can have an anomalously large negat…
It is known that the $(a,c)$ central charges in four-dimensional CFTs are linear combinations of the three independent OPE coefficients of the stress-tensor three-point function. In this paper, we adopt the holographic approach using AdS gravity as an effect field theory and consider higher-order corrections up to and including the cubic Riemann tensor invariants. We derive the holographic central charges and OPE coefficients and show that they are invariant under the metric field redefinition. We further discover a hidden relation among the OPE coefficients that two of them can be expressed i…
The stress tensor is a basic local operator in any field theory; in the context of AdS/CFT, it is the operator which is dual to the bulk geometry itself. Here we exploit this feature by using the bulk geometry to place constraints on the local energy density in static states of holographic $(2+1)$-dimensional CFTs living on a closed (but otherwise generally curved) spatial geometry. We allow for the presence of a marginal scalar deformation, dual to a massless scalar field in the bulk. For certain vacuum states in which the bulk geometry is well-behaved at zero temperature, we find that the bu…
The geometerization of the renormalization group flow triggered by the $T\bar{T}$ deformation of large $c$ conformal field theories in two dimensions is presented. This entails the construction of the off shell Einstein-Hilbert action in three dimensions from said renormalization group flow. The crucial ingredient to this construction will be the encoding of general covariance in the emergent bulk theory in a very particular form of the Wess--Zumino consistency conditions. The utilisation of the local renormalization group, which requires putting the theory under consideration on an arbitrary …
We explore a fruitful connection between the physics of conformal field theories (CFTs) in d-dimensional Minkowski spacetime and the extended gravitational thermodynamics of hyperbolic black holes in (d+1)-dimensional anti-de Sitter spacetime. The CFTs are reduced on a region bounded by a sphere. We show that Renormalization Group (RG) flows between CFTs are specific thermodynamic processes in the (p,V) plane, where the irreversibility of coarse-graining flows from the ultraviolet to the infrared corresponds to the Second Law of thermodynamics, preventing heat from flowing from low temperature…
We extend the holographic duality between 3d pure gravity and the 2d Ising CFT proposed in [Phys. Rev. D 85 (2012) 024032] to CFTs with boundaries. Besides the usual asymptotic boundary, the dual bulk spacetime now has a real cutoff, on which live branes with finite tension, giving Neumann boundary condition on the metric tensor. The strongly coupled bulk theory requires that we dress the well-known semiclassical AdS/BCFT answer with boundary gravitons, turning the partition function into the form of Virasoro characters. Using this duality, we relate the brane tensions to the modular S-matrix …
The number of asymptotically de Sitter (non-singular) solutions of 5d dilatonic gravity with positive cosmological constant is found. These solutions are similar to the previously known asymptotically AdS spaces where dilaton may generate the singularity. Using these solutions the consistent $c$-function is proposed in the same way as in AdS/CFT. The consistency of RG flow gives further support for dS/CFT correspondence. From holographic RG flow equations we calculate the holographic 4d conformal anomaly with dilatonic contributions. This conformal anomaly turns out to be the same as in AdS/CF…
We present a large class of holographic models where the boundary ${\mathbb R}^{2,1}$ dimensional conformal field theory has a thermal phase with a spontaneously broken global symmetry. The dual black branes in a Poincare patch of asymptotically $AdS_4$ violate the no-hair theorem.
We propose a holographic entanglement negativity conjecture involving the bulk geometry, for mixed states of adjacent intervals in $(1+1)$-dimensional dual conformal field theories through the $AdS/CFT$ correspondence. The holographic entanglement negativity is obtained from a specific algebraic sum of the geodesics anchored on respective intervals on the boundary which reduces to the holographic mutual information between them. Utilizing our conjecture we obtain the entanglement negativity of adjacent intervals in zero and finite temperature ($1+1$)-dimensional holographic conformal field the…
