Holographic complexity in general quadratic curvature theory of gravity
Ahmad Ghodsi, Saeed Qolibikloo, Saman Karimi
备注：
19 pages, 4 figures; v3. Subsection 2.5 modified and more discussions on the second singularity in new subsection 2.6 added. Accepted for publication in EPJC
In the context of CA conjecture for holographic complexity, we study the action growth rate at late time approximation for general quadratic curvature theory of gravity. We show how the Lloyd's bound saturates for charged and neutral black hole solutions. We observe that a second singular point may modify the action growth rate to a value other than the Lloyd's bound. Moreover, we find the universal terms that appear in the divergent part of complexity from computing the bulk and joint terms on a regulated WDW patch.
We study the structure of the higher-curvature gravitational densities that are induced from holographic renormalization in AdS$_{d+1}$. In a braneworld construction, such densities define a d-dimensional higher-curvature gravitational theory on the brane, which in turn is dual to a (d-1)-dimensional CFT living at its boundary. We show that this CFT$_{d-1}$ satisfies a holographic c-theorem in general dimensions (different than the g-theorem of holographic boundary CFTs), since at each and every order the higher-curvature densities satisfy c-theorems on their own. We find that, in these densit…
We study a fully backreacted holographic model of a four-dimensional superconductor by including a higher curvature interaction in the bulk action. We study how the critical temperature and the field theory condensate vary in this model and conclude that positive higher curvature couplings make the condensation harder. We also compute the conductivity, finding significant deviations from the conjectured universal frequency gap to critical temperature ratio.
In this paper we show that in addition to the known minimal surfaces which appear in the literature for computing the entanglement entropy there are other minimal surfaces with non-zero extrinsic curvature. We use the approach of regularization procedure for computing the quadratic and cubic curvature invariants on manifolds with squashed cones. The results can be used to find the leading and universal terms of the holographic entanglement entropy to understand which solution corresponds to the actual minimal surface.
Standard methods for calculating the black hole entropy beyond general relativity are ambiguous when the horizon is non stationary. We fix these ambiguities in all quadratic curvature gravity theories, by demanding that the entropy be increasing at every time, for linear perturbations to a stationary black hole. Our result matches with the entropy formula found previously in holographic entanglement entropy calculations. We explicitly calculate the entropy increase for Vaidya-like solutions in Ricci-tensor gravity to show that (unlike the Wald entropy) the holographic entropy obeys a Second La…
We derive a covariant expression for the renormalized holographic entanglement entropy for Conformal Field Theories (CFTs) dual to Quadratic Curvature Gravity in arbitrary dimensions. This expression is written as the sum of the bare entanglement entropy functional obtained using standard conical defect techniques, and a counterterm defined at the boundary of the extremal surface of the functional. The latter corresponds to the cod-2 self-replicating part of the extrinsic counterterms when evaluated on the replica orbifold. This renormalization method isolates the universal terms of the hologr…
We study the extremal surfaces of functionals recently proposed for the holographic calculation of entanglement entropy in general higher curvature theories, using New Massive gravity and Gauss-Bonnet gravity as concrete examples. We show that the entropy functionals admit closed extremal surfaces, which for black hole backgrounds can encircle the event horizon of the black hole. In the examples considered, such closed surfaces correspond to a lower value of the entropy functional than expected from CFT calculations, implying a seeming mismatch between the bulk and boundary calculations. For L…
The holographic entanglement entropy functional for higher-curvature gravities involves a weighted sum whose evaluation, beyond quadratic order, requires a complicated theory-dependent splitting of the Riemann tensor components. Using the splittings of general relativity one can obtain unambiguous formulas perturbatively valid for general higher-curvature gravities. Within this setup, we perform a novel rewriting of the functional which gets rid of the weighted sum. The formula is particularly neat for general cubic and quartic theories, and we use it to explicitly evaluate the corresponding f…
In this paper we place observational constraints on the interaction and spatial curvature in the holographic dark energy model. We consider three kinds of phenomenological interactions between holographic dark energy and matter, i.e., the interaction term $Q$ is proportional to the energy densities of dark energy ($ρ_Λ$), matter ($ρ_{m}$), and matter plus dark energy ($ρ_m+ρ_Λ$). For probing the interaction and spatial curvature in the holographic dark energy model, we use the latest observational data including the type Ia supernovae (SNIa) Constitution data, the shift parameter of the cosmic…
Renormalized entanglement entropy can be defined using the replica trick for any choice of renormalization scheme; renormalized entanglement entropy in holographic settings is expressed in terms of renormalized areas of extremal surfaces. In this paper we show how holographic renormalized entanglement entropy can be expressed in terms of the Euler invariant of the surface and renormalized curvature invariants. For a spherical entangling region in an odd-dimensional CFT, the renormalized entanglement entropy is proportional to the Euler invariant of the holographic entangling surface, with the …
We examine holographic entanglement entropy with higher curvature gravity in the bulk. We show that in general Wald's formula for horizon entropy does not yield the correct entanglement entropy. However, for Lovelock gravity, there is an alternate prescription which involves only the intrinsic curvature of the bulk surface. We verify that this prescription correctly reproduces the universal contribution to the entanglement entropy for CFT's in four and six dimensions. We also make further comments on gravitational theories with more general higher curvature interactions.
Recent works have demonstrated that one can construct a (d+2) dimensional solution of the vacuum Einstein equations that is dual to a (d+1) dimensional fluid satisfying the incompressible Navier-Stokes equations. In one important example, the fluid lives on a fixed timelike surface in the flat Rindler spacetime associated with an accelerated observer. In this paper, we show that the shear viscosity to entropy density ratio of the fluid takes the universal value 1/4πin a wide class of higher curvature generalizations to Einstein gravity. Unlike the fluid dual to asymptotically anti-de Sitter sp…
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…
We study (3+1)-dimensional holographic superconductors in Einstein-Gauss-Bonnet gravity both numerically and analytically. It is found that higher curvature corrections make condensation harder. We give an analytic proof of this result, and directly demonstrate an analytic approximation method that explains the qualitative features of superconductors as well as giving quantitatively good numerical results. We also calculate conductivity and $ω_g / T_c $, for $ω_g$ and $T_c$ the gap in the frequency dependent conductivity and the critical temperature respectively. It turns out that the `univers…
We use the holographic language to show the existence of the $a$-theorem for even dimensional CFTs, dual to the AdS space in general quadratic curvature gravity. We find the Wess-Zumino action which is originated from the spontaneous breaking of the conformal symmetry in $d\leq 8$, by using a radial cut-off near the AdS boundary. We also study the RG flow and (average) null energy condition in the space of the couplings of theory. In a simple toy model, we find the regions where this holographic RG flow has a monotonic decreasing behavior.
We consider the effects of higher curvature terms on a holographic dual description of boundary conformal field theory. Specifically, we consider three-dimensional gravity with a specific combination of Ricci tensor square and curvature scalar square, so called, new massive gravity. We show that a boundary entropy and an entanglement entropy are given by similar expression with those of the Einstein gravity case when we introduce an {\it effective} Newton's constant and an {\it effective} cosmological constant. We also show that the holographic g-theorem still holds in this extension, and we g…
The holographic entanglement entropy for the most general higher derivative gravity is investigated. We find a new type of Wald entropy, which appears on entangling surface without the rotational symmetry and reduces to usual Wald entropy on Killing horizon. Furthermore, we obtain a formal formula of HEE for the most general higher derivative gravity and work it out exactly for some squashed cones. As an important application, we derive HEE for gravitational action with one derivative of the curvature when the extrinsic curvature vanishes. We also study some toy models with non-zero extrinsic …
In this paper, the curvature structure of a (2+1)-dimensional black hole in the massive-charged-Born-Infeld gravity is investigated. The metric that we consider is characterized by four degrees of freedom which are the mass and electric charge of the black hole, the mass of the graviton field, and a cosmological constant. For the charged and neutral cases separately, we present various constraints among scalar polynomial curvature invariants which could invariantly characterize our desired spacetimes. Specially, an appropriate scalar polynomial curvature invariant and a Cartan curvature invari…
Motivated by the holographic principle, it has been suggested that the dark energy density may be inversely proportional to the area of the event horizon of the Universe. However, such a model would have a causality problem. In this paper, we propose to replace the future event horizon area with the inverse of the Ricci scalar curvature. We show that this model does not only avoid the causality problem and is phenomenologically viable, but also naturally solves the coincidence problem of dark energy. Our analysis of the evolution of density perturbations show that the matter power spectra and …
We model the soft pomeron in QCD using a scalar Polyakov string with extrinsic curvature in the bottom-up approach of holographic QCD. The overall dipole-dipole scattering amplitude in the soft pomeron kinematics is shown to be sensitive to the extrinsic curvature of the string for finite momentum transfer. The characteristics of the diffractive peak in the differential elastic $pp$ scattering are affected by a small extrinsic curvature of the string.
We present an holographic model of the Dvali-Gabadadze-Porrati scenario with a Gauss-Bonnet term in the bulk. We concentrate on the solution that generalises the normal Dvali-Gabadadze-Porrati branch. It is well known that this branch cannot describe the late-time acceleration of the universe even with the inclusion of a Gauss-Bonnet term. Here, we show that this branch in the presence of a Gauss-Bonnet curvature effect and an holographic dark energy with the Hubble scale as the infra-red cutoff can describe the late-time acceleration of the universe. It is worthwhile to stress that such an en…
We study the holographic superconductors in f(R) gravity, and show how the critical temperature and the condensate of the dual operator depend on the modifications to Einstein gravity. We first review the black holes with constant curvature, which are clear in f(R) gravity. The dependencies are represented with the modified Newtonian constant and the effective cosmological constant trivially. Then we consider a planar black hole solution for a specific f(R) without imposing the constant curvature, which is asymptotic to AdS spacetime. The corrections to the thermal dynamics of the black holes,…
In this paper, we report the results of constraining the holographic dark energy model with spatial curvature and massive neutrinos, based on a Markov Chain Monte Carlo global fit technique. The cosmic observational data include the full WMAP 7-yr temperature and polarization data, the type Ia supernova data from Union2.1 sample, the baryon acoustic oscillation data from SDSS DR7 and WiggleZ Dark Energy Survey, and the latest measurements of $H_0$ from HST. To deal with the perturbations of dark energy, we adopt the parameterized post-Friedmann method. We find that, for the simplest holographi…
We study the holographic renormalization group (RG) flow in the presence of higher-order curvature corrections to the $(d+1)$-dimensional Einstein-Hilbert (EH) action for an arbitrary interacting scalar matter field by using the superpotential approach. We find the critical points of the RG flow near the local minima and maxima of the potential and show the existence of the bounce solutions. In contrast to the EH gravity, regarding the values of couplings of the bulk theory, superpotential may have both upper and lower bounds. Moreover, the behavior of the RG flow controls by singular curves. …
We show that Thurston geometries are solutions to a large class of 3D quadratic curvature theories, where New Massive Gravity, which was studied in arXiv:2104.00754, is a special case.
We study the effects of Gauss-Bonnet corrections on some nonlocal probes (entanglement entropy, $n$-partite information and Wilson loop) in the holographic model with momentum relaxation. Higher-curvature terms as well as scalar fields make in fact nontrivial correction to the coefficient of universal term in entanglement entropy. We use holographic methods to study such corrections. Moreover, holographic calculation indicates that mutual and tripartite information undergo a transition beyond which they identically change their values. We find that the behavior of transition curves depends on …
For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulae for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmor…
We have reexamined the holographic dark energy model by considering the spatial curvature. We have refined the model parameter and observed that the holographic dark energy model does not behave as phantom model. Comparing the holographic dark energy model to the supernova observation alone, we found that the closed universe is favored. Combining with the Wilkinson Microwave Anisotropy Probe (WMAP) data, we obtained the reasonable value of the spatial curvature of our universe.
We investigate the neutral AdS black-hole solution in the consistent $D\rightarrow4$ Einstein-Gauss-Bonnet gravity proposed in [K. Aoki, M.A. Gorji, and S. Mukohyama, Phys. Lett. B {\bf 810}, 135843 (2020)] and construct the gravity duals of ($2+1$)-dimensional superconductors with Gauss-Bonnet corrections in the probe limit. We find that the curvature correction has a more subtle effect on the scalar condensates in the s-wave superconductor in ($2+1$)-dimensions, which is different from the finding in the higher-dimensional superconductors that the higher curvature correction makes the scalar…
In this note, we evaluate the Weyl-invariant quadratic curvature tensors for the particular Weyl's gauge field constructed in the $3+1$-dimensional noncompact Weyl-Einstein-Yang-Mills model. We subsequently extend the model to its higher curvature version. Here, we also compute Weyl-invariant extension of topological Gauss-Bonnet term for this specific choice of vector field.
We investigate the holographic fermions in the charged Gauss-Bonnet $AdS_{d}$ black hole background with the dipole coupling between fermion and gauge field in the bulk. We show that in addition to the strength of the dipole coupling, the spacetime dimension and the higher curvature correction in the gravity background also influence the onset of the Fermi gap and the gap distance. We find that the higher curvature effect modifies the fermion spectral density and influences the value of the Fermi momentum for the appearance of the Fermi surface. There are richer physics in the boundary fermion…
We propose a recipe - arguably the simplest - to compute the holographic type-B Weyl anomaly for general higher-derivative gravity in asymptotically AdS spacetimes. In 5 and 7 dimensions we identify a suitable basis of curvature invariants that allows to read off easily, without any further computation, the Weyl anomaly coefficients of the dual CFT. We tabulate the contributions from quadratic, cubic and quartic purely algebraic curvature invariants and also from terms involving derivatives of the curvature. We provide few examples, where the anomaly coefficients have been obtained by other me…
We study the effect of the Gauss-Bonnet term on the complexity growth rate of dual field theory using the "Complexity--Volume" (CV) and CV2.0 conjectures. We investigate the late time value and full time evolution of the complexity growth rate of the Gauss-Bonnet black holes with horizons with zero curvature ($k=0$), positive curvature ($k=1$) and negative curvature ($k=-1$) respectively. For the $k=0$ and $k=1$ cases we find that the Gauss-Bonnet term suppresses the growth rate as expected, while in the $k=-1$ case the effect of the Gauss-Bonnet term may be opposite to what is expected. The r…
We define the notion of energy, and compute its values, for gravitational systems involving terms quadratic in curvature. While our construction parallels that of ordinary Einstein gravity, there are significant differences both conceptually and concretely. In particular, for D=4, all purely quadratic models admit vacua of arbitrary constant curvature. Their energies, including that of conformal (Weyl) gravity, necessarily vanish in asymptotically flat spaces. Instead, they are proportional to that of the Abbott-Deser (AD) energy expression in constant curvature backgrounds and therefore also …
Partial results are obtained for Schwarzschild- like solutions in a gravity theory with action density (-g)^(1/2)[Rik^2+bR^2]. A seven parameter family of implicit solutions is found. A number of explicit solutions are also exhibited.
We construct the holographic p-wave superfluid in Gauss-Bonnet gravity via a Maxwell complex vector field model and investigate the effect of the curvature correction on the superfluid phase transition in the probe limit. We obtain the rich phase structure and find that the higher curvature correction hinders the condensate of the vector field but makes it easier for the appearance of translating point from the second-order transition to the first-order one or for the emergence of the Cave of Winds. Moreover, for the supercurrents versus the superfluid velocity, we observe that our results nea…
We construct the holographic superconductors away from the probe limit in the consistent $D\rightarrow4$ Einstein-Gauss-Bonnet gravity. We observe that, both for the ground state and excited states, the critical temperature first decreases then increases as the curvature correction tends towards the Chern-Simons limit in a backreaction dependent fashion. However, the decrease of the backreaction, the increase of the scalar mass, or the increase of the number of nodes will weaken this subtle effect of the curvature correction. Moreover, for the curvature correction approaching the Chern-Simons …
In this paper, we investigate AdS/BCFT for curvature-squared gravity. To warm up, we start with Gauss-Bonnet gravity. We derive the one point function of stress tensor and show that the central charge related to the norm of displacement operator is positive for the couplings obeying causality constraints. Furthermore, by imposing the null energy condition on the end-of-the-world brane, we prove the holographic g-theorem for Gauss-Bonnet gravity. This corrects a wrong point of view in the literature, which claims that the holographic g-theorem is violated for Gauss-Bonnet gravity. As a by-produ…
We study the effects of including higher-curvature corrections to the Einstein gravity bulk action on the holographic entanglement entropy (HEE) expression for geometries with hyperscaling violation (hvLf). For $θ< 0$ we show that one single new divergence arises for general curvature-squared gravities, which allows us to conjecture the general expression of HEE for any higher-order gravity action. For $0<θ<d$, we assume the hvLf geometry to arise above some intermediate scale $r_F$, becoming AdS in the UV and perform a similar analysis for $R^n$ gravities. For negative values of $θ$ we find t…
Quadratic Curvature Correction to the Euclidean Action of Rotating AdS Black Holes in General Dimensions
The International Joint Institute of Tianjin University, Fuzhou, Tianjin University（天津大学福州国际联合学院）；Center for Joint Quantum Studies, Department of Physics, School of Science, Tianjin University（天津大学理学院物理系联合量子研究中心）
We prove a holographic c-theorem for the a central charge in AdS/CFT where the bulk is described by a gravitational action built out of an arbitrary function f(R^{ab}_{cd}) of the Riemann tensor coupled to bulk matter. This theorem holds provided a generalized null energy condition involving both matter and higher curvature gravitational interactions is satisfied. As an example, we consider the case of a curvature-squared action, and find that generically the generalized null energy condition involves not just the bulk matter, but also the sign of R" where a prime denotes a radial derivative a…
We study the holographic entanglement entropy of deformed entangling regions in three-dimensional CFTs dual to Einstein-AdS gravity, using a renormalization scheme based on the addition of extrinsic counterterms. In this prescription, when even-dimensional manifolds are considered, the universal contribution to the entanglement entropy is identified as the renormalized volume of the Ryu-Takayanagi hypersurface, which is written as the sum of a topological and a curvature term. It is shown that the change in the renormalized entanglement entropy due to the deformation of the entangling surface …
We argue for an exponential bound characterizing the chaotic properties of modular Hamiltonian flow of QFT subsystems. In holographic theories, maximal modular chaos is reflected in the local Poincare symmetry about a Ryu-Takayanagi surface. Generators of null deformations of the bulk extremal surface map to modular scrambling modes -positive CFT operators saturating the bound- and their algebra probes the bulk Riemann curvature, clarifying the modular Berry curvature proposal of arXiv:1903.04493.
We provide new formulas for the energy of black hole solutions in the anti-de Sitter (AdS) sector of a generic gravity theory with curvature-squared terms. This is achieved by the addition of counterterms of the extrinsic type (\emph{Kounterterms}) which produces a finite variation of the total action. The procedure is compatible with the boundary conditions in asymptotically AdS spaces within the framework of Quadratic Curvature Gravity (QCG) theory. As a consequence, without resorting to background-substraction, or linearized methods, the conserved charges correctly reproduce the energy of s…
We investigate the combined effect of fluxes and higher-order curvature corrections, in the form of the Gauss-Bonnet term, on the existence of de Sitter vacua in a heterotic string inspired framework, compactified on spheres and tori. We first gain some intuition on the effects of these corrections by studying a perturbative expansion in the small Gauss-Bonnet coupling. Then, for choices of potential closer to the string theory predictions, we show that the inclusion of quadratic curvature corrections actually reduces the parametric likelihood of de Sitter solutions.
It is a known fact that the Kerr-Schild type solutions in general relativity satisfy both exact and linearized Einstein field equations. We show that this property remains valid also for a special class of the Kerr-Schild metrics in arbitrary dimensions in generic quadratic curvature theory. In addition to the AdS-wave (or Siklos) metric which represents plane waves in an AdS background, we present here a new exact solution, in this class, to the quadratic gravity in D-dimensions which represents a spherical wave in an AdS background. The solution is a special case of the Kundt metrics belongi…
In this thesis we explore general aspects of the entanglement entropy (EE) for Conformal Field Theories (CFTs) dual to Cubic Curvature Gravity. We derived a covariant expression for the EE by using a scheme inherited from the bulk renormalization method through extrinsic counterterms. We evaluate this functional in different entangling regions to calculate CFT data. In particular, we compute the $t_4$ coefficient of the 3-point function of the stress-tensor correlator by considering a deformed entangling region. We observe that there is a discrepancy between the outcomes attained through the e…
It is well known that, in single clock inflation, the curvature perturbation $ζ$ is constant in time on superhorizon scales. In the standard bulk description this follows quite simply from the local conservation of the energy momentum tensor in the bulk. On the other hand, in a holographic description, the constancy of the curvature perturbation must be related to the properties of the RG flow in the boundary theory. Here, we show that, in single clock holographic inflation, the time independence of correlators of $ζ$ follows from the cut-off independence of correlators of the energy momentum …
Palatini variational principle is implemented on a five dimensional quadratic curvature gravity model, rendering two sets of equations which can be interpreted as the field equations and the stress-energy tensor. Unification of gravity with electromagnetism and the scalar dilaton field is achieved through the Kaluza-Klein dimensional reduction mechanism. The reduced curvature invariant, field equations and the stress-energy tensor in four dimensional spacetime are obtained. The structure of the interactions among the constituent fields is exhibited in detail. It is shown that the Lorentz force…
We continue our study of gravity described by the action density (-g)^(1/2)(R_ik^2+bR^2); and look for cosmological solutions of gravity coupled to dust, for the closed isotropic model. There is a solution that at t approaches zero has for the radius, a(t)=t/3^(1/2); in the absence of dust this solution holds for all time.
