We investigate general features of the evolution of holographic subregion complexity (HSC) on Vaidya-AdS metric with a general form. The spacetime is dual to a sudden quench process in quantum system and HSC is a measure of the ``difference'' between two mixed states. Based on the subregion CV (Complexity equals Volume) conjecture and in the large size limit, we extract out three distinct stages during the evolution of HSC: the stage of linear growth at the early time, the stage of linear growth with a slightly small rate during the intermediate time and the stage of linear decrease at the late time. The growth rates of the first two stages are compared with the Lloyd bound. We find that with some choices of certain parameter, the Lloyd bound is always saturated at the early time, while at the intermediate stage, the growth rate is always less than the Lloyd bound. Moreover, the fact that the behavior of CV conjecture and its version of the subregion in Vaidya spacetime implies that they are different even in the large size limit.
We study holographic subregion volume complexity for a line segment in the AdS$_3$ Vaidya geometry. On the field theory side, this gravity background corresponds to a sudden quench which leads to the thermalization of the strongly-coupled dual conformal field theory. We find the time-dependent extremal volume surface by numerically solving a partial differential equation with boundary condition given by the Hubeny-Rangamani-Takayanagi surface, and we use this solution to compute holographic subregion complexity as a function of time. Approximate analytical expressions valid at early and at lat…
We investigate the evolution of the mutual information between two spatial subsystems in a compact 1+1 dimensional CFT after a quantum quench. To this end, we use the dual holographic process, given by the 2+1 dimensional Vaidya-BTZ spacetime in global coordinates, which describes the collapse of a spherically symmetric null shell. So, we first discuss the spacelike geodesic structure of this geometry and then we present the various behaviors of the holographic mutual information observed in this case. We also consider the analogous process in the adiabatic limit and compare these two cases fr…
We numerically investigate the evolution of the holographic subregion complexity during a quench process in Einstein-Born-Infeld theory. Based on the subregion CV conjecture, we argue that the subregion complexity can be treated as a probe to explore the interior of the black hole. The effects of the nonlinear parameter and the charge on the evolution of the holographic subregion complexity are also investigated. When the charge is sufficiently large, it not only changes the evolution pattern of the subregion complexity, but also washes out the second stage featured by linear growth.
Recently holographic prescriptions are proposed to compute quantum complexity of a given state in the boundary theory. A specific proposal known as `holographic subregion complexity' is supposed to calculate the the complexity of a reduced density matrix corresponding to a static subregion. We study different families of singular subregions in the dual field theory and find the divergence structure and universal terms of holographic subregion complexity for these singular surfaces. We find that there are new universal terms, logarithmic in the UV cutoff, due to the singularities of a family of…
In this paper, we use the "complexity equals action" (CA) conjecture to discuss growth rate of the complexity in a charged AdS-Vaidya black hole formed by collapsing an uncharged spherically symmetric thin shell of null fluid. Using the approach proposed by Lehner $et\, al.$, we evaluate the action growth rate and the slope of the complexity of formation. Then, we demonstrate that the behaviors of them are in agreement with the switchback effect for the light shock wave case. Moreover, we show that to obtain an expected property of the complexity, it is also necessary for the CA conjecture to …
We study the evolution of holographic subregion complexity under a thermal quench in this paper. From the subregion CV proposal in the AdS/CFT correspondence, the subregion complexity in the CFT is holographically captured by the volume of the codimension-one surface enclosed by the codimension-two extremal entanglement surface and the boundary subregion. Under a thermal quench, the dual gravitational configuration is described by a Vaidya-AdS spacetime. In this case we find that the holographic subregion complexity always increases at early time, and after reaching a maximum it decreases and …
We investigate subregion holographic complexity in the context of renormalization group flow geometries. We use both the Poincaŕe slicing and the Janus ansatz as holographic duals to renormalization group flows in the boundary conformal field theory. In the former metric, subregion complexity is computed for a disc and a strip shaped entangling region. For the disc shaped region, consistent emergence of length scales for flow to the deep infra-red is established. For strip shaped regions, we find that complexity cannot locate holographic phase transitions in a sharp domain wall scenario. For s…
Using holographic subregion complexity, we study the confinement-deconfinement phase transition of quantum chromodynamics. In the model we consider here, we observe a connection between the potential energy of probe meson and the behavior of its complexity. Moreover, near the critical point, at which the phase transition takes place, our numerical calculations indicate that we need less information to specify a meson in the non-conformal vacuum than in the conformal one, despite the fact that the non-conformal vacuum has larger energy!
In this paper, we have studied the holographic subregion complexity for boosted black brane for strip like subsystem. The holographic subregion complexity has been computed for a subsystem chosen along and perpendicular to the boost direction. We have observed that there is an asymmetry in the result due to the boost parameter which can be attributed to the asymmetry in the holographic entanglement entropy. The Fisher information metric and the fidelity susceptibility have also been computed using bulk dual prescriptions. It is observed that the two metrics computed holographically are not rel…
We study holographic subregion complexity in a spatially anisotropic field theory, which expresses a confinement-deconfinement phase transition. Its holographic dual is a five-dimensional anisotropic holographic model characterized by a Van der Waals-like phase transition between small and large black holes. We propose a new interpretation from the informational perspective to determine the stable and unstable thermodynamically solutions. According to this proposal, the states which need (more) less information to be specified characterize the (un) stable solutions. We similarly offer an inter…
We study holographically the zero and finite temperature behavior of the potential energy and holographic subregion complexity corresponding to a probe meson in a non-conformal model. We observe that in zero and low temperature non-conformality has a decreasing effect on the dimensionless meson potential energy. However, non-conformal corrections increase absolute value of the dimensionless holographic subregion complexity in both zero and finite temperature which means the non-conformal state needs less information to be specified. In other words, considering the effect of non-conformality, t…
We study the volume prescription of the holographic subregion complexity in a holographic 5 dimensional model consisting of Einstein gravity coupled to a scalar field with a non-trivial potential. The dual 4 dimensional gauge theory is not conformal and exhibits a RG flow between two different fixed points. In both zero and finite temperature we show that the holographic subregion complexity can be used as a measure of non-conformality of the model. This quantity exhibits also a monotonic behaviour in terms of the size of the entangling region, like the behaviour of the entanglement entropy in…
We propose that the definition of holographic subregion complexity (HSC) needs a slight modification for supergravity solutions with warped anti-de Sitter (AdS) factors. Such warp factors can arise due to the nontrivial dilaton profile, for example, in $AdS_6$ solutions of type IIA supergravity. This modified definition ensures that the universal piece of the HSC is proportional to that of the holographic entanglement entropy, as is the case for supergravity solutions without warp factors. This also means that the leading behaviour at large $N$ is the same for both these quantities, as we show…
We construct renormalized holographic entanglement entropy (HEE) and subregion complexity (HSC) in the CV conjecture for asymptotically AdS$_4$ and AdS$_5$ geometries under relevant perturbations. Using the holographic renormalization method developed in the gauge/gravity duality, we obtain counter terms which are invariant under coordinate choices. We explicitly define different forms of renormalized HEE and HSC, according to conformal dimensions of relevant operators in the $d=3$ and $d=4$ dual field theories. We use a general embedding for arbitrary entangling subregions and showed that any…
Recently, there has been much interest in holographic computations of two-point non-equilibrium Green functions from AdS-Vaidya backgrounds. In the strongly coupled quantum field theory on the boundary, the dual interpretation of the background is an equilibration process called a holographic quench. The two dimensional AdS-Vaidya spacetime is a special case, dual to conformal quantum mechanics. We study how the quench is incorporated into a Hamiltonian $H + θ(t) ΔH$ and into correlation functions. With the help of recent work on correlation functions in conformal quantum mechanics, we first r…
We consider the computation of volumes contained in a spatial slice of AdS$_3$ in terms of observables in a dual CFT. Our main tool is kinematic space, defined either from the bulk perspective as the space of oriented bulk geodesics, or from the CFT perspective as the space of entangling intervals. We give an explicit formula for the volume of a general region in the spatial slice as an integral over kinematic space. For the region lying below a geodesic, we show how to write this volume purely in terms of entangling entropies in the dual CFT. This expression is perhaps most interesting in lig…
The volume inside a Ryu-Takayanagi surface has been conjectured to be related to the complexity of subregions of the boundary field theory. Here, we study the behaviour of this volume analytically, when the entangling surface has a strip geometry. We perform systematic expansions in the low and high temperature regimes for AdS-Schwarzschild and RN-AdS black holes. In the latter regime, we point out spurious divergences that might occur due to the limitations of a near horizon expansion. A similar analysis is performed for extremal black holes, and at large charge, we find that there might be s…
AApplying the Born-Infeld Anti de Sitter charged black hole metric we calculate holographic entanglement entropy (HEE) by regarding the proposal of Ryu and Takanayagi. To do so we assume that time dependence of the black hole mass and charge to be as step function. Our work is restricted to small subregions where a collapsing null shell dose not penetrate the black holes horizon. To calculate time dependent HEE we use perturbation method for small subregions where turning point is much smaller than local equilibrium point of black hole. We choose two shape functions for entangled regions on th…
We investigate the evolution of holographic entanglement entropy (HEE) and holographic complexity (HC) under a thermal quench in Einstein-Maxwell-Axion theory (EMA), which is dual to a field theory with momentum relaxation on the boundary. A strip-shaped boundary geometry is utilized to calculate HEE and HC via `entropy=surface' and `complexity=volume' conjecture, respectively. By fixing other parameters we claim that either large enough black hole charge or width of the strip will introduce swallow-tail behaviors in HEE and multi-values in HC due to the discontinuity of the minimum Hubeny-Ran…
We examine the five-region holographic entropy cone inequalities for the special case of the AdS$_3$-Vaidya metric for a variety of boundary configurations. This is done by numerically solving the geodesic equation in the bulk for various boundary configurations. In all the cases we examine, we find that all the inequalities are satisfied when the bulk satisfies the null energy condition, while the inequalities are all violated when the bulk spacetime violates the null energy condition. A proof of the five-region holographic entropy cone inequalities for the dynamical bulk case remains an open…
The BPS D3 brane has a non-supersymmetric cousin, called the non-susy D3 brane, which is also a solution of type IIB string theory. The corresponding counterpart of black D3 brane is the `black' non-susy D3 brane and like the BPS D3 brane, it also has a decoupling limit, where the decoupled geometry (in the case we are interested, this is asymptotically AdS$_5$ $\times$ S$^5$) is the holographic dual of a non-conformal, non-supersymmetric QFT in $(3+1)$-dimensions. In this QFT we compute the entanglement entropy (EE), the complexity and the Fisher information metric holographically using the a…
We examine holographic complexity in time-dependent Vaidya spacetimes with both the complexity$=$volume (CV) and complexity$=$action (CA) proposals. We focus on the evolution of the holographic complexity for a thin shell of null fluid, which collapses into empty AdS space and forms a (one-sided) black hole. In order to apply the CA approach, we introduce an action principle for the null fluid which sources the Vaidya geometries, and we carefully examine the contribution of the null shell to the action. Further, we find that adding a particular counterterm on the null boundaries of the Wheeler…
Holographic multipartite entanglement dynamics in AdS$_3$-Vaidya
Institute for Advanced Study, Tsinghua University（清华大学高等研究院）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
We study shape-deformations of the entanglement entropy and the modular Hamiltonian for an arbitrary subregion and state (with a smooth dual geometry) in a holographic conformal field theory. More precisely, we study a double-deformation comprising of a shape deformation together with a state deformation, where the latter corresponds to a small change in the bulk geometry. Using a purely gravitational identity from the Hollands-Iyer-Wald formalism together with the assumption of equality between bulk and boundary modular flows for the original, undeformed state and subregion, we rewrite a pure…
We derive a general expression for obtaining Holographic subregion complexity for asymptotically $AdS$ spacetimes, pertubatively around pure $AdS$ using a variational technique. An essential step in finding subregion complexity is to identify the bulk minimal surface of the entangling subregion. Our method therefore heavily relies on solutions of an inhomogeneous version of Jacobi equation, used to study deformations of the entangling surface for perturbations of the bulk metric. Using this method we have obtained the change in complexity for a strip and a circular disk like subsystem for \emp…
We investigate the effect of massive graviton on the holographic thermalization process. Before doing this, we first find out the generalized Vaidya-AdS solutions in the de Rham-Gabadadze-Tolley (dRGT) massive gravity by directly solving the gravitational equations. Then, we study the thermodynamics of these Vaidya-AdS solutions by using the Minsner-Sharp energy and unified first law, which also shows that the massive gravity is in a thermodynamic equilibrium state. Moreover, we adopt the two-point correlation function at equal time to explore the thermalization process in the dual field theor…
We study holographic subregion complexity, and its possible connection to purification complexity suggested recently by Agón et al. In particular, we study the conjecture that subregion complexity is the purification complexity by considering holographic purifications of a holographic mixed state. We argue that these include states with any amount of coarse-graining consistent with being a purification of the mixed state in question, corresponding holographically to different choices of the cutoff surface. We find that within the complexity = volume and complexity = spacetime volume conjecture…
The back reaction imparted by a uniform distribution of heavy static fundamental quarks on large $N_c$ strongly coupled gauge theory can be holographically realized as a deformation in $AdS$ blackhole background. The presence of back reaction brings significant changes in to the entanglement structure of the strongly coupled boundary theory at finite temperature. Since the deformed blackhole geometry still remains asymptotically $AdS$, the $gauge/gravity$ duality allows us to explore the entanglement structure of back reacted plasma in a quantitative way by computing various measures, e.g holo…
Holographic entanglement entropy quantities for a three-party system are considered, namely tripartite information, total correlation and secrecy monotone. A holographic approach with Vaidya-AdS is used to calculate time evolution of the entanglement entropy during non-equilibrium heating. We study time dependence of these three entropy quantities.
We analyze the holographic subregion complexity in a $3d$ black hole with the vector hair. This $3d$ black hole is dual to a $1+1$ dimensional $p$-wave superconductor. We probe the black hole by changing the size of the interval and by fixing $q$ or $T$. We show that the universal part is finite across the superconductor phase transition and has competitive behaviors different from the finite part of entanglement entropy. The behavior of the subregion complexity depends on the gravitational coupling constant divided by the gauge coupling constant. When this ratio is less than the critical valu…
In this second part of the study initiated in arxiv:1804.07410, we investigate holographic complexity for eternal black hole backgrounds perturbed by shock waves, with both the complexity$=$action (CA) and complexity$=$volume (CV) proposals. In particular, we consider Vaidya geometries describing a thin shell of null fluid with arbitrary energy falling in from one of the boundaries of a two-sided AdS-Schwarzschild spacetime. We demonstrate how known properties of complexity, such as the switchback effect for light shocks, as well as analogous properties for heavy ones, are imprinted in the com…
We study the time evolution of holographic subregion complexity (HSC) in Vaidya spacetime with dS boundary. The subregion on the boundary is chosen to be a sphere within the cosmological horizon. It is found that the behaviour of HSC is similar to that in cases with flat boundary. The whole evolution can be divided into four stages: First, it grows almost linearly, then the growth slows down; After reaching a maximum it drops down quickly and gets to saturation finally. The linear growth rate in the first stage is found to depend almost only on the the mass parameter. As the subregion size app…
Thin-shell AdS-Vaidya spacetimes can be considered as holographic models of the thermalization process in strongly-coupled conformal field theories following a rapid injection of energy from an external source. While the expected thermalization time is the inverse temperature, Bhattacharyya and Minwalla have pointed out that bulk causality implies that expectation values of local field-theory observables actually take on their thermal values immediately following the injection. In this paper we study two-point functions, for which the causality argument does not apply. Specifically, we study t…
We study the portion of an asymptotically Anti de Sitter geometry's bulk where the metric can be reconstructed, given the areas of minimal 2-surfaces anchored to a fixed boundary subregion. We exhibit situations in which this region can reach parametrically far outside of the entanglement wedge. If the setting is furthermore holographic, so that the bulk geometry is dual to a state in a conformal field theory (CFT), these minimal 2-surface areas can be deduced from the expectation values of operators localized within the boundary subregion. This presents us with an alternative: Either the redu…
We study the subregion complexity in a semi-analytical holographic QCD model. Two cases with different warped factor are considered and both can realize confinement-deconfinement transition. By studying the behavior of the renormalized holographic complexity density $\hat{\cal C}$ versus the subregion length scale $\ell$, we find that for both cases, $\hat{\cal C}$ always experiences a discontinuity at certain critical value $\ell_c$ in confinement phases, while it is always continuous in deconfinement phases. This property may be seen as a signal to characterize confinement or deconfinement p…
Using the volume of the space enclosed by the Ryu-Takayanagi (RT) surface, we study the complexity of the disk-shape subregion (with radius R) in various (2+1)-dimensional gapped systems with gravity dual. These systems include a class of toy models with singular IR and the bottom-up models for quantum chromodynamics and fractional quantum Hall effects. Two main results are: i) in the large-R expansion of the complexity, the R-linear term is always absent, similar to the absence of topological entanglement entropy; ii) when the entanglement entropy exhibits the classic `swallowtail' phase tran…
The volume of the region inside the bulk Ryu-Takayanagi surface is a codimension-one object, and a natural generalization of holographic complexity to the case of subregions in the boundary QFT. We focus on time-independent geometries, and study the properties of this volume in various circumstances. We derive a formula for computing the volume for a strip entangling surface and a general asymptotically AdS bulk geometry. For an AdS black hole geometry, the volume exhibits non-monotonic behaviour as a function of the size of the entangling region (unlike the behaviour of the entanglement entro…
We study the holographic quantum error correcting code properties of a Sierpinski Triangle-shaped boundary subregion in $AdS_4/CFT_3$. Due to existing no-go theorems in topological quantum error correction regarding fractal noise, this gives holographic codes a specific advantage over topological codes. We then further argue that a boundary subregion in the shape of the Sierpinski gasket in $AdS_5/CFT_4$ does not possess these holographic quantum error correction properties.
We study various entanglement measures associated with certain non-conformal field theories. We consider non-conformal D$p$-brane backgrounds, which are dual to these field theories, for our holographic analysis. Restricting our interests in $p=1,2,4$, we explicitly compute properties of holographic entanglement entropy and entanglement wedge cross section, $\text{E}_{W}$, corresponding to two parallel strip shaped boundary subregions in these set ups. We study low and high temperature behaviours of these quantities analytically as well as using numerical methods. In all cases, the $\text{E}_{…
We investigate jet quenching of virtual gluons and thermalization of a strongly-coupled plasma with a non-zero chemical potential via the gauge/gravity duality. By tracking a charged shell falling in an asymptotic AdS$_{d+1}$ background for $d=3$ and $d=4$, which is characterized by the AdS-Reissner-Nordström-Vaidya (AdS-RN-Vaidya) geometry, we extract a thermalization time of the medium with a non-zero chemical potential. In addition, we study the falling string as the holographic dual of a virtual gluon in the AdS-RN-Vaidya spacetime. The stopping distance of the massless particle representi…
In this paper we study the propagation of entanglement entropy after a global instantaneous quench on the CFT boundary of AdS bulk. We consider the Gauss-Bonnet model as a higher curvature gravity model for which we correct the RT(HRT) proposal to compute the holographic entanglement entropy(HEE). To obtain an analytical solution we perform an approximation approach which bounds our computations to the small subregions and we compare its thermalization regimes to the result of large subsystem case. We can see tsunami picture where the evolution of entanglement breaks down for the large systems…
We study the evolution of entanglement entropy in a 2-dimensional equilibration process that has a holographic description in terms of a Vaidya geometry. It models a unitary evolution in which the field theory starts in a pure state, its vacuum, and undergoes a perturbation that brings it far from equilibrium. The entanglement entropy in this set up provides a measurement of the quantum entanglement in the system. Using holographic techniques we recover the same result obtained before from the study of processes triggered by a sudden change in a parameter of the hamiltonian, known as quantum q…
Dynamical evolution of thin shells composed by different kinds of degrees of freedom collapsing within asymptotically AdS spaces is explored with the aim of investigating models of holographic thermalization of strongly coupled systems. From the quantum field theory point of view this corresponds to considering different thermal quenches. We carry out a general study of the thermalization time scale using different parameters and space-time dimensions, by calculating renormalized space-like geodesic lengths and rectangular minimal area surfaces as extended probes of thermalization, which are d…
