Holographic complexity in general quadratic curvature theory of gravity

In the context of CA conjecture for holographic complexity, we study the action growth rate at late time approximation for the general quadratic curvature theory of gravity. We show how the Lloyd bound saturates for charged and neutral black hole solutions. We suggest a counter-term when there is a second singular point in neutral solutions. 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.


Introduction
In recent years, by combining the basic ideas of Quantum Information Theory (QIT) and AdS/CFT duality, a significant development in the area of black hole physics has occurred and we have witnessed the appearance of new paradigms in our quest to understand the quantum theory of gravity.
Holographic Entanglement Entropy (HEE) is a successful example and has been captured the imagination of many people in this area of research [1,2]. Meanwhile, it was claimed in [3] that the HEE is not enough to distinguish the degrees of freedom inside a black hole, because even if the space-time reaches the thermal equilibrium, the volume of the black hole continues to grow. Instead, complexity is an interesting proposal that characterizes these degrees of freedom.
In QIT, complexity is defined as the minimum number of quantum gates, essential to produce a particular state |ψ from a reference state |ψ 0 . In other words, complexity is a measure of how hard it is to construct a final state from an initial state [4][5][6].
In the context of AdS/CFT there are two main proposals on how to compute the complexity of a boundary state. The first one is the Complexity=Volume or (CV) conjecture [7][8][9] and the second one is the Complexity=Action or (CA) conjecture [10,11]. The CV duality states that the complexity of a holographic boundary state on a time slice Σ is given by where is a certain length scale of the geometry (for example the curvature scale or horizon radius) and B is the corresponding bulk surface. The CA conjecture claims that the complexity is given by the gravitational action evaluated on the Wheeler-DeWitt (WDW) patch. This patch is defined as the domain of dependence of the Cauchy surface in the bulk which asymptotically approaches the time slice Σ on the boundary C A (Σ) = I WDW π .
(1. 2) In CA conjecture we are dealing with evaluating the gravitational action on various spacetime regions including space/time-like boundaries, null boundary surfaces, and different types of joints at the intersection of boundaries [12]. Other than complexity, there are related interesting parameters which play major roles in the computation of CV and CA conjectures. For example complexity of formation ∆C, the structure of UV divergences of the complexity, the time dependence or the rate of complexity dC/dt, and the so-called Lloyd bound. In what follows we briefly review these parameters one by one.
Let us begin by introducing the complexity of formation. The thermofield double (TFD) state can be considered as the dual description of the full geometry of an eternal AdS black hole [13] |TFD = Z −1/2 where the corresponding asymptotic boundaries (denoted by L and R) are two copies of the same CFT. One can construct thermal density matrix of the CFT at temperature T , by integrating out either the left or right degrees of freedom in the above state. Moreover the entanglement between these two sets of degrees of freedom gives rise to the appearance of Einstein-Rosen bridge in the bulk [13,14]. The complexity of formation is the difference between complexity in the process of forming the entangled TFD state and preparing two individual copies of the vacuum state of the left and right boundary CFTs In [15] it was found that for boundary dimensions greater than two, the complexity of formation grows linearly with the thermal entropy at high temperatures ∆C ∼ S. The growth rate of action within the WDW patch at late time approximation or dC/dt is another important factor. One of the interesting features of this parameter is that it tends to a specific universal value. In QIT this limit has been found in the study of an arbitrary quantum system [16] (Lloyd bound). In the context of CA conjecture it was first studied in [10,11], where they show that the bound is violated for large enough Schwarzschild AdS black holes, and is saturated for small ones. Their results suggest that the growth rate would be bounded by the total energy of the system (1.5) Further exploration of this bound, in the case of rotating and charged black holes were carried out in [17,18], where a modified formula for the action growth rate was presented.
For example in the case of charged black holes where M and Q are mass and charge of the black hole and µ is the chemical potential. The ± signs represent the outer and inner horizons of the black hole. Since then, there has been extensive discussion on the preservation or violation of the Lloyd bound in different gravitational setups [19][20][21][22][23][24][25][26]. For example in [26], using Noether charge formalism of Iyer and Wald 1 , the authors have found a modified version of the Lloyd bound in multiple Killing horizons black hole for a higher curvature theory of gravity. Also, in [29] it was shown that strong energy condition is a sufficient condition to ensure the bound inequality (1.5) and it was argued that the equality (1.6) satisfies the bound (1.5).
Generally, there are two main methods for computation of the action growth rate in CA conjecture, the BRSSZ method introduced in [11], and the LMPS method [12]. In the latter paper, the authors show that the two methods give the same result for the action growth rate in Einstein gravity. Also in [30] it was pointed out that even in general higher curvature theories of gravity these two methods give the same result.
Since in the first approach one does not need to know the corresponding action of the null boundaries in general higher curvature theories of gravity, we use this method to compute the action growth rate. In the BRSSZ method for the computation of dC/dt, in addition to the bulk action, it is only necessary to know the action on the time/spacelike boundaries, which is essentially the Gibbons-Hawking-York (GHY) action and its generalizations (see section 2 for more details).
For a general higher curvature gravitational theory it is hard to find an appropriate surface term to make the variational principle well-posed [31], but the non-null surface terms have been developed for some gravitational theories, such as F (R), Gauss-Bonnet gravity and Lanczos-Lovelock theory [32][33][34][35][36][37] and other higher curvature theories [38].
Specifically for f (Riemann) theories of gravity by using the auxiliary field formalism these surface terms have been investigated in [39]. In this context, complexity has been studied for higher curvature theories of gravity of example see [40][41][42][43][44]. Inspired by this, we are going to consider the General Quadratic Curvature (GQC) theory of gravity in this paper and compute the action growth rate in this theory.
Another related subject is the structure of UV divergences of the complexity. In both complexity conjectures, we evaluate quantities which when extending to the asymptotic boundaries they become divergent. These divergences are related to the existence of shortscale correlations in the dual boundary CFTs. As it was shown in [45] the coefficients in these divergent terms can be written in terms of intrinsic and extrinsic boundary curvatures. The general structure of divergent terms in complexity in d dimensional space-time is given by where δ is the UV cut-off. Here the coefficients a i depend on the regularization scheme but the coefficients b i are universal (regulator independent). In section 3 we will study the structure of these divergences of the complexity and compute the universal terms of the complexity in GQC gravity.

The action growth rate of GQC gravity
Following [11] and [17] we are going to compute the action growth rate on a WDW patch associated with a two-sided black hole in GQC theory of gravity, see figure (1). It is believed that this would be dual to the rate of growth of the complexity of the boundary state. This black hole is a charged/neutral Schwarzchild AdS solution of the equations of motion. For charged solution (left diagram) we suppose outer and inner horizons at r = r ± . For neutral case (right diagram) we consider inside the black hole is limited from singularity to the future horizon.
According to the arguments of [11], the WDW patch is bounded by t L and t R on the left and right for a (un)charged black hole. When time passes on the left boundary from t L tot L , the WDW patch starts growing in the green region and shrinks in the red region. At late times, the whole contribution of action growth comes from the green region that now lives between inner and outer horizons at r = r ± of the charged black hole or between singularity at r = r s and future horizon of the neutral black hole. Therefore the total contribution of action comes from the bulk action in this region as well as the boundary actions on the space-like surfaces at r ± or the singularity and future horizon.

General quadratic curvature action
Let us first introduce the bulk and boundary actions of GQC. The bulk action consists of the Einstein-Hilbert action together with a cosmological term. Moreover, we add all quadratic curvature terms as the sum of Ricci and Ricci scalar curvature squared terms and the well-known Gauss-Bonnet (GB) terms. To study charged black holes we also consider a U (1) gauge field through a Maxwell term in our Lagrangian Figure 1: Penrose diagram of charged (left) and neutral (right) black holes in GQC gravity. In the charged case we have depicted the outer horizon r + and the inner horizon r − , together with two singularities, one at the center of space-time r = 0, and one behind the inner horizon, at r = r s , denoted by the wiggly arcs in this picture. In the neutral case, however, the inner horizon disappears and the singularities become space-like, in such a way that WDW patch ends on the singularity.
where κ 2 = 8πG. To write the action for boundary surfaces, which make the variational principle of the gravitational field well-defined, instead of using the auxiliary field formalism (see e.g. [39]), we consider three types of Gibbons-Hawking (GH) terms corresponding to the Einstein-Hilbert, GB and Ricci square terms. The GH term associated to the Einstein-Hilbert action is given by where h is the determinant of the induced metric on the boundary surfaces ∂M and K is the trace of the extrinsic curvature. The extrinsic curvature is defined by and n µ is a space-like unit vector, normal to the boundary. Moreover, there is a generalized GH action corresponding to the GB gravity [47] where G ab is the Einstein tensor constructed out of the induced metric. For Ricci squared terms of the action we use the following expression for GH term The total GH term therefore is the sum of these three parts

Charged solution in d dimension
To find a charged solution with asymptotic AdS symmetries we use the following metric in d dimensional space-time The field strength satisfies the Maxwell equation where q is the constant of integration, and is related to the electric charge Q by By variation of the Lagrangian (2.1) and inserting (2.7) and (2.8) for metric and field strength, we will find a couple of third and fourth order differential equations for f 1 (r) and f 2 (r). Finding an exact solution is a hard task to do, instead we find a solution which is linear in terms of the couplings of theory i.e. a 1 , a 2 and a 3 . The solution is given by where all the coefficients are given in terms of q and m To prevent the divergence in the gauge field strength (2.8), the function h(r) in relation (2.10b) should not be equal to zero. This requires to consider the following condition between couplings of the theory For the neutral solution as we send q → 0, both functions in (2.10a) and (2.10b) become equal or h(r) = 0 at this level of perturbation, but it can be shown that, f 1 (r) and f 2 (r) become separated when we consider next orders of perturbation into account.

Action growth rate in WDW patch
The growth rate of bulk action in the WDW patch at late-time approximation can be computed by inserting the solution (2.10a) and (2.10b) into bulk action (2.1). For more details of computations see relations (A.1a)−(A.1c) in appendix A where α 1 , ..., α 5 are given in equation (B.1) in appendix B. At late-time approximation, the growth rate of Gibbons-Hawking surface terms of WDW patch is given by computing the value of (2.6). To do this we insert the solution (2.10a) and (2.10b) into the equations (A.2) and (A.3) and we find where β 1 , ..., β 9 are given in equation (B.2). Finally, we perform the integration of (2.13) and add it to the relation (2.14). This gives the total action growth rate. The final result can be simplified by using the following steps: 1. Because we considered r + as the outer horizon, we can solve f 1 (r + ) = 0 to find a relation for Λ 0 .
2. Insert the value of Λ 0 from the first step into f 1 (r − ) = 0 to find a relation for m in terms of r + and r − .
3. Insert the relations of Λ 0 and m in the first and second steps into the total action growth rate. The final result is (2.15) Using the field strength relation (2.8) and the value of electric charge in (2.9), one can write the above expression as follow where we have supposed M as the mass of black hole and are the values of chemical potential on r ± horizons.
The result of equation (2.16) shows that the proposal introduced in [17] is correct for general quadratic curvature theory.

Neutral black hole in d dimension
To find the action growth rate in the neutral case we need to know the singular points of the geometry. To do this we first review the solutions of equations of motion for some special cases: • a 3 = 0: In this case the exact solution of equations of motion is given by the known Schwarzshcild AdS black hole, for example see [49] This solution has a singularity at r s = 0.
• a 1 = a 2 = 0: This case corresponds to the Gauss-Bonnet gravity. The exact solution is given by [17] This solution has two singularities. The first one, is where the metric becomes divergent and the other one, is the point where the metric or scalar curvature terms become imaginary (2.20) The above analysis for special cases helps us to find the singularity structure of the solution for general case. To make the analysis easier, it would be better to find a solution which reduces to solution (2.10a) when q → 0 and also reduces to (2.19a) when a 1 = a 2 = 0. In fact, we may assume that there is a general solution which can be written as f (r) = 1 + r 2 X(r) 1 − 1 + c + m r d−1 and solve equations of motion perturbativly to find c, m and X(r). To first order of perturbation both f 1 (r) and f 2 (r) functions are given by By an expansion around small couplings (2.21) would be equal to where in this expansion m = m 0 + δm (m 0 is a constant of integration) and We can see that (2.21) has the same the singularity structure as (2.19a) at least at this order of perturbation.

Action growth rate in WDW patch
To find the growth rate of actions for neutral Schwarzschild AdS black hole we carry out the following steps: 1. First we add results of equations (2.13) and (2.14) after taking the limit of q → 0, with a difference that here we compute it from r = r h , the location of future horizon to r = r s , the location of the singularity 2. The horizon is given by solving the equation We use this equation as a constraint between r h and other parameters. Suppose that the location of horizon from this equation can be found perturbativly by putting r h = r 0 + δr into (2.25). In this way we can find the following relations 3. Using the above equations we can simplify the growth rate (2.24) at r = r h dI q=0 (2.27) 4. In general the action growth rate at r = r s diverges when r s → 0. But there are special cases that its value is finite: • For a 3 = 0, as we already mentioned, the singularity is located at r s = 0. At this point equation (2.24) will be equal to therefore the total action growth rate is The last equality is written in terms of the mass of black hole for Ricci square theory of gravity. For example see [49] for all details of computation of this mass 2 . Note that (2.29) is a general result and for example as a special case it reduces the result of d dimensional critical gravity.
• For d = 4 the Gauss-Bonnet part of the theory is a topological term. The growth rate in this case is equal to 5. For general case when a 3 = 0 or d > 4 there is a second singularity at r s = 0. Although this non-zero value makes the equation (2.24) finite, but the final result gives rise to a wrong answer for late time complexity as a 1 , a 2 , a 3 → 0. This behavior also has been reported for d dimensional Gauss-Bonnet gravity in [17].
To solve this problem suppose a counter-term living on r = r s . We can construct it from the induced metric of the bulk. Due to the symmetries of the bulk metric we can choose the following action where g ef f is an effective coupling and R is the scalar curvature constructed from the induced metric γ ab . After some tedious algebra one can show that the following effective coupling gives a correct value for the action growth rate. For d = 5 g d=5 ef f = 34 15 and for d > 5 we find By considering the above results the total action growth rate would be where the last equality is coming from computation of the mass to linear order of couplings (at this order the GB part does not have a contribution to mass). 2 To translate computations of [49] to ours, we need to change a 1 ↔ a 2 , κ 2 → −κ 2 and replace l = 1,

WDW action for global AdS
In this section, we are going to compute the universal terms that appear in the divergent part of C A complexity in the GQC theory of gravity. All steps that we follow here, have been presented already in reference [46]. We will show how with some simple modifications we can find the universal coefficients of C A . Paper [46] begins with Lovelock theory with the following bulk and space/time-like boundary actions where X 2n is the Euler density and Q n is the generalized GH boundary term. The boundary terms make the variational principle well defined. For example for n = 1, 2, the relations for Q 1 and Q 2 are given in equations (2.2) and (2.4). After that, [46] computes the contribution from space/time-like joints by employing the Hayward smoothing method. These joints are co-dimension two surfaces which are made from the intersection of boundary surfaces. In this way [46] prove that the joint terms can be computed from the Lovelock boundary term in (3.1) and are given by where η = ±cosh −1 |n 1 .n 2 |, and n 1,2 are normal one-forms to each boundary that their intersection makes the joint C. In above equationX 2(n−1) is the Euler density constructed from the induced metric on the joint. To find the universal terms, one needs to compute the gravitational action on a regularized WDW patch. This patch contains a cut-off distance δ from the boundary (see figure (2)) 2δ δ δ ′ Figure 2: Two ways for regularizing the WDW patch.
By choosing this patch, [46] argues that despite of surface terms for null boundaries, the universal terms are just coming from the joint terms (3.2) and bulk action. The final form of joint term is given by where a = ± log |k 1 .k 2 /2| for joints between two null boundaries and a = ± log |n.k 1 | for joints between a time-like and a null boundary. Now consider the GQC action (2.1) without Yang-Mills term. Although the bulk action cannot be written as a sum of Euler terms but as it was shown in [48] or from computing equation (2.5) for a Global AdS solution, the total GH boundary term (2.6) can be written as Therefore the joint term (3.3) for GQC theory simplifies to whereR is the scalar curvature on the joint that is constructed from the induced metric. Following [45], we use the regulated graph depicted in figure (2), i.e. we change the WDW patch by an inward shift on the right and left edges. To compute the structure of divergences, we begin with the following metric which is asymptotically AdS d space-time with radiusL in GQC theory By two successive proper coordinate transformation i.e. z =L r and t = τL, and then z = 2L cos θ 1+sin θ we find the following metric In this metric, the null boundaries of the WDW patch (the right diagram of figure (2)) are given by Moreover the unit normal vectors to S + and S − null surfaces are where α 1,2 are normalization constants. By another change of coordinate, θ = π 2 − θ the location of joint becomes (τ, θ ) = (0, δ ). In this way the induced metric on the joint and scalar curvature tensor would be Also for a joint between two null boundaries in this coordinate By substituting the above values into the joint action (3.5), finally we obtain We can write the above expression in terms of original cut-off by using δ = 2L sin δ 1+cos δ . After expansion around δ = 0 we have where dots represent the power expansion terms. Therefore the universal term for GQC from the joint term for even d dimensional space-time is given by We can also find the universal terms which are coming from the computation of the bulk action on WDW patch. In d dimensional space-time these terms are computed in [46] C univ bulk = 4 π a * d logL δ for even d , where a * d is the a-anomaly, and for GQC theory it is equal to (see [48] for more discussions)

Summary and Discussion
In this paper we have used the CA conjecture (1.2) for studying the holographic complexity. In the bulk space-time, we assume a general quadratic action that includes both Riemann and Ricci curvature tensors up to the quadratic terms (2.1). This form of action allows us to generalize the ideas around the holographic complexity when one takes into account the higher curvature theories of gravity in d dimensional space-time.
In section two we examine the proposal of growth bound on complexity for two types of charged and neutral black holes of GQC which are asymptotically AdS space-time. In subsection 2.2 we first find a U (1) charged black hole solution and then in subsection 2.3 we obtain the action growth (2.15) by computing the bulk and boundary actions on WDW patch at late-time approximation. Our final result (2.16) confirms the proposal in [17], i.e. we can write the total action growth as a difference between the value of M − µQ on the outer and inner horizons of the black hole. This result has been already reported for various theories of gravity and here we observe that this bound is preserved even for a general theory such as GQC.
Despite this, the case of the neutral black hole is more challenging. The reason is the existence of the singularity as one of the surface boundaries of the WDW patch. In subsection 2.4 we learn how from a particular class of solutions we can guess and compute the singularity structure of geometry in GQC theory. Using this in subsection 2.5 we compute the action growth rate. We Show that for special cases such as a 3 = 0 or d = 4 where the singularity is located at r s = 0, the singularity does not produce any divergence and the Lloyd bound saturates at 2M .
On the other hand, when we consider the general case, a second singularity appears in the solution at r s = 0. Our calculation shows that the techniques for special cases do not work here and it produces extra terms with wrong results. To overcome this difficulty, we suggest and compute an extra counter-term with an effective coupling on the singularity surface (2.31). This term allows us to correctly reproduce the bound at 2M . It would be interesting for future works to find the root of these types of counter-terms on the singularity surfaces.
In section three we have looked at another interesting subject in the context of holographic complexity and computed the universal terms that appear in the divergent part of C A for the GQC theory of gravity. Usually, there are two types of these universal terms, one from the joint terms of regularized WDW patch and one from the bulk action. Using a simple trick, by introducing an effective GH term (3.4), we have found the joint terms from the techniques in paper [46]. In that paper, the joint terms were calculated for Lovelock theory. Although the GQC can not be written in terms of the Lovelock theory, its GH terms as we mentioned, are compatible with those in [46] technique. In this way, we have found the universal terms (3.14b) and (3.16).

B Action growth coefficients
The coefficients of bulk action growth in equation (2.13) are