Holographic Complexity of Anisotropic Black Branes

Holographic complexity of anisotropic black branes

Seyed Ali Hosseini Mansoori,1, 2, ∗ Viktor Jahnke,3, † Mohammad M. Qaemmaqami,4, ‡ and Yaithd D. Olivas3, §

1 Faculty of Physics, Shahrood University of Technology, P.O.Box 3619995161 Shahrood, Iran

2 School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran

3 Departamento de F´ısica de Altas Energias, Instituto de Ciencias Nucleares, Universidad Nacional Autónomade México Apartado Postal 70-543, CDMX 04510, México

4 School of Particles and Accelerators Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran We use the complexity = action (CA) conjecture to study the full-time dependence of holographic complexity in anisotropic black branes. We find that the time behavior of holographic complexity of anisotropic systems shares a lot of similarities with the behavior observed in isotropic systems. In particular, the holographic complexity remains constant for some initial period, and then it starts to change so that the complexity growth rate violates the Lloyd’s bound at initial times, and approaches this bound from above at later times. Compared with isotropic systems at the same temperature, the anisotropy reduces the initial period in which the complexity is constant and increases the rate of change of complexity. At late times the difference between the isotropic and anisotropic results is proportional to the pressure difference in the transverse and longitudinal directions. In the case of charged anisotropic black branes, we find that the inclusion of a Maxwell boundary term is necessary to have consistent results. Moreover, the resulting complexity growth rate does not saturate the Lloyd’s bound at late times.

I. INTRODUCTION A convenient gravity set-up to study complexity growth is a two-sided black hole geometry. This geometry has two asymptotic regions, which we call left (L) and right The gauge-gravity duality [1] provides a framework in (R) boundaries, and an Einstein-Rosen Bridge (ERB) which one can study the emergence of gravity from non- connecting the two sides of the geometry. The Penrose gravitational degrees of freedom. Within this framework, diagram of this geometry is shown in ﬁgure 1. From the gravitational theory lives in a higher dimensional the point of view of the boundary theory, the two-sided space M, usually called bulk, and the non-gravitational black hole is dual to a thermoﬁeld double (TFD) state, theory can be thought of as living on the boundary of constructed out of two copies of the boundary theory [12] M. Despite the existence of a dictionary [2, 3] relat- 1 −βEn ing bulk and boundary quantities, the description of the X −iEn(tL+tR) |TFDi = e 2 e |E i |E i , (1) Z1/2 n L n R black hole’s interior in terms of boundary degrees of free- n dom remains elusive. Recently, there has been progress in this direction, with the conjecture that the growth of where L and R label the quantum states of the left and the interior of a black hole is related to the quantum com- right boundary theories, respectively. The TFD state is putational complexity [76] of the states in the boundary invariant under evolution with a Hamiltonian of the form arXiv:1808.00067v3 [hep-th] 9 Sep 2019 theory. There are two main proposals relating the com- H = HL − HR, which means that the system is invariant

plexity to geometric quantities in the bulk, namely, the under the shifts tL → tL + ∆t, and tR → tR − ∆t. As Complexity = Volume (CV) [4, 5] and the Complexity a result, the TFD state only depends on the sum of the

= Action (CA) [6, 7] conjectures. In the CV conjecture, left and right boundary times t = tL + tR. the complexity is dual to the volume of a certain extremal surface in the bulk and provides an example of The ERB connecting the two sides of the geometry grows the recent proposal about the connection between tensor linearly with time. Classically, this behavior goes on for- networks and geometry [8–10], while in the CA conjecture ever. In [4] Susskind proposed that this behavior is dual the complexity is dual to the gravitational action evalu- to the growth of the computational complexity in the ated in certain region in the bulk. More details about boundary theory, which is known to persist for very long CA conjecture will be given in section III. times. Using the CV proposal, the authors of [5] showed that the late-time behavior of the rate of change of complexity is given by dCV /dt = 8πM/(d − 1), where M is the black hole’s mass and d is the number of dimensions ∗Electronic address: [email protected], [email protected] of the boundary theory. †Electronic address: [email protected] ‡Electronic address: [email protected] Despite having a qualitative agreement with the behavior §Electronic address: [email protected] of complexity for quantum systems, the CV conjecture is 2 defined in terms of an arbitrary length scale, which is scribes a renormalization group (RG) flow from an AdS usually taken to be of the order of the AdS radius. In geometry in the ultraviolet (UV) to a Lifshitz-like geom- order to avoid the ambiguity associated to the arbitrary etry in the infrared (IR). The transition is controlled by length scale the authors of [6, 7] proposed the CA con- the ratio a/T , where a is a parameter that measures the jecture. For neutral black holes, the late time behavior degree of anisotropy and T is the black brane’s tempera- of the rate of change of holographic complexity reaches ture. From the point of view of the boundary theory, this a constant value which is also proportional to the black parameter is small close to the UV fixed point and large hole’s mass close the IR fixed point. We would like to understand how the complexity rate changes as we move along this dC 2M lim A = . (2) RG flow and whether this system respects the Lloyd’s t→∞ dt ~π bound. As a first step towards this, we have considered This late-time behavior may be associated with the small deviations from the UV fixed point, i.e., small val- Lloyd’s bound on the rate of computation by a system ues of a/T , which can be incorporated by considering with energy M [13]. This saturation of the complexifica- an analytical black brane solution with small corrections tion bound lead to the conjecture that the black holes are due to anisotropy [78]. the fastest computers in nature [7]. It was later shown The DK model is a solution of 5-dimensional Einstein- that a more precise definition of CA requires the introduc- Maxwell gravity that is dual to the 4-dimensional N = 4 tion of joint and boundary terms, which were not present SYM theory in the presence of a background magnetic in the calculation of [6, 7]. In particular, it was shown field. This solution describes an RG flow between an that the CA proposal also have an ambiguity related AdS geometry in the UV to a BTZ ×R2 geometry in the to the parametrization of null surfaces [14]. Using the IR. The parameter controlling such transition is B/T 2, boundary and joint terms derived in [14, 15], the authors where B is the intensity of the magnetic field, while T is of [16, 17] showed that these ambiguities do not affect the black brane’s temperature. We would like to under- the late time behavior of dCA/dt, but they play a role at stand how the magnetic field affects the rate of change early times, leading to a violation of Lloyd’s bound. of complexity.

Therefore, both the CA and the CV proposals have am- The CGS model [27, 28] is a generalization of the biguities which (apparently) cannot be eliminated. This MT model to the charged case. The geometry is an is not a problem, however, because the same ambigui- anisotropic RN-AdS solution, being also affected by a ties were found in the definition of complexity for free charge parameter, q. When a 6= 0 and q → 0, the solution quantum field theories [18–20]. Moreover, the quantita- reduces to the MT geometry. When q 6= 0 and a → 0, the tive disagreement between the results obtained with the solution becomes an RN-AdS geometry. The motivation CA and CV proposals might be related to other ambigu- to consider this solution is to understand how the rate of ities in the definition of complexity, like the choice of the change of complexity is affected by the presence of un- reference state or the choice of the elementary gates. charged and charged matter fields, and whether the CA The Lloyd’s bound was shown to be violated even at late prescription can provide sensible results in the presence times by anisotropic systems, including the SYM the- of several matter fields. ory defined in a non-commutative geometry [21], and In the uncharged cases, we find that the time behav- Lifshitz and hyperscaling violating geometries [22–24]. ior of holographic complexity is qualitatively similar to This raises the question of whether there is a more gen- the behavior observed for isotropic systems, namely, the eral bound that is also respected by anisotropic systems. holographic complexity remains constant for some pe- With this in mind, in this paper we use the CA conjec- riod, and then it starts to change so that the rate of com- ture to study the holographic complexity of a class of plexity growth violates the Lloyd’s bound at initial times, anisotropic black branes [77]. More specifically, we con- and it approaches this bound from above at later times. sider the Mateos and Trancanelli (MT) model [25, 26], Additionally, we find that the net effect of anisotropy the D’Hoker and Kraus (DK) model [73], and the Cheng- is basically a vertical upward shift in the curves of the Ge-Sin (CGS) model [27, 28], and study the time depen- rate of change of holographic complexity versus time. dence of holographic complexity in thermofield double At later times, the difference between the isotropic and states which are dual to two-sided black brane geome- anisotropic results is proportional to the difference in tries. pressures in the longitudinal and transverse directions. In The MT model is a solution of type IIB supergravity the charged case, we find that the inclusion of a Maxwell that was designed to model the effects of anisotropy in boundary term is necessary to have consistent results. the quark-gluon plasma (QGP) created in heavy ion col- The remainder of paper is organized as follows. In sec- lisions. The anisotropy is present in the initial stages tion II we review the MT and DK solutions and present after the collision and it leads to different transverse and some of its thermodynamic properties. In section III we longitudinal pressures in the plasma. For our purposes, use the CA conjecture to study the full-time behavior the main motivation to consider this model is that it de- 3 of holographic complexity of thermofield double states the anisotropy parameter a as which are dual to two-sided anisotropic black branes so- 4 2 lutions. The case of charged anisotropic black branes is rH a h 2 2 2 F = 1 − 4 + 4 2 8r rH − 2rH (4 + 5 log 2) considered in section IV. We discuss our results in section r 24r rH V. We relegate some technical details of the calculations r2 i to the appendices A and B. + (3r4 + 7r4 ) log 1 + H + O(a4) (6) H r2 2 2 2 a 10rH rH 4 B = 1 − 2 2 2 + log 1 + 2 + O(a )(7) 24rH r + rH r 2 2 a rH 4 φ = − 2 log 1 + 2 + O(a ) . (8) 4rH r By requiring regularity of the Euclidean continuation of the above metric at the horizon, one obtains the Hawking temperature as

2 rH (5 log 2 − 2) a 4 II. GRAVITY SET-UP T = + + O(a ) . (9) π 48π rH The Bekenstein-Hawking entropy can be obtained from Anisotropic black branes: the MT model the horizon area as 3 2 rH 5a 4 S = 1 + 2 V3 + O(a ) . (10) The Mateos and Trancanelli (MT) model [25, 26] is a 4GN 16rH solution of type IIB supergravity whose effective action R in five dimensions can be written as where V3 = dxdydz is the volume in the xyz−directions. Using holographic renormalization, the Z √ h 1 5 12 1 2 stress tensor of the deformed SYM theory can be ob- S = d x −g R + 2 − (∂φ) 16πGN M L 2 tained as [26, 68] 1 2φ 2 i − e (∂χ) + SGH , (3) 2 Tij = diag(E,Pxy,Pxy,Pz) , (11) where φ, χ and gµν are the dilaton field, the axion field where and the metric respectively, GN is the five-dimensional 3π2N 2T 4 N 2T 2 Newton constant, and SGH is the Gibbons-Hawking term. E = + a2 + O(a4) , (12) The solution in Einstein frame takes the form 8 32

2 2 −φ(r)/2h 2 2 ds = L e − r F(r) B(r) dt is the energy density of the black brane and 2 dr 2 2 2 2 i 2 2 4 2 2 + + r dx + dy + H(r) dz , (4) π N T N T 2 4 r2F(r) Pxy = + a + O(a ) , (13) 8 32 π2N 2T 4 N 2T 2 with P = − a2 + O(a4) (14) z 8 32 χ = a z , φ = φ(r) , H = e−φ , (5) are the pressures along the transverse and longitudinal where (t, x, y, z) are the gauge theory coordinates and r directions, respectively. The mass of the black brane can is the AdS radial coordinate. Here L is the AdS radius, then be calculated as which we set to unity in the following [79]. The above 2 2 4 2 2 3π N T N T 2 4 solution has a horizon at r = rH and the boundary is M = EV3 = + a V3 + O(a ) , (15) located at r = ∞, where F = B = H = 1 and φ = 8 32 0. The axion is proportional to the z−coordinate and A more simple way of calculating the black brane’s mass this introduces an anisotropy into the system, which is is through the expression measured by the anisotropy parameter a. For a 6= 0, the above solution corresponds to the gravity dual of N = 4 Z Z rH dS(r ) M = T dS = T (r ) H dr SYM theory, with gauge group SU(N), deformed by a H dr H position-dependent theta term. When a = 0, the above 0 H V h r2 a2 i solution reduces to the gravity dual of the undeformed 3 4 H 4 = 3rH + (5 log 2 − 1) + O(a ) (16) SYM theory. The functions F, B, H and the dilaton φ 16πGN 4 can be determined analytically [80] for small values of 4 where the integral was calculated using the equations (9) and (10) for T (rH) and S(rH), respectively. Expressing rH as a function of the temperature T and using that 2 GN = π/(2N ), we recover the expression for the mass given in equation (15).

Note that the mass of the anisotropic black brane is larger than the mass of an isotropic black brane with the same temperature, or with the same horizon radius. For future reference, we note that V M(a) = M(0) + 3 (P − P ) + O(a4) . (17) 2 xy z

FIG. 1: Penrose diagram for the two-sided black branes we Magnetic black branes: the DK model consider. This geometry is dual to a thermoﬁeld double state constructed out of two copies of the boundary theory.

The D’Hoker and Kraus (DK) model [73] is a magnetic black brane solution of 5-dimensional Einstein-Maxwell gravity. The action of this model reads 1 Z √ S = d5x −g R + 12 − F F MN . (18) 16πG MN N The Penrose diagram is obtained as follows. We consider For very large values of the magnetic field (B/T 2 >> 1), a general general metric of the form given in Eq. (25). the solution takes the form [81] We first define Kruskal-Szekeres coordinates U and V as

2π (r −t) 2π (r +t) 2 U = +e β ∗ ,V = −e β ∗ (left exterior region) 2 2 2 2 dr ds = −3(r − rH)dt + 2π 2π 2 2 (r∗−t) (r∗+t) 3(r − rH) U = −e β ,V = +e β (right exterior region) B 2π 2π 2 2 2 2 (r∗−t) (r∗+t) −√ dx + dy + 3r dz . (19) U = +e β ,V = +e β (future interior region) 3 2π (r −t) 2π (r +t) U = −e β ∗ ,V = −e β ∗ (past interior region) with field strength F = B dx ∧ dy. (22) The Hawking temperature and the Bekenstein-Hawking where β is the black hole inverse temperature, and r∗ is entropy associated to the above solution are easily found the tortoise coordinate, which is defined in (27). In terms to be of these coordinates, the metric (25) becomes 2 3rH 3V3B rH 4π T = ,S = . (20) 2 − β r∗ 2 β e i j 2π 4GN ds = − G (UV )dUdV + G (UV )dx dx . 8π2 tt ij R (23) where V3 = dxdydz. The black brane’s mass can then be calculated as The Penrose diagram is obtained with one additional change of coordinates, U˜ = tan−1(U) and V˜ = tan−1(V ), Z V3 2 in terms of which the boundaries of the spacetime lie MB = T dS = × 3BrH . (21) 16πGN at finite coordinate distance. The Penrose diagram will have the form given in figure 1 as long as the blackening factor F(r) has a single root, and the tortoise satisfies three conditions, namely: (I) limr→∞ r∗(r) = 0; (II)

Penrose diagram limr→rH r∗(r) = −∞; (III) limr→0 r∗(r) = 0. Each point in the Penrose diagram is a three-dimensional space, with metric Gij. The fact that Gij is anisotropic does not af- Lastly, we comment that the above gravitationals solu- fect the diagram, because the diagram is only constructed tion can be extended to a two-sided eternal black brane out of the coordinates t and r∗. We explicitly checked geometry, with two asymptotic boundaries. See figure that both the MT and the DK models satisfy the above 1. The extended solution is dual to a thermofield dou- conditions. The Penrose diagram of the charged MT ble state constructed out of two copies of the boundary model, considered in section IV, is different because in theory. that case the blackening factor F(r) has two roots. 5

III. HOLOGRAPHIC COMPLEXITY

In this section we compute the holographic complexity using the complexity=action (CA) [6, 7]. Here we follow closely the analysis of [16], with adaptations for anisotropic systems. We consider neutral anisotropic black branes with a generic bulk action of the form 1 Z √ S = ddxdr −gL(r, x) , (24) 16πGN and metric

2 2 2 i j ds = −Gtt(r)dt + Grr(r)dr + Gij(r)dx dx (25) where r is the AdS radial coordinate and (t, xi) are the gauge theory coordinates. Here i = 1, 2, ..., d − 1. We take the boundary as located at r = ∞ and we assume the existence of a horizon at r = rH, where Gtt has a zero and Grr has a simple pole. We denote as G the determinant of Gij, i.e. G = det(Gij). In the computations of holographic complexity it is convenient to use coordinates that cover smoothly the two sides of the geometry. We use Eddington-Finkelstein coordinates

u = t − r∗(r) , v = t + r∗(r) , (26) where the tortoise coordinate is deﬁned as s Z r 0 ∗ 0 Grr(r ) r (r) = sgn(Gtt(r)) dr 0 . (27) Gtt(r )

The CA conjecture states that the quantum complexity of the state of the boundary theory is given by the gravitational action evaluated in a region of the bulk known as the Wheeler-DeWitt (WDW) patch FIG. 2: Penrose diagram and the WDW patch (blue region) I for the two-sided black brane we consider. (a) Configuration C = WDW . (28) A π at initial times (t ≤ tc) in which the WDW patch intersects ~ both the future and the past singularity. (b) Configuration at The WDW patch is the domain of dependence of any later times (t > tc) when the WDW patch no longer intersects spatial slice anchored at a given pair of boundary times the past singularity. The dashed lines represent the cutoff surfaces at r = rmax. (tL, tR). See figure 2. The gravitational action in the WDW patch is divergent because this region extends all the way up to the asymptotic boundaries of the spacetime. We regularize this divergence by introducing a cut- in the WDW patch can be written as off surface at r = rmax near the boundaries. We also introduce a cutoff surface r = 0 near to the past and fu- IWDW = Ibulk + Isurface + Ijoint , (29) ture singularities. Without loss of generality, we consider the time evolution of holographic complexity for the sym- where metric configuration t = t = t/2. More general cases L R 1 Z √ can be obtained from the symmetric configuration by us- I = dd+1x −gL(x) (30) bulk 16πG ing the fact that the system is symmetric under shifts N M t → t + ∆t and t → t − ∆t. The gravitational action L L R R is the bulk action and Isurface and Ijoint are surface and joint terms that are necessary to have a well-defined vari- ational principle when one considers a finite domain of 6 space-time [14]. The surface terms are given by Z Z 1 d p 1 d−1 √ Isurface = d x |h|K ± dλd θ γκ 8πGN B 8πGN B0 (31) where the first term, which is defined in terms of the trace of the extrinsic curvature K, is the well-known Gibbons- Hawking-York boundary term [69, 70]. This term is necessary when the boundary includes (smooth) space-like and time-like segments, which we denoted as B. The second term in the above equation includes the contribution of null segments. This term is defined in terms of the parameter κ, which measure how much the null surface B0 fails to be affinely parametrized. Here we follow [14] and set κ = 0, so that we do not need to consider these null FIG. 3: Critical time (normalized by isotropic result) versus boundary terms. This choice of κ correspond to affinely a/T . We consider increasing values of a, but we choose rH in parametrize the null boundary surfaces. such a way to keep fixed the temperature as T = 1/π.

The joint terms are necessary when the intersection of two boundary terms is not smooth. These terms can be written as

Z Z A. Behavior at initial times: 0 ≤ t ≤ tc 1 d−1 √ 1 d−1 √ Ijoint = d x ση + d x σa¯ 8πGN Σ 8πGN Σ0 (32) where the first term [82] corresponds to the intersec- For initial times 0 ≤ t ≤ tc the WDW patch intersects tion of two boundary segments which can be time-like or with both the future and past singularities. The contri- space-like, so the intersection can be of the type: time- butions for IWDW include: the bulk term, the GHY terms like/time-like, time-like/space-like or space-like/space- and the joint terms. In principle, the GHY terms include like. As the WDW patch do not include such intersection, contributions from the cutoff surfaces at r = rmax and we do not need to consider this first term. The second r = 0, as well as from the null boundaries of the WDW term includes the contribution of the intersection of a patch. However, since we affinely parametrize the null null segment with any other boundary segment, so it in- surfaces, we do not need to consider the surface contri- cludes contribution of the type: null/null, null/time-like butions from the null boundaries. The joint terms include and null/space-like. A more precise definition of the sur- contribution from the intersection of the null boundaries face and joint terms will be given throughout the text of the WDW patch with the cutoff surfaces at r = rmax along with the adopted conventions [83]. The quantitya ¯ and r = 0. We use the left-right symmetry of the WDW is defined in appendix A. patch to calculate IWDW for the right side of Penrose diagram and then multiply the result by two. As first pointed out in [19], at early times the WDW patch intersects both the future and the past singularity, To calculate the bulk contributions we split the right-side of the WDW patch into three parts: region I, region II and this causes IWDW to be constant for some period of and region III, which are shown in figure 2 (a). We then time 0 ≤ t ≤ tc. At later times, t > tc, the WDW patch calculate the bulk contribution as no longer intersects the past singularity, and IWDW starts to change with time. These two cases are illustrated in I (t ≤ t ) = 2 (I I + I II + I III ) , (34) figure 2. The time scales separating these two regimes bulk c bulk bulk bulk can be written as where [84] ∗ ∗ ∗ ∗ tc = 2 (r∞ − r (0)) , r∞ = lim r (r) (33) V Z rH √ t r→∞ I I = d−1 dr −g L(r) + r∗ − r∗(r) bulk 16πG 2 ∞ where we have used that t = t = t/2. Figure 3 shows N 0 L R V Z rmax √ how the critical time (33) behaves as a function of the II d−1 ∗ ∗ Ibulk = dr −g L(r) r∞ − r (r) anisotropy parameter in MT model. This figure shows 8πGN rH that, as compared to an isotropic system at the same V Z rH √ t I III = d−1 dr −g L(r) − + r∗ − r∗(r) temperature, the anisotropy reduces the critical time, i.e., bulk ∞ 16πGN 2 the complexity starts to change earlier in anisotropic sys- 0 (35) tems. R d−1 with Vd−1 = d x. Note that in the above expressions we are assuming that the on-shell Lagrangian L only de- 7 pends on r. Summing all the contributions we obtain where

Z rmax r 0 0 1 √ ∗ ∗ GttG Gtt G Ibulk = dr −g L(r) r∞ − r (r) . (36) G(r) = + . (44) 2πGN 0 Grr Gtt G

Note that Ibulk(t ≤ tc) does not depend on time. Now In the above expressions we have already multiplied the we turn to the computation of the GHY surface terms. results by two to account for the two sides of the WDW bdry These contribution come from the cutoff surfaces at r = patch. Note that Isurface does not depend on time. More- future past rmax on the two sides of the geometry and from the cutoff over, the time dependence of Isurface and Isurface cancel, so surfaces at r = 0 both at the past and future singulari- that the total surface contribution is time-independent ties. In either cases the surfaces are described by a rela- V tion of the form r = constant, and the outward-directed d−1 ∗ ∗ Isurface(t ≤ tc) = G(r)(r∞ − r (r)) r= normal vector are proportional to ∂µ(r − constant). We 4πGN 0 V write the corresponding normal as d−1 ∗ ∗ + G(r)(r∞ − r (r)) (45) 8πGN r=rmax nµ = (nt, nr, ni) = b (0, 1, 0) , (37) The only terms left to calculate are the joint contribu- where b is some normalization constant. We normalize tions that come from the intersections of the null bound- 2 r the normal vector as n = n nr = ±1, where the plus aries of the WDW patch with the cutoff surfaces at sign is for space-like vectors at the r = rmax cutoff surface, r = rmax and r = 0. The joint terms can be written and the minus sign if for the time-like vectors at the as r = 0 cutoff surface. We obtain sing bdry Ijoint = Ijoint + Ijoint , (46) (s) (s) (s) (s) p nµ = (nt , nr , ni ) = (0, Grr(rmax), 0) (38) where I sing includes the contributions from the past and (t) (t) (t) (t) p joint n = (n , n , n ) = (0, −G ( ), 0). (39) bdry µ t r i rr 0 future singularities and Ijoint corresponds to the contribution from the two asymptotic boundaries. In [33] it where the superscript (s) denotes space-like vectors, was shown that, for a large class of isotropic systems, while the superscript (t) denotes time-like vectors. The the contribution from the asymptotic boundaries do not trace of the extrinsic curvature of these r−constant sur- depend on time, while the contributions at r = 0 van- faces can be calculated as ish. We show in appendix A that this also happens in √ anisotropic systems. So we can write K = ∇ nµ = √1 ∂ ( −gnr) µ −g r r=0,rmax bdry h i Ijoint(t ≤ tc) = Ijoint , (47) = √ 1 ∂r Gtt + ∂r G (40) 2 ∓Grr Gtt G r=0,rmax bdry where Ijoint does not depend on time. where we use the minus sign for the r = surface and 0 Finally, as none of the terms I , I and I depend the plus sign for the r = r surface. Here G = det(G ) bulk surface joint max ij on time for 0 ≤ t ≤ t , the gravitational action evaluated is the determinant along the transverse coordinates xi, c on the WDW patch is constant for this period of time not the full determinant, which we denoted as g. dI The GHY surface contributions can then be written as WDW = 0 , for 0 ≤ t ≤ t . (48) dt c future past bdry Isurface(t ≤ tc) = Isurface + Isurface + Isurface (41) where the contributions from the cutoff surfaces at future and past singularities are given by B. Behavior at later times: t > tc

V t I future = d−1 G(r) + r∗ − r∗(r) surface ∞ 8πGN 2 r=0 For later times t > tc the WDW patch no longer inter- V t sects with the past singularity. In this case, there are no past d−1 ∗ ∗ Isurface = G(r) − + r∞ − r (r) (42) surface and joint terms related to the past singularity, 8πG 2 r= N 0 but there is an additional joint term that comes from the and the contributions from the cutoff surfaces at the two intersection of two null boundaries of the WDW patch. asymptotic boundaries read See figure 2 (b). Again, we calculate all the contribution for the right side of the WDW patch and multiply V the results by two to account for the two sides of the I bdry = d−1 G(r) r∗ − r∗(r) (43) surface ∞ 8πGN r=rmax geometry. To compute the bulk contribution, we again split the 8 right side of the WDW patch into three regions, which from the cutoff surface at the future singularity and a we call I, II and III. See figure 2 (b). We write the total contribution from the intersection of the two null bound- bulk contribution as aries of the WDW patch. The joint term can then be written as I II III Ibulk(t > tc) = 2 (Ibulk + Ibulk + Ibulk) , (49) bdry null Ijoint(t > tc) = Ijoint + Ijoint , (55) where now where I null is the contribution from the intersection of V Z rH √ t joint I d−1 ∗ ∗ the two null boundaries. This term reads Ibulk = dr −g L(r) + r∞ − r (r) 16πGN 2 0 1 Z √ Z rmax √ null d−1 II Vd−1 ∗ ∗ Ijoint = d x G a¯ (56) Ibulk = dr −g L(r) r∞ − r (r) 8πGN 8πGN rH V Z rH √ t wherea ¯ is defined in terms of the left and right null vec- I III = d−1 dr −g L(r) − + r∗ − r∗(r) bulk 16πG 2 ∞ tors that parametrize the null boundaries of the WDW N rm patch. These null vectors are given by (50) L ∗ R ∗ kµ = −α ∂µ(t − r ) , kµ = α ∂µ(t + r ) (57) where the only difference from the 0 ≤ t ≤ tc case is that the r−integral in the region III starts at the point r = rm, L R In terms of kµ and kµ the quantitya ¯ can be written as instead of starting at the cutoff surface r = 0 at the past singularity. The point rm determines the intersection of 1 L R Gtt(rm) the two past null boundaries of the WDW patch and it a¯ = log k · k = − log . (58) 2 α2 satisfies the equation Using the above expressions we can write t − r∗ + r∗(r ) = 0 . (51) 2 ∞ m null Vd−1 p Gtt(rm) Ijoint = − G(rm) log 2 . (59) which can be solved numerically. Note that we recover 8πGN α the equation that gives the critical time t when we take c L the limit r → 0 in the above equation. Summing where rm is given by equation (51). The null vectors kµ m R the above contributions we can write the bulk term at and kµ are defined in terms of an arbitrary normalization later times as the bulk term at initial times plus a time- constant α that introduces an ambiguity in the calcula- dependent term tion of IWDW. With the above results, the joint term can be written as

Ibulk(t > tc) = Ibulk(t ≤ tc) + Vd−1 p Gtt(rm) Z rm I (t > t ) = I (t ≤ t )− G(r ) log . Vd−1 √ t joint c joint c m 2 ∗ ∗ 8πGN α dr −g L(r) − r∞ + r (r) 8πGN 0 2 (60) (52) Note that for t > tc the gravitational action calculated in the WDW patch can be written as where Ibulk(t ≤ tc) is given in equation (36). For later times the GHY term includes contributions from the fu- IWDW(t > tc) = IWDW(t ≤ tc) + δI , (61) ture singularity and from the two asymptotic boundaries. The contributions from the cutoff surfaces at the asymp- where totic boundaries do not depend on time, and have the δI = δIbulk + δIsurface + δIjoint , (62) same value that they have for t ≤ tc. The contribution from the cutoff surface at the future singularity reads with V t future d−1 ∗ ∗ δI = I (t > t ) − I (t ≤ t ) Isurface = G(r) + r∞ − r (r) . (53) bulk bulk c bulk c 8πG 2 r= N 0 V Z rm √ δt = d−1 dr −g L(r) + r∗(r) − r∗(0) , The total surface term can be written as 8πGN 0 2 δI = I (t > t ) − I (t ≤ t ) I (t > t ) = I (t ≤ t ) + I future . (54) surface surface c surface c surface c surface c surface V δt = d−1 G(r) , r= where Isurface(t ≤ tc) is defined in equation (45). Finally, 8πGN 2 0 we turn to the computation of the joint terms. These δIjoint = Ijoint(t > tc) − Ijoint(t ≤ tc) terms include time-independent contributions from the Vd−1 p Gtt(rm) two asymptotic boundaries, which are equal to the corre- = − G(rm) log 2 . (63) 8πGN α sponding quantities for t ≤ tc, a vanishing contribution 9

It is convenient to work with the time variable δt = t−tc, which is related to rm as δt + r∗(r ) − r∗(0) = 0 . (64) 2 m Finally, the time derivative of each contribution reads

dδI V Z rm √ bulk = d−1 dr −g L(r) , (65) dt 16πGN 0 dδI V surface d−1 = G , (66) dt 16πGN r=0 dδI V r G joint d−1 0 = Gtt dt 16πGN GrrGtt r=rm r 1 Gtt 0 Gtt FIG. 4: rm/rH versus δt for the MT model (blue curve) and + G log 2 . (67) for the DK model (black curve). Here, for the MT model, we 2 GrrG α have ﬁxed rH = 1 and a/T = 0.314. For the DK model we The time derivative of I can then be computed as ﬁxed B = 3 and rH = 1. The curves obtained for another val- WDW ues of these parameters are indistinguishable from the above dI V Z rm √ results. WDW = d−1 dr −g L(r) + G(r) r= dt 16πGN 0 0 ! # 1r G G r G + tt G0 log tt + G0 . 2 tt 2 GrrG α GrrGtt r=rm (68)

Therefore, the time derivative of the holographic complexity can be obtained as Results for the MT model dC 1 dI A = WDW . (69) dt π~ dt Substituting the metric functions Gmn(r) and the on- shell Lagrangian L(r) for the MT model and expanding the above contributions for small anisotropies, we obtain

Z rH √ 4 5 2 2 4 dr −g L(r) = −2rH − rHa log 2 + O(0 log 0) , 0 6 rG G G0 G0 1 1. Late time behavior tt tt + = 4r4 + r2 a2 (5 log 2 − 1) H H Grr Gtt G r=0 6 4 +O(0 log 0) , In this section we now apply the formula (68) for the r G 1 MT and DK models to study the late time behavior of 4 2 2 ∂rGtt = 4rH + rHa (10 log 2 − 1) . G G r=r 3 the time-derivative of CA. We ﬁrst observe that, at later rr tt H (71) times, rm approaches rH. This can be seen in ﬁgure 4, where we plot rm versus δt. By summing the above contributions and taking the limit

Therefore, the late time behavior of dIWDW/dt is obtained 0 → 0, we ﬁnd by taking the limit rm → rH in the equation (68) dI V r2 a2 WDW = 3 6r4 + H (5 log(2) − 1) = 2M(a) , " rH r H dI V Z √ G dt 16πGN 2 WDW d−1 0 = dr −g L(r) + Gtt (72) dt 16πGN GrrGtt r=rH 0 where the mass of the black brane M(a) is given by equa- # rG G G0 G0 tion (16). Therefore, the late time behavior of the time tt tt + + . (70) derivative of holographic complexity reads Grr Gtt G r=0

dCA 2M(a) We have checked that the same late-time result for dCA = , (73) dt dt π can be obtained by following the approach developed by ~ Brown et al [7]. See appendix B. which saturates the Lloyd’s bound. 10

Results for the DK model anisotropy parameter rH such that T = 1/π 2M 0.00 1.0000 6.000 0.10 0.9997 6.005 For the DK model, we obtain the following results 0.15 0.9993 6.011 dIWDW V3 ˙ ˙ ˙ = δIbulk + δIsurface + δIjoint (74) TABLE I: black brane’s mass, measured in units of dt 16πGN V3/(16πGN), for several values of a and rH. Here we chosen where the contributions from bulk, surface and joint rH such that the Hawking temperature is ﬁxed T = 1/π. terms are given by

Z rH √ ˙ 2 δIbulk = dr −g L(r) = −6B rH , 0 rG G G0 G0 ˙ tt tt 2 δIsurface = + = 6B rH , Grr Gtt G r=0 r G δI˙ = G0 = 6B r2 . (75) joint tt H GttGrr r=rH With the above results, the late-time rate of change of holographic complexity reads

dCA 1 dIWDW V3 2 = = × 6B rH = 2MB , (76) dt π dt 16πGN which precisely saturates the Lloyd’s bound. This pro- FIG. 5: The time dependence of holographic com- vides another example where, despite the anisotropy, the plexity calculated with the CA proposal. The curves Lloyd’s bound is still respected. correspond to: (a, rH, 2M) = (0, 1, 6) (black curves), (a, rH, 2M) = (0.1, 0.9997, 6.005) (blue curves) and

(a, rH, 2M) = (0.15, 0.9993, 6.011) (red curves). The continu- 2 ous curves represent the results (in units of V3/(16π ~ GN)) for the time derivative of holographic complexity, while the 2. Full time behavior dashed horizontal lines represent 2M. We ﬁx the normalization of the null-vector in equation (57) by taking α = 0.1. The qualitative behavior is the same for other values of α. In this section we study the full time behavior of holographic complexity for the MT and DK models. We numerically solve the equation (64) to ﬁnd rm as a function Results for the DK model of δt and then we use the result in equation (68) to obtain

IWDW as a function of δt. The geometry of the DK model is controlled by the dimensionless parameter B/T 2, where B is the intensity of the magnetic field, while T is the black brane’s tem- Results for the MT model perature [85]. In figure 6 we show the full time behavior of the rate of change of complexity for different intensi- ties of the magnetic field. Just like in the MT model, The geometry in the MT model is controlled by the di- there is a violation of Lloyd’s bound at early times, and mensionless parameter arH, where a is the parameter of the result approach the bound from above at later times. anisotropy. The values of (a, rH,M) for which we study Moreover, the net effect of the magnetic field is just a the complexity growth are shown in table 1, and they ˙ vertical shift in the curves of CA versus δt. were chosen such that the temperature is fixed as we increase the anisotropy. In this table we can see that the black brane’s mass increases as we increase a while keeping T fixed. Figure 5 shows the time dependence IV. THE CHARGED CASE of the gravitational action in the WDW patch for the choice of parameters presented in table 1. The behavior of dCA/dt is qualitatively similar to the behavior ob- In this section we use the CA proposal to study the rate of served in isotropic systems. The anisotropy increase the change of complexity of charged anisotropic black branes. mass of the black brane and its effects on the rate of In particular, we consider the type IIB supergravity so- change of complexity seem to be just a vertical shift in lution found in [27, 28]. This solution is basically an ex- the curves of dCA/dt versus t. tension of the MT solution to the charged case, therefore, 11

where the O(q0) terms are " 1 2 2 2 f0(r) = − 4 2 8r rH − 2rH(4 + 5 log 2) 24r rH # r2 + (3r4 + 7r4 ) log 1 + H , H r2 2 2 1 10rH rH b0(r) = − 2 2 2 + log 1 + 2 , 24rH r + rH r 2 1 rH ϕ0(r) = − 2 log 1 + 2 , (80) 4rH r while second order terms are given by FIG. 6: Full time behavior of holographic complexity of mag- " netic black branes for different values of the magnetic field. 1 6 2 8 f2(r) = 6 2 2 2 6r rH + rH The curves correspond to B = 2 (black curves), B = 2.5 (blue 24r rH(r + rH) curves) and B = 3 (red curves). The continuous curves cor- + r4r4 (25 log 2 − 12)r2r6 (25 log 2 − 1) responds represent C˙A, while the dashed horizontal lines rep- H H 2 # resent 2MB . The results are given in units of V3/(16π GN). 2 2 2 6 2 4 6 rH We fixed the normalization of the null vector in equation (57) − (r + r )(6r + 7r r + 12r ) log 1 + , H H H r2 by taking α = 1.3. The qualitative behavior is the same for other values of α. 4 2 2 4 2 2r + 3r rH + 11rH 1 rH b2(r) = − 2 2 2 2 + 2 log 1 + 2 , 24r (r + rH) 12rH r 2 1 1 1 rH ϕ2(r) = − 2 − + 2 + log 1 + 2 . (81) 4r 4(r rH) 2 r

The field-strength and the associated chemical potential are given by √ 1 F = −Q Be3φ/4 dt ∧ dr , for conceptual clarity, we will refer to it as the charged r3 Q 5a2 MT model. The action of this model reads µ = 1 − log 2 . 2 24r2 Z H 1 5 √ h 12 1 2 S = d x −g R + 2 − (∂φ) (77) (82) 16πGN M L 2 1 1 i − e2φ (∂χ)2 − F F mn + S . The charged black brane’s mass is given by 2 4 mn GH a2r2 The solution in Einstein frame takes the form given by M(a, q) = 3r4 + H − 2 + 10 log 2 H 8 (4) and (5), with the metric functions given by h 5a2r2 i + q2 3r4 − H (−3 + 5 log 2) . (83) r4 r 6 r 4 H 8 F = 1 − H + H − H q2 + a2F (r, q) + O(a4) r4 r r 2 2 4 B = 1 + a B2(r, q) + O(a ) −φ(r) 2 4 H = e , with φ = a φ2(r, q) + O(a ) (78) A. Rate of change of complexity where the functions F2(r, q), B2(r, q) and φ2(r, q) now depend on the charge parameter q, which is related to the black brane’s charge as q ≡ Q√ . Here Q is the black The rate of change of complexity can be calculated as be- r3 2 3 H fore, by considering the on-shell action evaluated on the brane’s charge in units of V /(16πG ). For small values 3 N WDW patch, with the difference that now the Penrose of q, one can find an analytic solution of the form [27] diagram is modified by the fact that the black hole is charged. See figure 7. The total action is given by a sum F (r, q) = f (r) + f (r)q2 + O(q4) , 2 0 2 of four terms: the bulk contribution, the surface contri- 2 4 B2(r, q) = b0(r) + b2(r)q + O(q ) , bution, the joint contribution, and a boundary term for 2 4 φ2(r, q) = ϕ0(r) + ϕ2(r)q + O(q ) , (79) the Maxwell field. 12

Let us first evaluate the sum of the bulk and Maxwell given by contributions. Using the equations of motion that result Z r+ √ from the action (78), it is easy to show that the on-shell I Vd−1 t ∗ ∗ Ibulk = dr −gL(r) + r∞ − r (r) , lagrangian density is given by 16πGN 1 2 rm Z rmax √ II Vd−1 ∗ ∗ 1 mn I = dr −gL(r) r − r (r) , L(r) = −8 − FmnF . (84) bulk 8πG ∞ 6 N r+ Z r+ √ III Vd−1 t ∗ ∗ However, this on-shell Lagrangian density can be affected Ibulk = dr −gL(r) − + r∞ − r (r) . 16πGN 2 2 by the presence of a non-zero boundary term for the rm Maxwell field. For a gauge field action of the form (91) Z 1 d+1 √ mn By taking the time derivative of the above expressions IMaxwell = − 2 d x −gFmnF , (85) 4g M and using (89) one can check that

2 where g is the gauge coupling parameter [86]. The cor- Z rm dIbulk Vd−1 √ 1 − 2γ responding boundary term can be written as [74] = − dr −g 8 + F 2 . (92) dt 16πGN 1 6 rm γ Z I bdry = dΣ F mnA , (86) We now turn to the evaluation of the joint terms, corre- Maxwell g2 m m ∂M sponding to the future and past corners (the red dots in where γ is an arbitrary parameter that affects the late Figure 7). These contributions are given by time behavior of complexity. Later, we are going to V q G (r1,2) fix this parameter by requiring consistency with the un- I1,2 = − d−1 G(r1,2) log tt m (93) joint m 2 charged case. Using the equations of motion, one can 8πGN α show that whose time derivative gives Z √ bdry γ d+1 mn I = d x −gF F . (87) 1,2 r Maxwell 2 mn dI V G G on-shell 2g M joint = − d−1 tt G0(r) log tt 2 dt 16πGN GrrG α Therefore, taking into account the above contribution, r ! the on-shell Lagrangian density becomes G 0 + Gtt(r) . (94) GrrGtt 1 − 2γ r=r1,2 L(r) = −8 − F F mn . (88) m 6 mn Finally, the surface contributions coming from the two We now proceed to the evaluation of the bulk action cor- asymptotic boundaries do not depend on time, and hence responding to the above on-shell Lagrangian density. The do not contribute the rate of change of complexity. WDW patch is shown in Figure 7. The future and past Adding the above contributions, we find 1 2 corners are denoted as rm and rm, respectively. The " r2 r−coordinate of these points satisfy the following rela- dC V Z m √ 1 − 2γ π A = d−1 dr −g 8 + F 2 tions dt 16πGN 1 6 rm

t ∗ ∗ 1 t ∗ ∗ 2 r r ! r1 # + r − r (r ) = 0 , − r − r (r ) = 0 (89) Gtt Gtt G m ∞ m ∞ m − G0 log + G0 (95) 2 2 2 tt 2 GrrG α GrrGtt rm The time derivative of the above relations implies 1 2 At late times, rm → r− and rm → r+, we ﬁnd 1,2 r dr sgn(Gtt) Gtt m = ± . (90) " 1,2 Z r− dt 2 G r=rm dC V √ 1 − 2γ rr π A = d−1 dr −g 8 + F 2 dt 16πG 6 As before, we calculate the bulk contribution for half of N r+ r # the WDW patch, and then we multiply the ﬁnal result G r− 0 by two. The contributions for regions I, II and III are − Gtt(r) . (96) GrrGtt r+

In order to evaluate the above formula for the charged MT model, we parametrize the outer and inner horizon as a2 q2 r+ = rH , r− = rHq 1 + 2 5 log − 1 . (97) 48rH 4 13

parameter, however, can be fixed by requiring the q → 0 limit of (98) to be consistent with the uncharged case. This can be done by setting q = 0 in (98) and choosing γ such that the final result matches (72) [87]. By doing that, we find

3a2 γ = 2 . (99) 16rH Notice that γ = 0 for a = 0, which means that the contribution of the Maxwell boundary term is zero in the isotropic case. With this choice for γ, the ﬁnal result reads

dCA V3 2 4 π = 6(1 − q )rH dt 16πGN " #! a2r2 5 + H − 1 + 5 log 2 + q2 −1 + log 2 . 2 2 (100)

By construction, the q → 0 limit of the above result is consistent with the result for the uncharged case, given in (72). Furthermore, the a → 0 limit is consistent with previous results reported in the literature. See, for instance, (4.27) of [16].

Now let us discuss our result in the light of the bound proposed in [7], according to which the natural bound for states at a ﬁnite chemical potential is dC π A ≤ 2 (M − µQ) − 2 (M − µQ) , (101) dt gs where the second term correspond to the ground state (gs) value of (M − µQ). As the charged MT model does not have an extremal limit, the ground state is the vac- uum solution (M = Q = 0). Using the formula (82) for the chemical potential, and taking into account that the black brane’s charge is Q = V3Q , we can see that 16πGN

dC a2q2r2 π A − 2 (M − µQ) = H (−17 + 20 log 2) < 0 FIG. 7: Penrose diagram of a charged asymptotically AdS dt 4 black hole. We split half of the WDW patch into three regions, (102) 1 I, II and III. At later times, the future corner, rm, approaches which shows that the bound (101) can be saturated in 2 the inner horizon, r−, while the past corner, rm, approaches the a → 0 limit, but it is no longer saturated once we the outer horizon, r+. turn on the anisotropy parameter.

Specializing (96) for the charged MT model, and using (97), we ﬁnd " dCA V3 4 2 V. DISCUSSION π = rH(3 − 2γ) 2(1 − q ) + dt 16πGN

2 # a 2 We have used the CA conjecture to study the + 2 1 + 5q (−1 + log 2) + 10 log 2 (98). 12rH time-dependence of holographic complexity for three anisotropic black brane solutions, namely, the MT model, The result depends on the arbitrary parameter, γ. This the DK model, and the charged MT model. 14

MT model longitudinal directions, namely

dCA 2M(0) V3 4 = + (Pxy − Pz) + O(a ) . (103) The MT solution is dual to the N = 4 SYM theory de- dt π~ π~ formed by a position-dependent theta-term that breaks isotropy and conformal invariance. The background ge- This can be seen from equations (73) and (17). ometry is controlled by the ratio a/T , where a is the parameter of anisotropy, and T is the Hawking temperature. DK model Similarly to the case of isotropic systems, the rate of change of complexity in anisotropic systems is zero for The behavior of holographic complexity in the MT model t ≤ tc, and it is non-zero for t > tc, with this critical time given by equation (33). Figure 3 shows the behav- is very similar to the behavior observed in magnetic branes. By using the CA conjecture, we studied the time ior of tc as a function of the anisotropy parameter. In this figure we consider increasing values of the anisotropy behavior of holographic for the magnetic black brane so- parameter, while keeping fixed the temperature. As com- lution found by D’Hoker and Kraus in [73]. In this model pared with an isotropic system with the same tempera- one introduces a constant magnetic field that breaks the rotational symmetry of the background. The geometry ture, the holographic complexity of anisotropic systems 2 remains constant for a shorter period, i.e., the effect of is controlled by the ratio, B/T , between the magnetic field and the temperature squared. For very large val- the anisotropy is to reduce tc. ues of values of B/T 2, this system has a simple solu- In section III B 1 we study the late-time behavior of tion, which is given in (19). For this configuration, the the holographic complexity and find an expression for Lloyd’s bound is violated at early times, but it is satu- dIWDW/dt in terms of the metric functions. See equation rated at later times. This provides another example of (70). For simplicity, let us first consider the isotropic a system that breaks the rotational symmetry without case, in which a = 0. In this case the MT solution re- violating the Lloyd’s bound at later times. This should duces to the five-dimensional black brane solution that is be contrasted with the bound for η/s, which is known dual to the undeformed N = 4 SYM theory. From pre- to be violated in anisotropic systems. This suggests that vious works [7, 16, 17], we know that the Lloyd’s bound the violation of Lloyd’s bound [21–24] in the case of neu- should be respected in this case. As we turn on a small tral black holes is not due to anisotropy, but rather to anisotropy parameter, all the metric functions get correc- the presence of a conformal anomaly. As neither the tions up to the second order in a and this leads to a larger D’Hoker & Kraus nor the Mateos & Trancanelli model black brane’s mass (see equation (16)). In this case, we display a conformal anomaly (up to second order in the expect the formula (70) to provide the result for a = 0, anisotropy), this would explain why the Lloyd’s bound plus corrections up to the second order in the anisotropy is not violated in these two models. We are currently parameter. Applying our formulas for the MT model investigating whether this last statement is true. we find that the late time rate of change of complexity matches the Lloyd’s bound, i.e., dCA/dt = 2M(a)/π~. This is a highly non-trivial match, because it means that the anisotropy increases the value of 2M and the late Charged anisotropic black branes time value of dIWDW/dt precisely in the same amount.

The full-time behavior of dCA/dt can be seen in fig- In section IV we use the CA conjecture to study the ure 5. The results share a lot of similarities with the late-time behavior of holographic complexity for a gen- previous results obtained for isotropic systems [16]. In eralization of the MT model to the charged case. In this particular, dCA/dt violates the Lloyd’s bound at initial case the geometry is not only controlled by the parame- times, and approaches this bound (from above) at later ter a/T , but also by the dimensionless charge parameter times. In this figure we consider increasing values of the q. Following [74] we consider the inclusion of a Maxwell anisotropy parameter, while keeping fixed the tempera- boundary term (see equation (86)), which introduces an ture. The resulting black brane’s mass increases as we arbitrary parameter γ that affects the late-time rate of increase the anisotropy parameter, and the overall effect change of complexity. This new boundary term turned of the anisotropy is a vertical upward shift [88] in the out to be necessary to make the q → 0 limit of the final curves of dCA/dt versus δt. At later times, the difference result consistent with the result for neutral anisotropic 3a2 between the anisotropic and isotropic results is propor- black branes. We find γ = 2 , which suggests that the 16rH tional to the difference in pressures in the transverse and Maxwell boundary term is generically necessary when we have non-trivial matter fields besides the Maxwell field. Having fixed the value of γ, we find that the charged MT model respects the bound (101) proposed in [7]. 15

Conclusions and Future directions the background geometry. In isotropic geometries, one can assume the ansatz Xm = (v(λ), r(λ), x, y, x), where v = t + r∗. The xyz-rotational symmetry of this ansatz We have considered three different models in which mat- results in a simple form for the volume functional (104), ter fields break the rotational symmetry, and we studied which can be easily extremized. In anisotropic systems, how this affects the holographic complexity. In neutral one no longer has this rotational symmetry, because black holes, the formula for holographic complexity only Gxx 6= Gzz, and that results in a more complicated form depends on the metric components, having no explicit de- for Xm and V. pendence on the matter fields (the on-shell Lagrangian is just a constant). In other words, the matter fields Acknowledgements: We are grateful to Hugo Marro- only affect the holographic complexity through their ef- chio, Hesam Soltanpanahi and Alberto Güijosafor very fect on the geometry. This should be contrasted with the useful discussions and comments. We also thank Hamid charged case, in which the electric charge appears explic- Rajaian for useful correspondence and for comments on itly in the formulas for the holographic complexity, as the manuscript. We are indebted to an anonymous well as in the metric components. referee for helpful suggestions and comments. MMQ is supported by Institute for Research in Fundamen- We have observed that the holographic complexity of tal Sciences(IPM). VJ and YDO were supported by anisotropic systems increases as compared to isotropic Mexico’s National Council of Science and Technology systems at the same temperature. This happens because (CONACyT) grant CB-2014/238734. VJ is also par- the matter fields increase the mass of the black hole and, tially supported by the Basic Science Research Pro- for the Lloyd’s bound to be respected, the holographic gram through the National Research Foundation of Ko- complexity also has to increase. rea(NRF) funded by the Ministry of Science, ICT & Fu- ture Planning(NRF2017R1A2B4004810) and GIST Re- We have studied the effects of anisotropy on the complex- search Institute(GRI) grant funded by the GIST in 2019. ity growth considering the case of small anistropies. Our results are valid up to O(a2). It would be interesting to extend our results to higher anisotropies, because in this case the MT model displays a conformal anomaly [89], which might cause a violation of the Lloyd’s bound. Be- sides that, the MT gravitational solution can be thought of as describing a renormalization group (RG) flow from a AdS geometry in the ultraviolet (UV) to a Lifshitz ge- Appendix A: Joint terms at the r = rmax and r = 0 ometry in the infrared (IR). The parameter controlling cutoff surfaces this transition is the ratio a/T , which is small close to the UV fixed point and large close to the IR fixed point. It would be interesting to study how the complexity growth In this appendix we briefly review how to calculate the behaves under this RG flow. Moreover, as Lifshitz ge- joint terms at the asymptotic boundaries and at the sin- ometries were known to violate the Lloyd’s bound [24], gularities. We show that the contributions from the we expect such a violation to occur in the MT model at asymptotic boundaries are time-independent, while the higher anisotropies. contributions from the singularities vanish.

Another interesting extension of this work would be to A joint term for a corner involving the connection of at study the effects of the anisotropy in the holographic least one null surface has the form [15] complexity calculated using the CV conjecture. Al- though this calculation is relatively easy for isotropic sys- 1 Z √ I = dd−1x σ a¯ (A1) tems [5, 16, 17], the extension for anisotropic systems is joint 8πG non-trivial, because in this case the ansatz for the maxi- N mum volume surface is more complicated, preventing the where σ is the induced metric on the surfaces anda ¯ is use of the techniques used in [5, 16, 17]. More specifically, defined as the volume functional of the co-dimension one surface can  log |k · n(t)| for spacelike-null joints be generically written as  (s) Z a¯ = ± log |k · n | for timelike-null joints d p  + − V = d σ det(gab) , (104) log |k · k /2| for null-null joints

+ − where σ and g = ∂ Xm∂ XnG are the coordinates where k and k are outward directed null normal a ab a b mn (t) (s) and the induced metric along the surface, respectively. vectors, while n (n ) are outward directed timelike m (spacelike) normal vectors. The overall sign depends on Here X (σa) are the embedding functions describing the orientation of the normal vectors. For more details, the surface, while Gmn are the metric components of see the appendix A of [15]. The relevant normal vectors 16 can be written as

(t) (t) (t) (t) p nµ = (nt , nr , ni ) = (0, −Grr(0), 0) , (A2) (s) (s) (s) (s) p nµ = (nt , nr , ni ) = (0, Grr(rmax), 0) , (A3) ± ∗ kµ = ±α ∂µ(t ± r ) . (A4)

With the above deﬁnitions, the joints term coming from the singularities can be written as Z √ sing 1 d−1 (t) Ijoint = d x σ log |k · n | (A5) 8πGN V d−1 = − G(r) log |Gtt(r)| . 8πGN r=0

sing 3 For the MT model, one can show that Ijoint ∼ 0 log 0. Therefore, the contribution from this joint term vanishes in the limit 0 → 0. The joint terms coming from the FIG. 8: Change in the WDW patch as the time evolves in asymptotic boundaries are given by the left boundary Z √ bdry 1 d−1 (s) Ijoint = d x σ log |k · n | (A6) 8πGN V d−1 = G(r) log |Gtt(r)| . 8πGN r=rmax

bdry For the MT model Ijoint gives rise to a divergent contribution that is independent of time, because it only depends on quantities calculated on the outside of the black brane, and this region has a time-translation symmetry. There- fore, this term do not contribute to the rate of change of holographic complexity. FIG. 9: Piece of the WDW patch that contributes to the rate of change of complexity at late times.

Appendix B: Comparison with Brown et al while it decreases in the region shown in light-blue. To calculate the corresponding variation of the WDW patch, The CA conjecture was proposed by Brown et al in [6, 7]. the authors of [7] argue as follows: In those papers the authors find a clever way of calculating the late time rate of change of complexity without having to take into account the contributions from joint • the parts of the WDW patch that lie outside of and null boundary terms. In a later work, Myers et al the horizon are time-independent because this re- [14] derive the expressions for the joint and null bound- gion has a time-translation symmetry. As a conse- ary terms and showed how to include the corresponding quence, these parts do not contribute to the rate of contributions to the rate of change of holographic com- change of complexity; plexity. Myers et al find a perfect match with the results of Brown et al at later times and carefully explain the • the part of the WDW patch that lies inside the past reasons behind the agreement in [14]. In this appendix horizon contributes at early times, but it is highly we briefly review the approach of Brown et al and we suppressed at later times. Hence, at later times, the show that it gives the same results obtained in section only contribution for the rate of change of complex- III using the approach of Myers et al [14, 15]. ity comes from the region of the WDW patch that lies inside the future horizon. This region is shown In the approach of Brown et al it is more convenient to in figure 9; consider the time evolution of the WDW patch when we increase the time in the left boundary, while keeping fixed • under time evolution the surface B is replaced by 0 the time in the right boundary, as shown in figure 8. the surface B , while the corners HB and SB are replaced by the corners HB0 and S0B, respectively. Figure 8 shows that, as the time evolves in the left bound- The surfaces B and B0 are related by a time- ary, the WDW patch increases in the region shown in red, translation symmetry and so their contributions 17

cancel. The same cancellation occurs between the where we have used (40) to express K in terms of the contributions coming from HB and HB0 and be- metric functions. The time-derivative reads tween the contributions coming from SB and B0S; dI V h Z rH √ WDW = d−1 dr −g L(r) dt 16πGN With the above cancellations the only terms left to be 0 r G rG G G0 G0 computed are the bulk contribution and the surface con- 0 tt tt + Gtt + + , tributions coming from the horizon H and from the sin- GttGrr r=rH Grr Gtt G r=0 gularity S. Therefore, the gravitational action evaluated (B5) in the WDW patch can be written as where we have used that Gtt/Grr vanishes at the horizon

IWDW = Ibulk + Isurface , (B1) to simplify the expression for the GHY term at the horizon. The above results for the late-time rate of change where the bulk contribution reads of IWDW precisely coincides with the result (70) obtained 1 Z √ with the approach of Myers et al [14, 15]. The reason for I = dd+1x −gL(x) (B2) the agreement is the following: both approaches contain bulk 16πG N M identical bulk contributions and identical surface contri- while the GHY surface contribution reads butions coming from the future singularity. The only difference is that in the calculation of Brown et al there is 1 h Z Z i a GHY-like term for the horizon, while in the calculation I = ddxp|h|K + ddxp|h|K surface of Myers et al there is no such term, but there is instead 8πGN r=rH r=0 (B3) a joint contribution coming from a corner that lies just where r = rH indicates the boundary surface at the hori- behind the past horizon. Surprisingly, these two terms zon and r = 0 indicates the boundary surface at the precisely coincides and both approaches give the same singularity. For the general action and metric given in result. A more detailed explanation for the agreement (24) and (25) we can write between the two approaches can be found in [14]. " Z Z rH Vd−1 √ IWDW = dt dr −g L(r) 16πGN 0 Z rG G G0 G0 tt tt + dt + Grr Gtt G r=rH # Z rG G G0 G0 tt tt + dt + , (B4) Grr Gtt G r=0

[1] J. M. Maldacena, “The large N limit of superconformal [7] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle ﬁeld theories and supergravity,” Adv. Theor. Math. Phys. and Y. Zhao, “Complexity, action, and black 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [hep- holes,” Phys. Rev. D 93, no. 8, 086006 (2016) th/9711200]. doi:10.1103/PhysRevD.93.086006 [arXiv:1512.04993 [2] S. S. Gubser, I. R. Klebanov, A. M. Polyakov, “Gauge [hep-th]]. theory correlators from noncritical string theory,” Phys. [8] B. Swingle, “Entanglement Renormalization and Lett. B428, 105-114 (1998) [hep-th/9802109]. Holography,” Phys. Rev. D 86, 065007 (2012) [3] E. Witten, “Anti-de Sitter space and holography,” Adv. doi:10.1103/PhysRevD.86.065007 [arXiv:0905.1317 Theor. Math. Phys. 2, 253-291 (1998) [hep-th/9802150]. [cond-mat.str-el]]. [4] L. Susskind, “Computational Complexity and [9] G. Vidal, “Entanglement Renormalization,” Black Hole Horizons,” [Fortsch. Phys. 64, 24 Phys. Rev. Lett. 99, no. 22, 220405 (2007) (2016)] Addendum: Fortsch. Phys. 64, 44 (2016) doi:10.1103/PhysRevLett.99.220405 [cond- doi:10.1002/prop.201500093, 10.1002/prop.201500092 mat/0512165]. [arXiv:1403.5695 [hep-th], arXiv:1402.5674 [hep-th]]. [10] T. Hartman and J. Maldacena, “Time Evolution of [5] D. Stanford and L. Susskind, “Complexity and Entanglement Entropy from Black Hole Interiors,” Shock Wave Geometries,” Phys. Rev. D 90, no. JHEP 1305, 014 (2013) doi:10.1007/JHEP05(2013)014 12, 126007 (2014) doi:10.1103/PhysRevD.90.126007 [arXiv:1303.1080 [hep-th]]. [arXiv:1406.2678 [hep-th]]. [11] S. Aaronson, “The Complexity of Quantum States [6] A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Transformations: From Quantum Money to Black and Y. Zhao, “Holographic Complexity Equals Bulk Ac- Holes,” arXiv:1607.05256 [quant-ph]. tion?,” Phys. Rev. Lett. 116, no. 19, 191301 (2016) [12] J. M. Maldacena, “Eternal black holes in anti-de Sitter,” doi:10.1103/PhysRevLett.116.191301 [arXiv:1509.07876 JHEP 0304, 021 (2003) [hep-th/0106112]. [hep-th]]. [13] S. Lloyd, “Ultimate physical limits to compu- 18

tation,” Nature 406 (2000), no. 6799 10471054 doi:10.1103/PhysRevD.92.126009 [arXiv:1509.06614 [arXiv:9908043[quant-ph]] [hep-th]]. [14] L. Lehner, R. C. Myers, E. Poisson and [30] M. Alishahiha, A. Faraji Astaneh, A. Naseh R. D. Sorkin, “Gravitational action with null bound- and M. H. Vahidinia, “On complexity for F(R) aries,” Phys. Rev. D 94, no. 8, 084046 (2016) and critical gravity,” JHEP 1705, 009 (2017) doi:10.1103/PhysRevD.94.084046 [arXiv:1609.00207 doi:10.1007/JHEP05(2017)009 [arXiv:1702.06796 [hep- [hep-th]]. th]]. [15] D. Carmi, R. C. Myers and P. Rath, “Comments [31] M. Alishahiha and A. Faraji Astaneh, “Holographic on Holographic Complexity,” JHEP 1703, 118 (2017) Fidelity Susceptibility,” Phys. Rev. D 96, no. 8, doi:10.1007/JHEP03(2017)118 [arXiv:1612.00433 [hep- 086004 (2017) doi:10.1103/PhysRevD.96.086004 th]]. [arXiv:1705.01834 [hep-th]]. [16] D. Carmi, S. Chapman, H. Marrochio, R. C. My- [32] J. Couch, W. Fischler and P. H. Nguyen, “Noether ers and S. Sugishita, “On the Time Dependence of charge, black hole volume, and complexity,” JHEP Holographic Complexity,” JHEP 1711, 188 (2017) 1703, 119 (2017) doi:10.1007/JHEP03(2017)119 doi:10.1007/JHEP11(2017)188 [arXiv:1709.10184 [hep- [arXiv:1610.02038 [hep-th]]. th]]. [33] S. Chapman, H. Marrochio and R. C. Myers, “Com- [17] K. Y. Kim, C. Niu, R. Q. Yang and C. Y. Zhang, plexity of Formation in Holography,” JHEP 1701, 062 “Comparison of holographic and field theoretic com- (2017) doi:10.1007/JHEP01(2017)062 [arXiv:1610.08063 plexities by time dependent thermofield double states,” [hep-th]]. arXiv:1710.00600 [hep-th]. [34] R. G. Cai, S. M. Ruan, S. J. Wang, R. Q. Yang [18] R. Jefferson and R. C. Myers, “Circuit complexity and R. H. Peng, “Action growth for AdS black holes,” in quantum field theory,” JHEP 1710, 107 (2017) JHEP 1609, 161 (2016) doi:10.1007/JHEP09(2016)161 doi:10.1007/JHEP10(2017)107 [arXiv:1707.08570 [hep- [arXiv:1606.08307 [gr-qc]]. th]]. [35] A. R. Brown, L. Susskind and Y. Zhao, “Quantum [19] S. Chapman, M. P. Heller, H. Marrochio and Complexity and Negative Curvature,” Phys. Rev. D 95, F. Pastawski, “Toward a Definition of Com- no. 4, 045010 (2017) doi:10.1103/PhysRevD.95.045010 plexity for Quantum Field Theory States,” [arXiv:1608.02612 [hep-th]]. Phys. Rev. Lett. 120, no. 12, 121602 (2018) [36] A. R. Brown and L. Susskind, “The Second Law of Quan- doi:10.1103/PhysRevLett.120.121602 [arXiv:1707.08582 tum Complexity,” arXiv:1701.01107 [hep-th]. [hep-th]]. [37] W. J. Pan and Y. C. Huang, “Holographic complexity [20] R. Khan, C. Krishnan and S. Sharma, “Circuit Complex- and action growth in massive gravities,” Phys. Rev. D 95, ity in Fermionic Field Theory,” arXiv:1801.07620 [hep- no. 12, 126013 (2017) doi:10.1103/PhysRevD.95.126013 th]. [arXiv:1612.03627 [hep-th]]. [21] J. Couch, S. Eccles, W. Fischler and M. L. Xiao, “Holo- [38] R. Q. Yang, “Strong energy condition and complex- graphic complexity and noncommutative gauge theory,” ity growth bound in holography,” Phys. Rev. D 95, JHEP 1803, 108 (2018) doi:10.1007/JHEP03(2018)108 no. 8, 086017 (2017) doi:10.1103/PhysRevD.95.086017 [arXiv:1710.07833 [hep-th]]. [arXiv:1610.05090 [gr-qc]]. [22] M. Alishahiha, A. Faraji Astaneh, M. R. Moham- [39] D. Momeni, S. A. H. Mansoori and R. Myrza- madi Mozaffar and A. Mollabashi, “Complexity Growth kulov, “Holographic Complexity in Gauge/String with Lifshitz Scaling and Hyperscaling Violation,” Superconductors,” Phys. Lett. B 756, 354 (2016) arXiv:1802.06740 [hep-th]. doi:10.1016/j.physletb.2016.03.031 [arXiv:1601.03011 [23] B. Swingle and Y. Wang, “Holographic Complexity [hep-th]]. of Einstein-Maxwell-Dilaton Gravity,” arXiv:1712.09826 [40] D. Momeni, M. Faizal, S. Bahamonde and R. Myrza- [hep-th]. kulov, “Holographic complexity for time-dependent [24] Y. S. An and R. H. Peng, “The effect of Dilaton on the backgrounds,” Phys. Lett. B 762, 276 (2016) holographic complexity growth,” arXiv:1801.03638 [hep- doi:10.1016/j.physletb.2016.09.036 [arXiv:1610.01542 th]. [hep-th]]. [25] D. Mateos and D. Trancanelli, “The anisotropic N=4 su- [41] R. Q. Yang, C. Niu and K. Y. Kim, “Surface Coun- per Yang-Mills plasma and its instabilities,” Phys. Rev. terterms and Regularized Holographic Complexity,” Lett. 107, 101601 (2011) [arXiv:1105.3472 [hep-th]]. JHEP 1709, 042 (2017) doi:10.1007/JHEP09(2017)042 [26] D. Mateos and D. Trancanelli, “Thermodynamics and [arXiv:1701.03706 [hep-th]]. Instabilities of a Strongly Coupled Anisotropic Plasma,” [42] R. G. Cai, M. Sasaki and S. J. Wang, “Action JHEP 1107, 054 (2011) [arXiv:1106.1637 [hep-th]]. growth of charged black holes with a single hori- [27] L. Cheng, X. H. Ge and S. J. Sin, “Anisotropic plasma zon,” Phys. Rev. D 95, no. 12, 124002 (2017) at finite U(1) chemical potential,” JHEP 1407, 083 doi:10.1103/PhysRevD.95.124002 [arXiv:1702.06766 [gr- (2014) doi:10.1007/JHEP07(2014)083 [arXiv:1404.5027 qc]]. [hep-th]]. [43] E. Bakhshaei, A. Mollabashi and A. Shirzad, “Holo- [28] L. Cheng, X. H. Ge and S. J. Sin, “Anisotropic graphic Subregion Complexity for Singular Sur- plasma with a chemical potential and scheme- faces,” Eur. Phys. J. C 77, no. 10, 665 (2017) independent instabilities,” Phys. Lett. B 734, 116 (2014) doi:10.1140/epjc/s10052-017-5247-1 [arXiv:1703.03469 doi:10.1016/j.physletb.2014.05.032 [arXiv:1404.1994 [hep-th]]. [hep-th]]. [44] F. J. G. Abad, M. Kulaxizi and A. Parnachev, “On Com- [29] M. Alishahiha, “Holographic Complexity,” plexity of Holographic Flavors,” arXiv:1705.08424 [hep- Phys. Rev. D 92, no. 12, 126009 (2015) th]. 19

[45] J. Tao, P. Wang and H. Yang, “Testing Holographic Con- with a probe string,” arXiv:1807.01088 [hep-th]. jectures of Complexity with Born-Infeld Black Holes,” [67] J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal arXiv:1703.06297 [hep-th]. and U. A. Wiedemann, “Gauge/String Duality, Hot [46] W. D. Guo, S. W. Wei, Y. Y. Li and Y. X. Liu, “Complex- QCD and Heavy Ion Collisions,” book:Gauge/String ity growth rates for AdS black holes in massive gravity Duality, Hot QCD and Heavy Ion Collisions. Cam- and f(R) gravity,” arXiv:1703.10468 [gr-qc]. bridge, UK: Cambridge University Press, 2014 [47] K. Nagasaki, “Complexity of AdS5 black holes with a doi:10.1017/CBO9781139136747 [arXiv:1101.0618 rotating string,” arXiv:1707.08376 [hep-th]. [hep-th]]. [48] Y. G. Miao and L. Zhao, “Complexity/Action du- [68] V. Jahnke, A. S. Misobuchi and D. Trancanelli, ality of shock wave geometry in a massive gravity “Holographic renormalization and anisotropic black theory,” Phys. Rev. D 97, no. 2, 024035 (2018) branes in higher curvature gravity,” JHEP 1501, 122 doi:10.1103/PhysRevD.97.024035 [arXiv:1708.01779 (2015) doi:10.1007/JHEP01(2015)122 [arXiv:1411.5964 [hep-th]]. [hep-th]]. [49] M. M. Qaemmaqami, “Complexity growth in min- [69] J.W. York, Jr., “Role of conformal three geometry in imal massive 3D gravity,” Phys. Rev. D 97, no. the dynamics of gravitation,” Phys. Rev. Lett. 28 (1972) 2, 026006 (2018) doi:10.1103/PhysRevD.97.026006 1082. [arXiv:1709.05894 [hep-th]]. [70] G.W. Gibbons, S. W. Hawking, “Action Integrals and [50] L. Sebastiani, L. Vanzo and S. Zerbini, “Action growth Partition Functions in Quantum Gravity,” Phys. Rev. D for black holes in modified gravity,” arXiv:1710.05686 15 (1977) 2752. [hep-th]. [71] G. Hayward, “Gravitational action for space-times with [51] W. Cottrell and M. Montero, “Complexity is Simple,” nonsmooth boundaries,” Phys. Rev. D 47 (1993) 3275. arXiv:1710.01175 [hep-th]. [72] D. Brill and G. Hayward, “Is the gravitational ac- [52] M. Moosa, “Evolution of Complexity Following a Global tion additive?,” Phys. Rev. D 50, 4914 (1994) Quench,” arXiv:1711.02668 [hep-th]. doi:10.1103/PhysRevD.50.4914 [gr-qc/9403018]. [53] S. A. Hosseini Mansoori and M. M. Qaemmaqami, “Com- [73] E. D’Hoker and P. Kraus, “Magnetic Brane Solutions plexity Growth, Butterfly Velocity and Black hole Ther- in AdS,” JHEP 0910, 088 (2009) doi:10.1088/1126- modynamics,” arXiv:1711.09749 [hep-th]. 6708/2009/10/088 [arXiv:0908.3875 [hep-th]]. [54] X. H. Ge, S. J. Sin, Y. Tian, S. F. Wu and [74] K. Goto, H. Marrochio, R. C. Myers, L. Queimada S. Y. Wu, “Charged BTZ-like black hole solutions and the and B. Yoshida, “Holographic Complexity Equals diffusivity-butterfly velocity relation,” arXiv:1712.00705 Which Action?,” JHEP 1902, 160 (2019) [hep-th]. doi:10.1007/JHEP02(2019)160 [arXiv:1901.00014 [hep- [55] S. J. Zhang, “Complexity and phase transitions in a holo- th]]. graphic QCD model,” arXiv:1712.07583 [hep-th]. [75] J. Jiang and M. Zhang, “Holographic complexity in [56] M. Moosa, “Divergences in the rate of complexification,” higher curvature gravity,” arXiv:1905.07576 [hep-th]. arXiv:1712.07137 [hep-th]. [76] The quantum computational complexity is a state- [57] A. Ovgun and K. Jusufi, “Complexity growth rates for dependent quantity that measures how difficult is to pre- AdS black holes with dyonic/ nonlinear charge/ stringy pare a given state. More precisely, one starts from some hair/ topological defects,” arXiv:1801.09615 [gr-qc]. reference state and defines the complexity of the target [58] C. A. Agon, M. Headrick and B. Swingle, “Subsystem state as the minimal number of simple unitary operations Complexity and Holography,” arXiv:1804.01561 [hep-th]. required to prepare it. For a more precise definition, see [59] R. Auzzi, S. Baiguera and G. Nardelli, “Volume and com- the review [11]. plexity for warped AdS black holes,” arXiv:1804.07521 [77] Some previous work on holographic complexity include, [hep-th]. for instance, [29–66]. [60] Y. S. An, R. G. Cai and Y. Peng, “Time Dependence [78] To investigate the entire RG flow it is necessary to con- of Holographic Complexity in Gauss-Bonnet Gravity,” sider arbitrary values of a/T . This leads to technical dif- arXiv:1805.07775 [hep-th]. ficulties because the solution needs to be calculated nu- [61] K. Bamba, D. Momeni and M. A. Ajmi, “Holo- merically in these cases. We are currently investigating graphic Entanglement Entropy, Complexity, Fidelity these more general cases. π Susceptibility and Hierarchical UV/IR Mixing Prob- [79] Note that L = 1 implies that GN = 2N2 (see e.g. [67]), lem in AdS2/open strings,” Int. J. Mod. Phys. A where N is the rank of the gauge group SU(N) of the 33, 1850100 (2018) doi:10.1142/S0217751X18501002 dual field theory. [arXiv:1806.02209 [hep-th]]. [80] For generic values of the anisotropy parameter, the met- [62] H. Ghaffarnejad, E. Yaraie and M. Farsam, “Complexity ric functions can be determined numerically. For more growth and shock wave geometry in AdS-Maxwell-power- details, see the appendix A of [26]. Yang-Mills theory,” arXiv:1806.07242 [gr-qc]. [81] For small and intermediate values of B/T 2, the geometry [63] H. Ghaffarnejad, M. Farsam and E. Yaraie, “Effects of can be found numerically. See [73] for more details. quintessence dark energy on the action growth and but- [82] This contribution is known as the Hayward joint term terfly velocity,” arXiv:1806.05735 [hep-th]. [71, 72]. [64] R. Auzzi, S. Baiguera, M. Grassi, G. Nardelli and [83] Here we adopted the conventions found in the appendix N. Zenoni, “Complexity and action for warped AdS black A of [15]. I holes,” arXiv:1806.06216 [hep-th]. [84] Here we√ are using that Ibulk = [65] R. Fareghbal and P. Karimi, “Complexity Growth in 1 R dd+1x −g L(x) = 16πGN I √ ∗ Flatland,” arXiv:1806.07273 [hep-th]. 1 R d−1 R R v1−r (r) d x dr GttGrrG L(r) dt, where [66] K. Nagasaki, “Complexity growth of rotating black holes 16πGN I 0 20

∗ 2 v1 = tR + r∞ defines the future null boundary of [86] In the charged MT model, one has g = 16πGN. the right side of the WDW patch. In region II, for [87] Such tunning of γ was also observed to be necessary instance, the integral over the time coordinate is in [75]. ∗ R v1−r (r) ∗ ∗ ∗ 1 dCA ∗ dt = 2 r∞ − r (r) , where u1 = tR − r∞ [88] We have checked that the curves of M(a) dt versus δt are u1+r (r) defines the past null boundary of the right side of the indistinguishable for different (and small) anisotropies. WDW patch. This confirms that the basic effect of the anisotropy is a dCA [85] Differently from the MT model, the temperature of the upward shift in the curves of dt versus δt. We thank DK solution does not depend on the intensity of the mag- Alberto Güijosafor suggesting this comparison.

netic ﬁeld (see (20)), so we don’t need to change rH to [89] In the MT model the conformal anomaly appears at order 4 keep the temperature ﬁxed while we vary B. O(a ).