Holographic complexity for nonlinearly charged Lifshitz black holes
HHolographic complexity for nonlinearly charged Lifshitz black holes
Kai-Xin Zhu , , ∗ Fu-Wen Shu , , , † and Dong-Hui Du , ‡ Department of Physics, Nanchang University, Nanchang, 330031, China Center for Relativistic Astrophysics and High Energy Physics,Nanchang University, Nanchang 330031, China Center for Gravitation and Cosmology,Yangzhou University, Yangzhou, China
Abstract
Using “complexity=action” proposal we study the late time growth rate of holographic complexity fornonlinear charged Lifshitz black hole with a single horizon or two horizons. As a toy model, we consider twokinds of such black holes: nonlinear charged Lifshitz black hole and nonlinear logarithmic charged Lifshitzblack hole. We find that for the black hole with two horizons, the action growth bound is satisfied. But forthe black hole with a single horizon, whether the Lloyd bound is violated depends on the specific value ofdimensionless coupling constants β , β , spacetime dimension D and dynamical exponent z . ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] a r X i v : . [ h e p - t h ] J u l . INTRODUCTION Many significant concepts emerged in the exploration of the unification of general relativity andquantum mechanics. One of them is the work of holographic principle [1, 2], which states that degreesof freedom of a higher dimensional gravitational system can be characterized by those of a lower di-mensional quantum system. This principle is widely regarded as a fundamental principle of quantumgravity with the discovery of AdS (Anti-de Sitter)/CFT (Conformal field theory) correspondence [3].Based on this duality, the so-called holographic dictionary is proposed [4, 5] and it offers us a newway to calculate the physical quantities on the field theory side. In addition, increasing evidencesshow that it may play a crucial role in the understanding the nature of space-time [6, 7]. In partic-ular, as implied by the holographic entanglement entropy [8, 9], there is a fundamental connectionbetween quantum information theory and gravitational physics [10–18]. The quantum complexity isanother significant concept of the connection and thus has attracted a lot of attention in the pastfew years. It is defined by the minimal number of quantum gates needed to build a target statefrom a reference state within small tolerance (cid:15) . Two proposals on holographic duality of quantumcomputational complexity have emerged. First, the complexity = volume (CV) conjecture [19, 20],which states that complexity of the boundary state was proportional to the maximum volume ofcodimension one hypersurface bounded by the CFT slices, C V ∼ V ( B ) G N (cid:96) , (1)where G N is the Newton’s constant and (cid:96) is a arbitrary length scale. In order to avoid this unclearlength scale appearing in the CV conjecture, the conjecture of complexity = action (CA) was pro-posed [21, 22], which states that complexity on the boundary CFT is related to the on-shell actionof the Wheeler-DeWitt(WDW) patch in the bulk, which is defined as the domain of dependence ofCauchy surface that is anchored on the boundary state C A = I W DW π (cid:126) . (2)Both these conjectures favor a statement that there exists a bound on the growth rate of complex-ity for neutral black holes which is known as the Lloyd bound [23]. It suggests that the rate ofcomputation is bounded by the total energy of the system d C dt ≤ Mπ (cid:126) , (3)where M is the mass of the black hole. For charged or rotated AdS black hole the bound is modified[24] as, d C dt = 1 π (cid:126) (cid:20) ( M − Ω J − µQ ) + − ( M − Ω J − µQ ) − (cid:21) (4)2here µ and Ω are black hole chemical potential and angular velocity, J and Q are black hole angularmomentum and electric charge, “+” and “-” stand for outer horizon and inner horizon of the blackhole, respectively. The validity of this upper bound in different types of black hole have been checkedin many literatures[25–31]. However, there also have many cases that violate the Lloyd bound [32–34]. The violation of the Lloyd bound in some systems implies that the CV or CA conjecture shouldbe modified in these systems. Hence, to test how common the CV or CA conjecture is deservesfurther study.For this sake, in this paper we focus on the holographic complexity in the non-realistic system,as all the aforementioned cases are restricted to the relativistic systems. Specifically, we discusswhether spacetime anisotropy will have an influence on the validity of the action growth bound ofholographic complexity, under the framework of Lifshitz holography, which was first addressed in[35, 36]. In the Lifshitz system, the time and space coordinates rescale with different weight t (cid:55)→ λ z t, (cid:126)x (cid:55)→ λ(cid:126)x (5)where the constant z is called the dynamical critical exponent. In the anisotropic systems, theLloyd bound was shown to be violated at late times for some cases, such as in the non-commutativegeometry [37], and the Einstein-Maxwell-Dilaton theory defined in the Lifshitz and hyperscalingviolating geometries [38–41]. However, all of these cases are discussing about the neutral black hole.The holographic complexity in the charged Lifshitz black hole, which has much richer structure, isstill lack.In this paper we consider a toy model. We use the CA conjecture to study the holographiccomplexity of the nonlinearly charged Lifshitz black hole in the Einstein-Proca-Maxwell model.Such non-trivial black hole solution in the Einstein-Proca-Maxwell gravity was obtained in [42], andlater was developed in [43, 44]. Related thermodynamics and CV duality have been investigated in[45–47]. More specifically, we consider the Einstein-Proca-Maxwell model in Ref [43]. One featureof this Lifshitz system is that the dynamical critical exponent z could be choose arbitrary for z > z ≤ z > β , β , we could get the charged black hole with a singlehorizon or two horizons. In this way, we could check the validity of the general charged black holewith two horizons and the more specific case where the black hole only has a single horizon, aswhat did in the relativistic system in [28, 48]. Interestingly, if we choose the dynamical exponent z = D −
2, charged black hole will become logarithmic decay supported by a specific logarithmicelectrodynamics. So we can check whether the holographic complexity is continuous for z , in otherwords, whether the logarithmic behavior will influence the holographic complexity.Our results show that the nonlinear charged Lifshitz black hole with two horizons the upperbound is always satisfied regardless of the value of parameters of the model, while for those with asingle horizon, the Lloyd bound could be violated depending on different value of the model. Wealso find in the side of gravity when z = D − z (cid:54) = D − z = D −
2. Thermodynamicsof these two black holes is investigated respectively. We then calculate action growth of nonlinearlycharged Lifshitz black holes for z (cid:54) = D − z = D −
2. The conclusions anddiscussions will be presented in section V.
II. NONLINEARLY CHARGED LIFSHITZ BLACK HOLES
In this section we consider the charged Lifshitz black holes for any exponent z > I = (cid:90) d D x √− g (cid:20) κ ( R − λ ) − H µν H µν − m B µ B µ − P µν F µν + H ( P ) (cid:21) , (6)where B µ is a massive vector field (Proca field) with strength H µν = ∂ u B ν − ∂ ν B µ , m is its mass.Usually the last two terms in the action are the functions of F = F µν F µν and describe the nonlinearbehavior of the electromagnetic field A µ with field strength F µν = ∂ u A ν − ∂ ν A µ . Using Legendretransformation H ( P ) this Lagrangian could be rewriten as a function of conjugate antisymmetrictensor P µν and electromagnetic field A ν [49], where H ( P ) is the so-called structural function depend-ing on the invariant P = P µν P µν . After variation of the action with respect to P µν , the strengthfield of the original nonlinear electromagnetic field can be express as F µν = H P P µν , (7)where H P = ∂ H /∂P . When H ( P ) = P it reduces to standard linear Maxwell theory. In general, thewell-behaved nonlinear electrodynamics need express as a polynomial of the invariants F µν [50, 51].The Lifshitz black holes with generic dynamical exponent z can be obtained by choosing the followingstructural function [43]: H ( P ) = − [2 z − Dz + 2( D − κl β (cid:112) − l P − ( z − z − D + 2) κz β P + ( D − κl β (cid:0) − l P (cid:1) z D − , (8)where β and β are dimensionless coupling constants. Varying the action (6) with respect to B µ , A µ and g µν , one gets the equations of motion, respectively, for Proca field, electromagnetic field andmetric filed ∇ µ H µν = m B ν ; (9) ∇ µ P µν = 0; (10) G µν + λg µν = κ (cid:20) H µα H αν − g µν H αβ H αβ + m (cid:18) B µ B ν − g µν B α B α (cid:19) + H P P µα P αν − g µν (2 P H P − H ) (cid:21) . (11)4hese equations admit the following charged Lifshitz black hole solution [43] ds = − r z l z f ( r ) dt + l r dr f ( r ) + r d(cid:126)x , (12) P µν = 2 δ t [ µ δ rν ] Ql D − (cid:18) lr (cid:19) D − z − , (13) B t ( r ) = (cid:114) z − zκ (cid:16) rl (cid:17) z f ( r ) , (14)The expression for f ( r ) depends on the parameters z and D and its explicit form will be given later. A. Nonlinearly charged Lifshitz black holes with exponent z > and z (cid:54) = D − For nonlinearly charged Lifshitz black hole with exponent z > z (cid:54) = D −
2, we have[43] f ( r ) = 1 − M (cid:18) lr (cid:19) D − + M (cid:18) lr (cid:19) z , (15) λ = − z + ( D − z + ( D − l , (16) M = β | Q | l D − , M = β (cid:18) | Q | l D − (cid:19) zD − , (17)where the integration constants M and M are related to the mass of the black hole, and Q isrelated to the electric charge.Before calculating the holographic complexity, we investigate the thermodynamics of the nonlin-early charged Lifshitz black holes with exponent z > z (cid:54) = D −
2. The entropy per unit volumeof the horizon is given, as usual, by the Bekenstein-Hawking formula S = 2 πr D − κ . (18)The temperature of the event horizon can be obtained by using the standard Wick-rotation method,yielding the result T = r z +1+ πl z +1 f (cid:48) r = r + = 14 πl (cid:16) r + l (cid:17) z (cid:20) ( D − M (cid:18) lr + (cid:19) D − − zM (cid:18) lr + (cid:19) z (cid:21) . (19)The electric charge density reads Q e = 1Ω k (cid:90) d Ω k r D − n µ P µν u ν , (20)where n µ and u ν are the unit spacelike and timelike normals to a sphere of radius ru ν = 1 √− g tt dt = l z r z √ f dt, n µ = 1 √ g rr dr = r √ fl dr. (21)Substituting the above formulae, one obtains Q e = Q. (22)5s a consequence, similar as [46], the electric potential reads A t ( r ) = 2 z − Dz + 2( D − zκ M l D − Q ( rl ) z − ( z − z − D + 2) zκ M l D − Q ( rl ) z − D +2 − z κ M l D − Q ( rl ) D − , (23)Φ e = − A ( r + ) = l D − κQ (cid:20) − ( D − M (cid:16) r + l (cid:17) z + zM (cid:16) r + l (cid:17) D − + 2( z − z − D + 2) z M M (cid:21) . (24)On the other hand, the mass of the black hole can be computed through the quasilocal method asdescribed in Refs[45, 52] with the following relation˜ Q ( ξ ) = (cid:90) B d D − x µν (cid:18) ∆ K µν ( ξ ) − ξ [ µ (cid:90) ds Θ ν ] (cid:19) , (25)where Θ ν is the surface term, ∆ K µν ( ξ ) ≡ K µνs =1 ( ξ ) − K µνs =0 ( ξ ) denotes the variation of the Noetherpotential from the vacuum solution to the black hole, and dx µν represents the integration over thecodimension-two boundary B . As shown in (15), the solution depends continuously on the integrationconstants M and M , and the Lifshitz black hole has the mass as the only charge. Hence we canintroduce the parameter sM with s ∈ [0 ,
1] in the space of solutions to define the conserved chargein the interior region (not in the asymptotic region) [53–56]. In our case (6), the involved quantitiesare [57] Θ µ = 2 √− g (cid:20) P µ ( αβ ) γ ∇ γ δg αβ − δg αβ ∇ γ P µ ( αβ ) γ + 12 ∂ L ∂ ( ∂ µ A ν ) δA ν + 12 ∂ L ∂ ( ∂ µ B ν ) δB ν (cid:21) , (26) K µν = √− g (cid:20) P µνρσ ∇ σ ξ σ − ξ σ ∇ ρ P µνρσ − ∂ L ∂ ( ∂ µ A ν ) ξ σ A σ − ∂ L ∂ ( ∂ µ B ν ) ξ σ B σ (cid:21) , (27)where P µνρσ = ∂ L ∂R µνρσ . In the present case, the last expression becomes (cid:90) ds Θ r = l D − κ (cid:20) (cid:16) rl (cid:17) z M − (cid:16) rl (cid:17) D − M − z − z − D + 2) z + D − M M (cid:21) , (28)∆ K rt = l D − κ (cid:20) − (cid:16) rl (cid:17) z M + 2 (cid:16) rl (cid:17) D − M + 2( z − z − D + 2) z M M (cid:21) . (29)From these expressions, we obtain the mass of the nonlinearly charged Lifshitz black holes withexponent z > z (cid:54) = D − M = l D − κ Ω D − ( z − z − D + 2) z D − z + D − M M , (30)and it is easy to check that the first law holds dM = T dS + Φ e dQ e . (31)The Smarr formula turns out to be M = D − z + D − T S + Φ e Q e ) . (32)6t is worth mentioning when the structural coupling constant β vanishes and β is fixed, the lasttwo terms in the action (6) becomes power law. This kind of power-law Lagrangian has been studiedin detail in Ref [45] where they get the same Smarr formula. On the other hand, when z = 2( D − β vanishes, the black hole mass in both cases turnsto zero.It is important to note that the values and signs of the structural coupling constants β , β are notlimited. This makes it possible to encode different kinds of black holes, each black hole is associatedwith a specific electrodynamics behavior as in [43]. We will divide it into single horizon and twohorizons and calculate their holographic complexity, respectively, in section III. B. Nonlinearly charged logarithmic Lifshitz black hole for exponent z = D − When z = D − D −
2, and the above solution can be obtained by redefining the integrationconstants ( M , M ) (cid:55)→ (( z − D + 2) M , M − M ), then the gravitational potential becomes [43] f ( r ) = 1 − (cid:18) lr (cid:19) D − (cid:104) M ln (cid:16) rl (cid:17) − M (cid:105) . (33)Meanwile the structural constants ( β , β ) should change to (( z − D + 2) β , β − β ), then thestructural function (8) is rewriten as H ( P ) = 12 κl (cid:110)(cid:104) ( D −
2) ln (cid:112) − l P − D + 8 (cid:105) β + ( D − β (cid:111) (cid:112) − l P − ( D − D − κ β P. (34)The integration constants can be expressed as M = β | Q | l D − , M = (cid:18) β + β D − | Q | l D − (cid:19) | Q | l D − . (35)Note that this logarithmic behavior is the limit of z → D − x ) ≡ lim α → x α − α to(15), other example for logarithmic black hole is produced by using higher curvature gravity [58–60].Now let us turn to the thermodynamics of the nonlinearly charged Lifshitz black hole withexponent z = D −
2. From (19) we obtain the temperature T = r D − πl D − f (cid:48) r = r + = 14 πl (cid:26) ( D − (cid:20) M ln (cid:16) r + l (cid:17) − M (cid:21) − M (cid:27) . (36)The electric potential is A t ( r ) = 12 κ l D − Q (cid:26) ( D − (cid:20) M ln( rl ) − M (cid:21) ( rl ) D − + 2( D − D − M ( rl ) D − − D − D − M ln( rl ) − M rl D − (cid:27) , (37)7 e = − A t ( r + ) = l D − κQ (cid:26) − ( D − (cid:20) M ln (cid:16) r + l (cid:17) − M (cid:21) (cid:16) r + l (cid:17) D − + M (cid:16) r + l (cid:17) D − + 2( D − D − M M (cid:27) . (38)The surface term and the Noether potential could be calculated by using (26) and (27) (cid:90) ds Θ r = l D − κ (cid:26) (cid:20) M ln (cid:16) rl (cid:17) − M (cid:21) (cid:16) rl (cid:17) D − − D − D − M M (cid:27) , (39)∆ K rt = l D − κ (cid:26) − (cid:20) M ln (cid:16) rl (cid:17) − M (cid:21) (cid:16) rl (cid:17) D − + 2( D − D − M M (cid:27) . (40)Now using (25), one finally obtains M = l D − κ Ω D − D − D − M M . (41)Again we can easily check that the first law holds dM = T dS + Φ e dQ e , (42)and the Smarr formula is given by M = 12 ( T S + Φ e Q e ) . (43)Actually, substituting z = D − III. ACTION GROWTH IN NONLINEARLY CHARGED LIFSHITZ BLACK HOLE WITHEXPONENT z > AND z (cid:54) = D − In this section we would like to calculate the action growth rate in the nonlinearly charged Lifshitzblack hole with exponent z > z (cid:54) = D − A. The case with a single horizon
According to the CA conjecture, the complexity is proportional to the on-shell action in theWDW patch of some time slice. The total action in presence of unsmooth boundaries is given by I tot = (cid:90) M d D x √− g ( L bulk )+ 1 κ sign (Σ) (cid:90) Σ d D − x (cid:112) | h | K + 1 κ sign ( N ) (cid:90) N d D − x √ ση + 1 κ sign (Σ (cid:48) ) (cid:90) Σ (cid:48) dλd D − θ √ γχ + 1 κ sign ( B ) (cid:90) B d D − x √ σa. (44)Here Σ is spacelike or timelike boundary, Σ (cid:48) is null boundary and K is the Gibbons-Hawking term. η is the Hayward joint term [62], while a is the null joint term. χ measures the failure of λ to be8n affine parameter on the null generators. Here we choose affine parametrization so that the nullboundary has no contribution. The signatures sign (Σ) , sign ( N ) , sign (Σ (cid:48) ) , sign ( B ) are determinedby the additivity rules. δt r = 0 r = 0 r = ∞ r = ∞ δtt L t R r = ǫr = ǫ u = − ∞ u = ∞ u = u u = u u = u + δ t b b v = v v = v + δ t b V V B B ′ C B ′ B b b bb CC ′ u = u + δ t u = u v = v v = v + δ t u = u v = v V V r = ∞ r = ∞ r = r = r = r + r = r − r = r − r = r + t L t R δtNN ′ N ′ N FIG. 1: A WDW patch and its change due to an infinitesimal time shift δt at the left boundary, for a chargedLifshitz black hole with a single horizon (left panel), and for a charged Lifshitz black hole with two horizons(right panel). Following [61], we introduce the null coordinates u and v by u := t + r ∗ , v := t − r ∗ , (45)where r ∗ = (cid:82) ( lr ) z +1 1 f dr . Under the null coordinates the metric can be written as ds = − (cid:16) rl (cid:17) z f du + 2 (cid:16) rl (cid:17) z − dudr + r d (cid:88) i =1 dx i , (46)or ds = − (cid:16) rl (cid:17) z f dv + 2 (cid:16) rl (cid:17) z − dvdr + r d (cid:88) i =1 dx i . (47)For the choices of ( t, r ),( u, r ) and ( v, r ), we have (cid:90) √− gd D x = Ω D − (cid:90) r D + z − l z − drdw. (48)9here w = { t, µ, υ } . In this case, the WDW patch is shown in the left panel of Fig. 1. As analyzedin the [43], when β ≥ , β ≤ β > , β > , < z < D − β < , β < , z > D − z > z (cid:54) = D − t to t + δt on the left boundary, and as shown in Ref [63] that the joints at the UV cutoffsurface and future singularity are unchanged under the time translation. So we get δI = I V − I V − κ (cid:90) S Kd Σ + 1 κ (cid:90) B (cid:48) adS − κ (cid:90) B adS. (49)With the equations of motion (11) the Lagrangian in the bulk has the form L bulk = − ( z + D −
2) + zM (cid:0) lr (cid:1) D − − ( D − M (cid:0) lr (cid:1) z κl . (50)Thus the action in region V is I V = 12 κ l D − Ω D − (cid:90) u + δtu du (cid:90) ρ ( u ) (cid:15) (cid:16) rl (cid:17) z + D − − z + D −
2) + 2 zM (cid:0) lr (cid:1) D − − D − M (cid:0) lr (cid:1) z l dr = 12 κ l D − Ω D − (cid:90) u + δtu du (cid:20) − (cid:16) rl (cid:17) z + D − + 2 M (cid:16) rl (cid:17) D − − M (cid:16) rl (cid:17) z (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ρ ( u ) (cid:15) . (51)Similarly, the contribution of V to the action is I V = 12 κ l D − Ω D − (cid:90) v + δtv dv (cid:90) ρ ( v ) ρ ( v ) (cid:16) rl (cid:17) z + D − − z + D −
2) + 2 zM (cid:0) lr (cid:1) D − − D − M (cid:0) lr (cid:1) z l dr = 12 κ l D − Ω D − (cid:90) v + δtv dv (cid:20) − (cid:16) rl (cid:17) z + D − + 2 M (cid:16) rl (cid:17) D − − M (cid:16) rl (cid:17) z (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ρ ( v ) ρ ( v ) . (52)Now we take the limit (cid:15) →
0, using a variables change: u = u + v + δt − v , then terms involving ρ ( u ) and ρ ( v ) cancel out. So we get I V − I V = 12 κ l D − Ω D − (cid:90) v + δtv dv (cid:20) − (cid:16) ρ l (cid:17) z + D − + 2 M (cid:16) ρ l (cid:17) D − − M (cid:16) ρ l (cid:17) z (cid:21) . (53)The variable ρ ( v ) varies from r B to r B (cid:48) as v increases from v to v + δt . But the variation is small,and one has r B (cid:48) = r B + O ( δt ) so that Eq. (53) reduces to I V − I V = 12 κ l D − Ω D − (cid:20) − (cid:16) r B l (cid:17) z + D − + 2 M (cid:16) r B l (cid:17) D − − M (cid:16) r B l (cid:17) z (cid:21) δt. (54)Next we calculate the contribution from the spacelike surface S at r = (cid:15) . The future-directedunit normal is given by n α = lr | f | − ∂ α r , then the extrinsic curvature is K = ∇ α n α = − l z − r z + D − ddr (cid:18) r z + D − l z | f | (cid:19) . (55)Meanwhile the volume element is given by d Σ = Ω D − r z + D − l z | f | dt (56)10ecause r = (cid:15) (cid:28) r + , approximatively f (cid:39) − M (cid:0) lr (cid:1) D − + M (cid:0) lr (cid:1) z , one obtains I S = − κ (cid:90) S Kd Σ= 12 κ l D − Ω D − δt (cid:20) ( − z − D + 2) M (cid:16) rl (cid:17) z + ( z + 2 D − M (cid:16) rl (cid:17) d − (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r = (cid:15) = 0 . (57)Finally we calculate the contribution of the joint terms at B, B (cid:48) . Following [61], the null jointrule states that a = ln (cid:18) − k · k (cid:19) . (58)Under affine parametrization the vectors k α and k α can be expressed as k α = − c∂ α v = − c∂ α ( t − r ∗ ) , k α = c∂ α u = c∂ α ( t + r ∗ ) , (59)where c and c are arbitrary positive constants. With these choices, we have k · k = 2 cc l z r z f , then weget a = − ln (cid:18) − r z fl z cc (cid:19) . (60)In the end we have I B (cid:48) B = 12 κ (cid:18) (cid:73) B (cid:48) adS − (cid:73) B adS (cid:19) = 1 κ Ω D − [ h ( r B (cid:48) ) − h ( r B )] , (61)where h ( r ) := − r D − ln (cid:16) − r z fl z cc (cid:17) . By making a Taylor expansion of h ( r ) around r = r B and using dr = − f (cid:0) rl (cid:1) z +1 δt , one obtains I B (cid:48) B = − κ Ω D − f (cid:16) rl (cid:17) z +1 dhdr (cid:12)(cid:12)(cid:12)(cid:12) r = r B δt = Ω D − κ r z + D − l z +1 (cid:20) zf + r dfdr + ( D − f ln (cid:18) − r z l z fcc (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r = r B δt. (62)Collecting the formulae (54),(57) and (62), and taking the late time limit r B → r + we have dI on − shell dt = 12 κ l D − Ω D − (cid:26)(cid:20) − (cid:16) rl (cid:17) z + D − + 2 M (cid:16) rl (cid:17) D − − M (cid:16) rl (cid:17) z (cid:21) + r z + D − l z + D − (cid:20) zf + r dfdr + ( D − f ln (cid:18) − r z l z fcc (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r = r + (cid:27) = 12 κ l D − Ω D − (cid:20) ( D − (cid:16) r + l (cid:17) z + D − + ( D − − z ) M (cid:16) r + l (cid:17) D − (cid:21) . (63)Combining (24) and (30) we finally get dI on − shell dt = z + D − D − M − Q e Φ e . (64)11ote that the WDW patch of charged Lifshitz black hole with a single horizon is similar to theone of neutral black hole as shown in Fig. 1. Therefore the action growth rate of charged Lifshitzblack hole with a single horizon (64) is also similar to the one of neutral black hole. From (64) wesee that, under the CA proposal in Lifshitz system, the late time action growth rate relates to thedynamics critical exponent z and spacetime dimension D . In [41] the action growth rate also hasan explicit dependence on z and d , that is dI on − shell /dt = 2( z + d − M/d . In their case, there isa continuous limit as z → M as z = 1. In our case, however, we don’t have a continuous limit in the sense that we cannotchoose the dynamics exponent z = 1 because Proca field disappears when z = 1. As a consequencein the limit z →
1, the usual Lloyd bound 2 M cannot be recovered from (64). This implies morenontrivial features of our model and may provide more nontrivial informations about the validity ofthe CA duality as one applies it to test the Lloyd bound. In what follows we turn to this issue byexamining whether the Lloyd bound is violated or not in our case.To proceed, let us first consider a special case where β = 0. In this case the action reduces to thepower-law electrodynamic and the mass of black hole vanishes. Obviously, in order to have a well-defined horizon, β must be less than zero and the action growth rate is dI on − shell dt = − zM ( r + l ) D − ,which indicates that the Lloyd bound is violated identically. In the same way, if we let β = 0so β >
0, we find that the Lloyd bound is also violated. Thermodynamics of these cases werestudied in Refs [45, 64]. For the more general situation, when β > , β < , z > D − β > , β > , < z < D − β < , β < , z > D −
2, the Lloyd bound is violated forpart of the parameter space as shown in Fig. 2, and the range where the bound is violated increaseas z increases and decreases as D increases. For the case of β > , β < , < z < D − β , β and it has no influence on the range where the bound isviolated when we change the value of D and z as shown in Fig. 3 . B. The case with two horizon
As analyzed in the [43], when β > , β > , z > D − β < , β < , < z < D − z > z (cid:54) = D − I V − I V becomes I V − I V = 12 κ l D − Ω D − (cid:20) − (cid:16) rl (cid:17) z + D − + 2 M (cid:16) rl (cid:17) D − − M (cid:16) rl (cid:17) z (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r + r − δt. (65)12 - - - - - - - - - - - - β β z =
3, D = G - - - - - - - - - - - - β β z =
3, D = G - - - - - - - - - - - - β β z =
3, D = G - - - - - - - - - - - - β β z =
3, D = FIG. 2: The case of β < , β < , z > D −
2. The blue part represent the regions where the the Lloydbound is violated for any value of β , β . The yellow part represent the limitations of 1 > r + /l >
0, imposingconstraints on the allowed values of β , β through the formula (15) for different z and D . So, the overlappingpart(grey part) in the parameter space is the regions where the Lloyd bound is violated. The point G isa critical point. It denotes the maximum that | β β | can take within the region where the Lloyd bound isviolated. There are four joints contributions with joints B (cid:48) , B inside the past horizon and joints C, C (cid:48) insidethe future horizon. Then we have I B (cid:48) B + I C (cid:48) C = − κ Ω D − f (cid:16) rl (cid:17) z +1 dhdr (cid:12)(cid:12)(cid:12)(cid:12) r + r − δt = Ω D − κ r z + D − l z +1 (cid:20) zf + r dfdr + ( D − f ln (cid:18) − r z l z fcc (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r + r − δt. (66)13 - - - - β β D = = G - - - - β β D = = G - - - - β β D = = G - - - - β β D = = FIG. 3: The case of β > , β < , < z < D −
2. The blue part represent the regions where the the Lloydbound is violated for any value of β , β . The yellow part represent the limitations of 1 > r + /l >
0, imposingconstraints on the allowed values of β , β through the formula (15) for different z and D . So, the overlappingpart(grey part) in the parameter space is the regions where the Lloyd bound is violated. The point G isa critical point. It denotes the maximum that | β β | can take within the region where the Lloyd bound isviolated. dI on − shell dt = 12 κ l D − Ω D − (cid:26)(cid:20) − (cid:16) rl (cid:17) z + D − + 2 M (cid:16) rl (cid:17) D − − M (cid:16) rl (cid:17) z (cid:21) + r z + D − l z + D − (cid:20) zf + r dfdr + ( D − f ln (cid:18) − r z l z fcc (cid:19)(cid:21)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) r + r − = 12 κ l D − Ω D − (cid:20) ( D − M (cid:16) rl (cid:17) z − zM (cid:16) rl (cid:17) D − (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r + r − = T + S + − T − S − . (67)Where T − ( T + ) and S − ( S + ) are the temperature and entropy of the inner(outer) horizon. We find thelate time action growth rate for the charged Lifshitz black hole with exponent z > z (cid:54) = D − dI on − shell dt = ( M − Q e Φ e ) + − ( M − Q e Φ e ) − . (68)This shows that the growth rate of the action for nonlinear charged Lifshitz black hole satisfiesthe universal expression in [24]. In other words, the statement that the action growth bound isindependent of the charged black hole size[24] is also valid for some cases of the Lifshitz system. IV. ACTION GROWTH IN NONLINEARLY CHARGED LIFSHITZ BLACK HOLE WITHEXPONENT z = D − In this section we would like to investigate the action growth rate of nonlinearly charged loga-rithmic Lifshitz black hole with exponent z = D − A. The case with a single horizon
The logarithmic Lifshitz black hole has a single horizon when β <
0. The Lagrangian nowbecomes L bulk = 1 κl (cid:20) − D −
2) + M (cid:18) lr (cid:19) D − − ( D − M (cid:18) lr (cid:19) D − + ( D −
2) ln (cid:16) rl (cid:17) M (cid:18) lr (cid:19) D − (cid:21) . (69)The on-shell action in region V is I V = 12 κ l D − Ω D − (cid:90) u + δtu du (cid:90) ρ ( u ) (cid:15) (cid:16) rl (cid:17) D − (cid:20) − D −
2) + 2 M (cid:18) lr (cid:19) D − − D − M (cid:18) lr (cid:19) D − + 2( D −
2) ln (cid:16) rl (cid:17) M (cid:18) lr (cid:19) D − (cid:21) dr = 12 κ l D − Ω D − (cid:90) u + δtu du (cid:20) − (cid:16) rl (cid:17) D − − M (cid:16) rl (cid:17) D − + 2 ln (cid:16) rl (cid:17) M (cid:16) rl (cid:17) D − (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ρ ( u ) (cid:15) . (70)15imilarly, the contribution of V to the action is I V = 12 κ l D − Ω D − (cid:90) u + δtu du (cid:20) − (cid:16) rl (cid:17) D − − M (cid:16) rl (cid:17) D − + 2 ln (cid:16) rl (cid:17) M (cid:16) rl (cid:17) D − (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ρ ( v ) ρ ( v ) . (71)In the late time, (70) together with (71) lead to I V − I V = 12 κ l D − Ω D − δt (cid:20) − (cid:16) rl (cid:17) D − − M (cid:16) rl (cid:17) D − + 2 ln (cid:16) rl (cid:17) M (cid:16) rl (cid:17) D − (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r + (cid:15) = 0 . (72)The contribution of the spacelike surface S is I S = l D − κ Ω D − δt (cid:20) (2 D − (cid:16) rl (cid:17) D − f + 12 r D − l D − f (cid:48) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r = (cid:15) = 0 . (73)The joint term is I B (cid:48) B = 12 κ Ω D − r D − l D − rf (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) r = r + = l D − κ Ω D − (cid:26) ( D − (cid:16) r + l (cid:17) D − (cid:20) M ln (cid:16) r + l (cid:17) − M (cid:21) − (cid:16) r + l (cid:17) D − M (cid:27) (74)Putting the term (72),(73) and (74) together, we have dI on − shell dt = l D − κ Ω D − (cid:26) ( D − (cid:16) r + l (cid:17) D − − (cid:16) r + l (cid:17) D − M (cid:27) = 2 M − Q e Φ e . (75)It has the same form as the late time action growth rate (64) if we let z = D −
2. This means thatthe logarithmic behavior doesn’t influence the late time action growth rate for charged nonlinearLifshitz black hole with a single horizon. On the other hand, although the requirement being a singlehorizon imposes a constraint on β by β <
0, there is no limitation on the value of β . For the caseof β ≥
0, the Lloyd bound is violated and the black hole mass is less than or equal to zero. For thecase of β < D increases. B. The case with two horizon As β > I V − I V = 12 κ l D − Ω D − δt (cid:20) − (cid:16) rl (cid:17) D − − M (cid:16) rl (cid:17) D − + 2 ln (cid:16) rl (cid:17) M (cid:16) rl (cid:17) D − (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r + r − . (76)As shown in Fig. 1, four joints B (cid:48) , B and C, C (cid:48) also have contributions to the action, which totallyis given by I B (cid:48) B + I C (cid:48) C = 12 κ Ω D − r D − l D − rf (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) r + r − = l D − κ Ω D − (cid:26) ( D − (cid:16) rl (cid:17) D − (cid:20) M ln (cid:16) rl (cid:17) − M (cid:21) − (cid:16) rl (cid:17) D − M (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) r + r − . (77)16 - - - - - - - - β β D = - - - - - - - - - β β D = FIG. 4: The case of β < , β <
0. The blue part represent the regions where the the Lloyd bound is violatedfor any value of β , β . The yellow part represent the limitations of 1 > r + /l >
0, imposing constraints on theallowed values of β , β through the formula (15) for different z and D . So, the overlapping part(grey part)in the parameter space is the regions where the Lloyd bound is violated. Combining (76) and (77), we have dI on − shell dt = l D − κ Ω D − (cid:26) ( D − (cid:16) r + l (cid:17) D − (cid:20) M ln (cid:16) r + l (cid:17) − M (cid:21) − (cid:16) r + l (cid:17) D − M (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) r + r − = T + S + − T − S − . (78)Using the Smarr formula (32), we can obtain the growth rate of action similar as the one for theAdS-RN black hole [22, 24] dI on − shell dt = ( M − Q e Φ e ) + − ( M − Q e Φ e ) − . (79)This means that the logarithmic behavior has no effect on the late time action growth rate of thenonlinear charged Lifshitz black holes with two horizons. V. CONCLUSIONS AND DISCUSSIONS
In this paper, we use the CA conjecture to investigate the late time complexity growth rate ofnonlinearly charged Lifshitz black holes and nonlinearly charged logarithmic Lifshitz black holeswith a single horizon and two horizons. In these cases, Proca field is the essential structure ofthe anisotropic Lifshitz vacuum. Since the mass of the Proca field is determined by the dynamicalexponent z and spacetime dimension D through (14), z and D could precisely reflect the effect of theProca field to the holographic complexity. Our results show that the value of dynamics exponent z ,17 ABLE I: The effect of different parameter choices on the Lloyd boundDynamic exponent Dimensionless constants Black hole mass Lloyd bound z > z (cid:54) = D − β = 0 , β < z > z (cid:54) = D − β > , β = 0 zero violated z > D − β > , β < < z < D − β > , β > z > D − β < , β < < z < D − β > , β < z = D − β < , β = 0 zero violated z = D − β < , β > z = D − β < , β < spacetime dimension D and dimensionless coupling constants β , β will influence the late time actiongrowth rate (63). Comparing the nonlinearly charged Lifshitz black holes and those logarithmic ones,we find although on the side of gravity when z = D − z > z (cid:54) = D −
2, however, we find that black holes with smaller mass are more likely to violate the Lloydbound. Actually, from Figs. 2 and 3 we see that there is a critical point (the point G). It denotesthe maximal value that | β β | can take to let the Lloyd bound be violated. It relates to the blackhole mass through definition (30) and (17). In this way we can define a critical mass (denoted by M c ) of black hole, above which there is no violation of the Lloyd bound. We discretely compute thecritical mass M c for different z when z > D −
2. We plot them as a function of z as shown in the leftpanel of Fig. 5. For the case of 1 < z < D −
2, the critical point G are same for different z and D .The relationship between the critical mass M c and z now can be ploted as a continuous function asshown in the right panel of Fig. 5. For the positive mass black hole with z = D −
2, however, thereis no obvious relationship between the violation of the Lloyd bound and the black hole mass (41).Although | β | can only choose small values ( | β | < | β | .Therefore we could get black hole with arbitrary mass such that the Lloyd bound is still violated asshown in the Fig. 4. In other words, there is no critical mass for this case.It would be very interesting to check whether the Lloyd bound is still violated for those values ofparameters when we use the CV conjecture to our model. Especially, there is an improved version18 = = z M c z > D - D = = z M c < z < D - FIG. 5: The critical mass M c as a function of z for D = 4 and D = 5. Left panel is the case of z > D − < z < D − of the CV conjecture—the“complexity=volume 2.0” (CV 2.0) — which was proposed in [65]. It wasfound that the CV 2.0 would not violate the Lloyd bound for many cases where the Lloyd boundis violated in the framework of the CA conjecture. The same thing deserves to do for the so-calledCA2.0, which is a modified version of CA conjecture proposed in[66]. Very more recently, there aretwo more versions of CV conjecture [67, 68] which show that the Lloyd bound holds under theirframework even for the cases where the bound is violated in the original CA or CV conjecture. Weexpect the system considered in the present paper, due to its non-relativistic and nonlinear nature,provides a very good probe to test their validity or rationality of these improved conjectures. Weleave these investigations to the future study. Acknowledgements
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