Complexity of holographic superconductors

We study the complexity of holographic superconductors (Einstein-Maxwell-complex scalar actions in d + 1 dimension) by the “complexity = volume” (CV) conjecture. First, it seems that there is a universal property: the superconducting phase always has a smaller complexity than the unstable normal phase below the critical temperature, which is similar to a free energy. We investigate the temperature dependence of the complexity. In the low temperature limit, the complexity (of formation) scales as Tα, where α is a function of the complex scalar mass m2, the U(1) charge q, and dimension d. In particular, for m2 = 0, we find α = d−1, independent of q, which can be explained by the near horizon geometry of the low temperature holographic superconductor. Next, we develop a general numerical method to compute the time-dependent complexity by the CV conjecture. By this method, we compute the time-dependent complexity of holographic superconductors. In both normal and superconducting phase, the complexity increases as time goes on and the growth rate saturates to a temperature dependent constant. The higher the temperature is, the bigger the growth rate is. However, the growth rates do not violate the Lloyd’s bound in all cases and saturate the Lloyd’s bound in the high temperature limit at a late time.


Introduction
The quantum information theory has been playing an important role in investigating the quantum gravity and quantum field theory.The AdS/CFT correspondence or gauge/gravity duality set this idea in a holographic framework: some geometric quantities in the bulk spacetime is related to the entanglement properties of the boundary field theory.One of the most studied ideas is the entanglement entropy and the Ryu-Takayanagi formula [1,2], from which the spacetime geometry may emerge (e.g., see [3][4][5][6][7]).However, it does not cover the spacetime region inside the black hole horizon.As a candidate to explore inside the black hole horizon, another important concept, the "complexity", has been introduced from quantum computation theory [8][9][10][11][12][13].
In the CV conjecture, the complexity of a state |ψ(t L , t R ) corresponds to the maximal volume of the codimension-one surface connecting the codimension-two time slices (denoted by t L and t R ) at two AdS boundaries: where Σ is all possible codimension-one surface connecting t L and t R , G N is the Newton's constant, and is some length scale such as the horizon, AdS radius or something related with the bulk geometry.In the CA conjecture, the complexity of a state |ψ(t L , t R ) corresponds to the action in the Wheeler-DeWitt (WDW) patch where the WDW patch is the collection of all space-like surface connecting t L and t R bounded by the null sheets coming from t L and t R .There have been many works on holographic complexity.For example, see [9-12, 38, 57-69].
In this paper, we analyze the complexity of the holographic superconductors for the first time.For our method, we only focus on the CV conjecture, leaving the CA conjecture and field theoretic methods as future research directions.A main difficulty of the CA conjecture is due to the region of the singularity involved in the on-shell action computation.In the cases where analytical solutions are available, we may deal with the singularity easily.Otherwise, with only numerical solutions, it is subtle and difficult to study the geometry and matter fields around the singularity.Such a difficulty does not appear in the CV conjecture because the extremal hyper-surface will not touch the singularity.Thus, in this paper, we only investigate the CV conjecture.Our main subjects are i) the complexity of formation and ii) the time-dependent complexity.
First, the complexity of formation was defined in [60] by the complexity of a thermal state such as the AdS d+1 -Schwarzschild or AdS d+1 -RN blackhole geometry from the pure AdS d+1 spacetime at t = 0.For the AdS d+1 -Schwarzschild case, the complexity of formation scales as T d−1 .For the AdS d+1 -RN case, it scales as T d−1 in the high temperature limit and ln T in the low temperature limit.In this paper, as another thermal state, we consider holographic superconductors.We find that the change of the complexity of formation at the critical temperature is not smooth.In superconducting phase, in the low temperature limit, the complexity of formation scales as T α , where α is a function of the complex scalar mass m 2 , the U (1) charge q, and dimension d.It has something to do with the near horizon geometry of the low temperature holographic superconductor.In particular, for m 2 = 0 we find α = d − 1, independent of q.It is due to the fact that, in the zero temperature limit, the near horizon geometry of the holographic superconductor with m 2 = 0 is just the AdS d+1 -Schwarzschild.
Next, we compute the full time evolution of holographic complexity.The holographic time-dependent complexity for the AdS d+1 -Schwarzschild or AdS d+1 -RN blackhole geometry has been studied in [64].These works focus on the late time behavior and rely on the analytic solutions.To compute the full time evolution for the holographic superconductor, we first develop a general numerical method to compute time-dependent complexity.By using this method we compute the complexity of the holographic superconductor in both normal and superconducting phase.The complexity always increases as time goes on and the growth rate saturates to a temperature dependent constant.The higher the temperature is, the bigger the growth rate is.However, the growth rates do not violate the Lloyd's bound (intuitively meaning the fastest computing time [70][71][72]) n all cases and saturates the Lloyd's bound in the high temperature limit at late time.This result is a nontrivial support for the conjecture that the Schwarzshield black is the fastest quantum computer of the same energy.Let us recall that the absolute value of the complexity growth rate in the CA conjecture can be arbitrarily large and does not satisfy the Lloyd's bound [38,64]. 3Thus, our results indicate that, from the perspective of the Lloyd's bound, the CV conjecture is more suitable than the CA conjecture as a definition of the holographic complexity.
The paper is organized as follows: In section 2, we briefly review our holographic superconductor model (Einstein-Maxwell-complex scalar action).In section 3, using the CV conjecture, we study the time-independent complexity of formation of the holographic superconductor.In particular, we investigate the low temperature behavior of the complexity of formation in both normal and superconducting phase.In section 4, we develop a general method to compute time dependent holographic CV conjecture.By this method, we analyze the full time evolution of the complexity of the holographic superconductor, from normal to superconducting phase.We investigate the growth rate of the complexity and compare it with the Lloyd's bound.We conclude in section 5.

Holographic superconductor model: a quick review
Let us first introduce a holographic superconductor model that we consider in this paper.It is the first holographic superconductor model proposed by Hartnoll, Herzog, and Horowitz [74,75].Here, we describe only essential features of the model necessary for our study and refer to [74,75] for more details.The action reads where D = ∇ − iqA.F = dA is a field strength of a bulk U (1) gauge field A and encodes a finite chemical potential or density in field theory.Φ is a complex scalar field with mass m and has something to do with the order parameter of the superconducting phase transition.From here, we set AdS = 1.The equations of motion obtained from this action are where T µν is the energy-momentum tensor given by If we consider the following ansatz for metric and matter fields with four functions; f (z), χ(z), A t (z) and φ(z), the equations of motion (2.2)-(2.4)boil down to where a prime denotes a derivative with respect to z.The first two equations in Eq. (2.7) come from the Einstein equation (2.4) 4 .The third and fourth are the Maxwell and scalar equations respectively.There are two classes of the solutions for Eq.(2.7): i) φ = 0 and ii) φ = 0.They correspond to the normal phase and the superconductor phase respectively.This identification will be explained below Eq. (2.12).
For φ = 0, Eq. (2.7) allows an analytic solution (2.8) For φ = 0, there is no analytic solution for Eq.(2.7) so we have to solve them numerically.For a concrete numerical computation, we will set d = 3 from here.In order to solve Eq. (2.7) we need total six initial or boundary conditions.
Because we are interested in the black brane solutions which have a regular event horizon at z h , we set f (z h ) = 0.The Hawking temperature then can be expressed as 5 In addition, one must require A t (z h ) = 0 in order for g µν A µ A ν to be finite at the horizon.As the horizon is regular, all functions should have finite values and admit Taylor's expansions in terms of z h − z when z → z − h .Then, at the horizon, we have the following relations At the boundary z → 0 + , we want the space-time to asymptote to the AdS spacetime in Poincaré coordinate.Thus, we impose a boundary condition χ(0) = 0.It also turns out that the asymptotic solutions for φ and A t are where ∆ ± = − 3 2 ± √ 4m 2 + 9/2.By the AdS/CFT correspondence the leading term φ + corresponds to a source.We turn off the source, i.e. φ + = 0 to describe a spontaneous symmetry breaking.Then, φ − corresponds to condensate.Thus, if φ − = 0 the state is superconducting while if φ − = 0 (or φ = 0) the state is normal.In the asymptotic form of A t , the leading term µ is chemical potential and the sub-leading coefficient ρ is charge density.If we consider the system in the grand canonical ensemble, we need to fix the chemical potential at the boundary.Near the boundary, we have the following asymptotic behaviors for f and χ, 5 In normal phase, Figure 1.Superconducting phase solutions for (d, m 2 , q) = (3, 0, 2.5).The various colors represent different temperature.i.e.T /T c = 0.99, 0.86, 0.61, 0.41, 0.17, 0.05 (red, orange, yellow, green, blue, purple).In (b), the dashed black line is the AdS-Schwarzschild blackening factor: for some constants χ 0 , χ 1 and f 0 .For numerical analysis, we used the shooting method and pseudo-spectral method independently and have cross-checked our results.For example, we show the numerical solutions of (2.7) for (d, m 2 , q) = (3, 0, 2.5) in Fig 1, where z := z/z h .

Complexity of formation
The complexity of formation by the finite temperature was first investigated by Ref. [60].
It is the complexity between the AdS-Schwarzschild black brane and pure AdS space-time for a very special maximum space-like slice.In this section, we consider the complexity of formation for the holographic superconductor in both normal phase and superconducting phase.The complexity of formation in normal phase also has been reported in [64].Here, we revisit it by a little different analytic method, and our results agree with [64].Before starting, let us introduce a new notation for the CV conjecture, different from (1.1).
V = max where we set G N = = 1 for convenience.
The maximal volume slice B connecting the two boundaries at t L = t R = 0 in one patch of the AdS-RN black hole.

Normal phase
Following [60], we consider the complexity of the thermal state defined on the time slice at t L = t R = 0.Because of the symmetry, the maximal volume is given by the t = 0 slice in bulk, i.e., the straight line connecting the two boundaries through the outer bifurcation horizon in the Penrose diagram shown in Fig. 2.
The volume integral then simplifies to: where z m is the cut-off near the boundary and Ω d−1 is the dimensionless area of the spatial geometry when z and t are fixed.With the general asymptotic behavior (2.13), we find that V is infinite when z m → 0. To evaluate the complexity of formation coming from finite temperature and chemical potential, we will subtract from this integral, the corresponding contribution from (two copies of) the vacuum AdS background, where The emblackening factor f (z) in Eq. (2.8) and the temperature(2.9)read where we define T := T z h and μ := µz h .The dimensionless variables with tilde denote the variables scaled by z h and convenient for numerical analysis.However, for physical interpretation it is better to choose the chemical potential as our scale.Thus we also define the dimensionless variables with bar to denote the variables scaled by the chemical potential.For example, where the indices 'n' and 'd' of ∆ Vn d means 'n'ormal state in 'd' spatial dimension.In the definition of ∆ Vn d we included the trivial volume factor Ω d−1 .The complexity of formation ∆ Vn d is a function of T , because μ is a function of T for a given dimension d by the relation (3.9).zm will be taken to be zero at the end of the day.
Eq. (3.8) does not allow an analytic expression in general, so we perform the integration numerically.The result is shown in Fig. 3.As the temperature goes to zero and goes to infinity, the complexity of formation diverges in both cases.There is a particular temperature in the middle where the complexity of formation is minimum.This has been first observed in [64].
To analyze the divergence behavior at low and high temperature, we consider the expansion in terms of T .First, in these limits the chemical potential reads from (3.9) so the emblackening factor in (3.6) behaves as follows: 1) . (3.13) In the high temperature limit( T 1), by using (3.12) and (3.10), (3.8) becomes  near the horizon(z = 1) as follows: Since f (z) goes to zero quadratically near the horizon, the integral (3.8) diverges logarithmically at the zero temperature limit.
To deal with this divergence separately we introduce ã such that zm < ã < 1 and divide the integral range as where To have a better understanding of the singular term in (3.16), we define the followings: where Γ( T ) is chosen such that h(z, T ) is regular and well-defined at z = 1 and T = 0, i.e.
h(1, 0) := lim Let us explain how to determine (3.19).The first condition in (3.20) may be rephrased as where (3.18) and (3.15) are used.To have a regular h(1, 0) we need to require The second condition of (3.20) implies which yields (3.24) By using (3.18), we rewrite (3.16) as follows: where S(ã, d, T ) is a singular term and F 2 (ã, d, T ) is another finite term defined as where For example, if d = 3, with zm = 0 and Note that F 3 (ã, 3) is independent of the arbitrary choice of ã.

Superconducting phase
In superconducting phase, the non-trivial complex scaler field plays a role and the low temperature behavior will be different from the RN-AdS black holes.At low temperature below some critical temperature T c , the superconductor state with φ = 0 has the lower free energy than a normal state with φ = 0 so it becomes the ground state.For example, for q = 1, m 2 = −2, the dimensionless critical temperature T c /µ 0.0208.In this section, we study the complexity of formation in the superconducting phase.For φ = 0, the geometry outside of the event horizon is still regular, so the causal structure outside of z h is similar to the RN black hole.However, in this case the inner horizon disappears and a new singularity appears.Due to the existence of the Cauchy horizon in the RN-AdS black hole, the complete description of the inner geometry of the charged black hole after the scalar hair appears is subtle.Some analyses show that the Cauchy horizon will become a "weak singularity" if the scalar field is infinitesimal [76,77].As the scalar field increases, it is not clear how such a new "weak singularity" will evolve.In Fig. 5, we show a schematic Penrose diagram in the connected region which contains an Einstein-Rosen bridge between two AdS boundaries.Though the structure of the singularity is not clear, the computation of the CV conjecture is not affected by the detailed structure of singularity.However, for the CA conjecture, as the WdW patch touches the singularity in general, we have to know the detailed properties of the singularity.Partly because of this subtlety, in this paper, we only focus on the CV conjecture.
The maximal volume slice B for t L = t R = 0 is the same as the RN black hole.Thus, the complexity for t L = t R = 0 in superconducting phase is still given by (3.2).Taking into account the general asymptotic solution for metric shown in (2.13) at the boundary z = 0, we find that the V diverges but the complexity of formation ∆V in (3.4) and ∆ Vn d in (3.8) are still valid also for the superconducting phase.

∆ Vs
where we changed the superscript 'n' to 's' to make clear we are considering the superconducting phase.Contrary to normal phase, in this case, there is no analytic formula for the emblackening factor f (z) is available so we resort to the numerical methods 6 .Note that the formula (3.34) is valid for both normal phase and superconducting phase, so we will use the subscript 'n' and 's' only we want to emphasize which phase we are looking at.transition.For T /T c > 1 the system is in normal phase and the complexity is the same as Fig. 3.At T /T c = 1 there is a kink and the transition is not smooth.For T /T c < 1 the complexity monotonically decreases contrary to the normal phase.The complexity in superconducting phase is smaller than normal phase similar to the free energy.We also have analyzed many cases with different masses m 2 and charges q, and found a universal property: the superconducting phase always has the smaller complexity than the unstable normal phase below the critical temperature.It seems that the complexity may play a role of the free energy and indicates the phase transition7 .It will be interesting to investigate if this property is indeed universal, and understand why, if it is.At zero temperature the complexity of formation ∆ Vs d ( T = 0) is nonzero and it is the complexity of the zero temperature superconductor state from the AdS vacuum state.We may also define the complexity of formation from the zero temperature superconductor not from the AdS vacuum state: Indeed, this new complexity of formation has a more natural and phenomenological interpretation because the ground state of the superconductor is zero temperature superconductor not a vacuum.To distinguish this new complexity of formation, we will call it "thermal complexity" from here.In Fig. 6(a) it looks that at low temperature the thermal complexity satisfies the power law δ Vs d ( T ) ∝ T α .
(3.37)This has been checked in Fig. 6(b) and α 2. To what extend is this integer exponent robust and what is a physical picture behind it if any?To answer these questions we have first performed more numerical analysis and investigated the dependence of α on the charge q.
The red dots in Fig. 7 are numerical data of the exponent α which belongs to the range 2 ± 2 × 10 −3 for the charges q = 2.5, 3, 3.5, 4, 4.5, 5, 6.Note that q 2 > 3/4 so that the scalar field can condense spontaneously [81].In addition, we consider higher dimensions.In Fig. 7 the green dots are for d = 4 and the blue dots are for d = 5 case.Based on our numerical results, we may deduce that the exponent α is independent of the q and α = d − 1 . (3.38) Indeed, this simple and universal exponent can be understood by the near horizon geometry of the holographic superconductor at low temperature.For m 2 = 0, it is shown to be the AdS-Schwarzschild black hole [82]: where χ 0 and φ 0 are constant.It can be also seen from our numerical solutions in Fig. 1(b), where the dashed black curve is the AdS-Schwarzschild blackening factor, f (z) = 1 − z3 .We see that as the temperature goes down the curves approach f (z) = 1 − z3 near horizon.Therefore, the thermal complexity of the superconductor with m 2 = 0 can be approximately computed as the complexity of formation of the AdS-Schwarzschild black hole: which has been computed in [38,60].Note that the geometry (3.39) is valid only near horizon so the volume integral based on (3.39) is not precise for (3.36).Here, we are assuming that the correction due to this discrepancy are common to both terms in (3.36) and cancelled out.This is justified a posteriori by (3.38).

Complex scalar mass:
What if we consider m 2 = 0? In this case, there is no known analytic small temperature geometry for holographic superconductor so far.Thus, for m 2 = 0 it is not easy to have analytic understanding such as (3.38) and (3.40).However, the zero temperature solution for holographic superconductor has been analyzed in [81].It shows that the ground state geometry depends on d, q and m 2 in general.Therefore, we may expect that the scaling behavior of the thermal complexity depends on d and q for a given m 2 .Note that only if m 2 = 0, the zero and small temperature geometry of the holographic superconductor is independent of q as shown in (3.39).It is why the thermal complexity in that case does not depend on q and only a function of d.
Even though we don't have an analytic intuition from the ground state geometry for m 2 = 0, we can still analyze the scaling behavior of the thermal complexity at low temperature numerically and try to understand physics behind it.
To be specific, let us consider d = 3.For a negative mass squared we choose m 2 = −2 and for a positive mass squared we choose m 2 = 1/2.First, for m 2 = −2 Fig. 8(a) shows the complexity of formation and thermal complexity with q = 1 and from Fig. 8(b) we can read off the exponent α.We repeat this computation by changing q and the results are summarized in Fig. 9(a).Next, for m 2 = 1/2, by the same procedure, we obtain the exponent α, which is shown in Fig. 9(b).The numerical data in Fig. 9 can be fitted as which are represented as a dashed line in Fig. 9(a) with (α 1 , α 2 ) = (2.2,1/20) and (α 3 , α 4 ) = (1.998,1/7).We have checked that these behaviors are qualitatively true for other values of m 2 : For a negative mass squared α increases as q increases while for a   positive mass squared α decreases as q increases.It seems that α 1 and α 3 goes to d − 1 and α 2 and α 4 goes to zero as m 2 goes to zero, reproducing the results in section 3.2.1.
Let us conclude this subsection by a remark on the possible low temperature geometry of the holographic superconductor with m 2 > 0. When m 2 > 0, it is known that the extremal geometry is Lifshitz geometry: where and ζ is the lifshitz dynamical exponent satisfying ) is reduced to the case in [81].
To follow the same logic as m 2 = 0 case in section 3.2.1,we may assume that the low temperature solution is obtained by the emblackening factor: The complexity of formation for this Lifshitz black hole is ∆V ∼ T where (3.45) is used.
which is different from our numerical fitting (3.42): α 3 is similar but the sign of α 4 is opposite.This is because the simple emblackening factor (3.46) is not the low temperature solution of the holographic superconductor for m 2 > 0.

Time-dependent complexity
In this section, we study the time-dependent complexity or complexity of formation.We consider the same ansatz as (2.6): From the CV conjecture perspective, it means that we evaluate the volume of the maximum codimension-one bulk surface (B or B ) in Fig. 10 instead of the horizontal surface in Fig. 2 or Fig. 5. 8 The green curves B or B are schematic ones.As the metric (4.1) has the time translation symmetry the volume of B and B are the same, if

Method of computation
To compute the volume of the codimension-one bulk surface let us first introduce the null coordinate V (t, z): where v(t, z) is the infalling Eddington-Finkelstein coordinate and β is determined by the near horizon expansion of the equations of motion.In this coordinate system, the metric (4.1) becomes By considering the induced metric on the green curves parameterized by λ in Fig. 10.
we have the volume of the codimension-one bulk surface where Ω d−1 ≡ dΩ d−1 is the volume of the spatial geometry.We obtain only one Euler-Lagrangian equation from (4.6) However, we need two equations for two independent fields(z(λ), V (λ)).To resolve this issue, we introduce the auxiliary field ε(λ) to the volume integral.
from which, we have three Euler-Lagrangian equations: (4.10)Among these, only two are independent and we will choose the second and third equations as independent ones.From here, we take ε(λ)=1, recovering the original variational problem.The two independent equations read We will solve the equation of motions (4.11) and (4.12) numerically starting from the horizon at (V (0), z(0)) = (0, z h ).See the red dot for λ = 0 in Fig. 10.Near this point the series solutions are obtained as where β introduced in (4.3) is determined as which is nothing but 2πT as expected in (2.10).
From here we introduce again z = z/z h defined in (3.5) without loss of generality.For a given initial values (V (1) , z (1) ) the solutions (z(λ), V (λ)) are numerically determined.Thus, at the cut-off zm (see Fig. 10), (λ R , λ L ) can be obtained as Note that λ R > 0 and λ L < 0. Once we know the parameters (λ R , λ L ) at the cut-off zm , the complexity (4.7) is computed straightforwardly as follows.
By repeating this procedure for different temperatures, T /T c = 5, 3, 1, 0.5, 0.3 we obtain the red, orange, yellow, blue, and purple curves respectively.Note that our numerical method works also for the normal phase (T > T c ).The complexity of formation at ∆t = 0  agrees with the colored dots in Fig. 8(a), which serves as a consistency check of our numerical analysis.
To see the growth rate of the complexity9 , we make the plots for the time derivatives of the complexity in Fig. 13.The higher the temperature is, the bigger the growth rate is.Thus, for a clear presentation we make two plots: (a) normal phase and (b) superconducting phase.We find that there are the upper bounds on the time derivative of complexity for both the normal phase and superconducting phase.T /T c = 100, 5, 3, 1, 0.7, 0.5, 0.3 (black, red, orange, yellow, green, blue, purple).The black curve at high temperature is approaching to the case of AdS-Schwarzschild as shown in [38].
To compare the upper bound with the Lloyd's bound, we first need to define the Lloyd's bound, which is proportional to the 'mass' (or energy) of the black hole.However, it is subtle how to define mass in curved spacetimes.For an asymptotic AdS spacetime, there are several different ways to define mass, such as the ADM mass (M ADM ), the Komar mass (M Komar ) and the mass (M h ) based on the holographically renormalized stress tensor.These three definitions give the same results for normal phase but different results for superconducting phase.See appendix A for details.In many cases, M h is used to define the mass of a boundary state.It has an advantage that the counterterms are fixed according to the "holographic renormalization" [83,84].However, M h may not be appropriate in the context of the Lloyd's bound because it can be negative in general.In the context of the Lloyd's bound, the other two masses have their own advantages.In the proofs of the "positive energy theorem" [85], the mass of an asymptotic AdS black hole is M ADM .In Ref. [73], it was shown that the mass corresponding to the Lloyd's bound in the late time limit for the CA conjecture is M Komar .
It is not clear which mass corresponds to the mass in the Lloyd's bound.We check the Lloyd's bound by all three definitions of mass.In our model it turns out that M ADM = M h ≤ M Komar (See appendix A).Thus, if the Lloyd's bound is satisfied by the ADM mass, it will be satisfied by any of three masses.For this reason we focus on the ADM mass.The Lloyd's bound based on ADM mass is given by There is an ambiguity in the definition of the Lloyd's bound.Essentially, it only says ĊLloyd ∼ M ADM and the proportionality constant remains to be fixed.We choose it as 8π/(d − 1), which is the maximum value for the Schwarzschild blackhole case [38].The factors in the definition of ĊL is introduced for the same normalization as (3.8) and the factor '2' is necessary to reflect the contribution from two sides(left and right) of the blackhole.
The ratio of the growth rate of the complexity to the Lloyd's bound( ĊLloyd ) is presented in Fig. 14: (a) is for normal phase and (b) is for superconducting phase.We find that the growth ratio of the complexity becomes smaller than the Lloyd's bound as temperature goes down.This tendency has been observed in normal phase in Fig. 13 of [64], where the complexity for the normal phase of AdS-RN in (4+1) dimension was studied.Here, we find that it continues to be true also for superconducting phase.At a very high temperature (the black curve in Fig. 14(a)) the growth rate of AdS-RN case saturates to the Lloyd's bound, which is similar to the case of the AdS-Schwarzschild as shown in [38].It can be understood by the fact that the high temperature(T /µ 1) corresponds to a small chemical potential close to the AdS-Schwarzschild case.The Lloyd's bound in the context of the holography was first introduced in the study of the CA conjecture.The complexity by the CA conjecture seemed to saturate to the Lloyd's bound at late time.It was taken as the first nontrivial evidence supporting the CA conjecture.However, more recent studies [38,64,[86][87][88] show that such a bound can be violated in many cases.Only in the late time limit the CA conjecture saturates to the Lloyd's bound from above.It was first reported by [38,64] independently.It has also been shown that the Lloyd's bound is not satisfied even in AdS-Schwarzschild black holes when time t L and t R are small [38,64].
Our results show that from the perspective of the Lloyd's bound the CV conjecture is more suitable than the CA conjecture as a definition of the holographic complexity.For AdS-Schwarzschild black holes, RN-AdS black holes and the black holes with nonzero complex scalar hair, we found that the complexity growth rate always satisfies the Lloyd's bound during the whole evolution.Based on these observation we may propose a conjecture: under a few suitable conditions on matter fields or causal structures of spacetime, the complexity by the CV conjecture will always satisfy the Lloyd's bound.

Conclusions
In this paper, we have investigated the holographic complexity for the holographic superconductor model in asymptotically AdS d+1 spacetime by using the CV conjecture.
We first study a time-independent complexity of formation ∆ Vd ( T ) as a function of temperature (T ) at fixed chemical potential (µ), i.e.T = T /µ.A typical behavior of ∆ Vd ( T ) is shown in Fig. 6(a).In many cases with different masses m 2 and charges q, we find that a universal property: the superconducting phase always has the smaller complexity than the unstable normal phase below the critical temperature.Thus, it seems that the complexity may play a role of the free energy, indicating the phase transition.It will be interesting to investigate if this property (higher complexity for unstable phase) is universal for other cases, and understand the reason behind it.
In superconducting phase, the complexity of formation always decreases as temperature goes down10 .In the high temperature limit, the complexity of formation scales as ∆ Vd ( T ) ∼ T d−1 , which is consistent with the AdS-Schwarzschild Black brane geometry.In the low temperature limit, if the system were in normal phase, the complexity of formation would behave as ∆ Vd ( T ) ∼ log T as shown in Fig. 3.However, at low temperature, the system must be in the superconducting phase and in this case it again scales as δ Vd ( T ) ∼ T α , where α is a function of the dimension d, the complex scalar mass squared m 2 , and the U (1) charge q 11 .
In particular, if m 2 = 0, α = d − 1 independent of q, which is the same as the high temperature limit.i.e.
δ Vd ( T ) ∼ T d−1 . ( It can be understood by the fact that the near horizon geometry of the holographic superconductor with m 2 = 0 at low temperature is nothing but the AdS-Schwarzschild Black brane geometry.For m 2 = 0, our numerical analysis suggests the following behavior (see for example Fig. 9.).
where α 1 and α 3 goes to d − 1 and α 2 and α 4 goes to zero as m 2 goes to zero, reproducing the results for m 2 = 0.For m 2 = 0 it is not easy to have a geometric understanding for α because there is no available near horizon analytic geometry at low temperature.We leave a better physical understanding of (5.2) and ( 5.3) as a future work.Next, we investigate the full time evolution of the complexity of formation for the holographic superconductor.We first have developed a general method to compute the time-dependent complexity of formation in section 4.1.By this method, we have computed the time dependent complexity of formation for the holographic superconductor, of which results are shown in Fig. 12.
In both normal and superconducting phase, the complexity of formation increases as time goes on.Furthermore, it increases linearly in time in the late time limit, which can be seen in Fig. 13.In other words, the growth rate of the complexity is constant in the late time limit.The lower the temperature is, the smaller the growth rate of the complexity is.Furthermore, we compare the growth rate of the complexity in the late time limit with the Lloyd's bound, defined by the ADM mass.If the temperature is high (T T c ), the growth rate of the complexity saturates to the Lloyd's bound [58,70,73].This is consistent with the result for the AdS-Schwarzschild Black brane case.As the temperature goes down, the growth rate of the complexity becomes smaller and smaller than the Lloyd's bound.Thus the complexity of the holographic superconductor does not violate the Lloyd's bound.
While the Lloyd's bound was first introduced in the study of the CA conjecture, recent studies [38,64,87,88] show that the CA conjecture violates the Lloyd's bound in many cases.It was for the first time reported by [38,64] independently that even in the late time limit the CA conjecture violates the the Lloyd's bound because it saturates to the bound from above.However, based on our observation for the CV conjecture in this paper, we may propose a conjecture: under a few 'suitable' conditions on matter fields or causal structures of spacetime, the complexity by the CV conjecture may always satisfy the Lloyd's bound.It will be interesting to clarify what 'suitable' means in the conjecture.We leave it as a future work.
It will be also interesting to study the complexity of formation and the time-dependent complexity for other models including additional matter fields such as axion or dilaton fields.Moreover, the complexity of the system at finite magnetic field or with higher derivative gravity model is other important directions to explore.In this paper, we focused on the CV conjecture only, but it will be interesting to study the complexity of superconductor by the holographic CA conjecture or other field theoretic methods, and compare them with our result.We leave these issues as future works.
seperate the metric into two parts, g µν = ḡµν + h µν , (A.4) where we assume that h µν vanishes at infinity.Let us denote the covariant derivative corresponding to ḡµν by ∇µ .To facilitate this decomposition, we will use the convention in this subsection that all operations such as index moving or differentiation are with respect to ḡµν .We can sperate the left hand side of Eq. (2.4) into three pieces, i.e., the piece independent of h µν which vanishes because ḡµν itself satisfies the vacuum Einstein's equation, the piece linear in h µν and the piece containing all quadratic and higher order terms in h µν .Then we can write Eq. (2.4) as Here − T µν /2 contains all the quadratic and higher order terms of h µν in the left hand side of Eq. (2.4).The subscript L refers to the terms linear in h µν .Physically, T µν is the energy-momentum tensor of the gravitational field and T µν total is the total energymomentum.Because the left hand side of Eq. (A.5) obeys the background Bianchi identity R µν L − 1 2 ḡµν R L + Λh µν can lead that ∇µ T µν total = 0 .(A.6) Note that in Eq. (A.6) the derivative is a background covariant derivative not an ordinary derivative.To construct a conserved current corresponding to Eq. (A.6), we can use the background Killing vector field ξµ which satisfies that ∇µ ξν + ∇ν ξµ = 0. Then one can check that the combination ξ ν T µν total is a conserved current and the corresponding conserved charge is Here Σ is the t =constant hypersurface and d Σν is induced volume element (corresponding to ḡµν ).Now according to Sec.2 in Ref. [92], Eq. (A.5) can be expressed as Here K µανβ is called superpotential, which is defined as We see that the superpotential has the same index symmetry as the Riemann tensor.The

3 nFigure 3 .
Figure 3.The complexity of formation in normal phase.The numerical plot of (3.8) with d = 3.

Figure 4 .
Figure 4.The low and high temperature behavior of the complexity of formation in normal phase.The blue curve is the numerical plot of (3.8) with d = 3.

Figure 5 .
Figure 5.The schematic Penrose diagram in the connected region which contains an Einstein-Rosen bridge to connect two AdS boundaries.The maximal volume slice B for t L = t R = 0 is the same as AdS RN black brane.When complex scalar field appears, the Cauchy horizon (inner horizon) will be taken place by a singularity.

3. 2 . 1 Fig. 6 (Figure 6 .
Fig.6(a) is the numerical plots of the complexity formation for the system with m 2 = 0, q = 2.5.The horizontal axis is T /T c where T c is the critical temperature of the phase

Figure 8 .
Figure 8.The complexity of formation and thermal complexity of the holographic superconductor with m 2 = −2 and q = 1.T c /µ 0.0208 is the critical temperature.

Figure 9 .
Figure 9.The exponent α vs q for m 2 = 0. Red dots are numerical data.The dashed lines are the fitting curves in (3.41) and (3.42).

Figure 10 .
Figure10.The maximal volume surface in the AdS-Schwarzschild geometry.At two boundaries, t L and t R stand for two states dual to the states in TFD.z h is the horizon radius.λ is used to parametrize z(λ) and V (λ).B and B are schematic representation of the maximum codimensionone bulk surfaces connecting t L and t R in the left plot, − ∆t 2 and ∆t 2 in the right plot, respectively.

Figure 14 .
Figure 14.The comparison between the growth rate of the complexity for the holographic superconductor model and the Lloyd's bound.Various colors represent different temperatures.i.e.T /T c = 100, 5, 3, 1, 0.7, 0.5, 0.3 (black, red, orange, yellow, green, blue, purple).The black curve at high temperature is approaching to the case of AdS-Schwarzschild as shown in[38].