aa r X i v : . [ h e p - t h ] M a y Kibble-Zurek scaling in holography
Makoto Natsuume ∗ KEK Theory Center, Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization, Tsukuba, Ibaraki, 305-0801, Japan
Takashi Okamura † Department of Physics, Kwansei Gakuin University, Sanda, Hyogo, 669-1337, Japan (Dated: October 8, 2018)The Kibble-Zurek (KZ) mechanism describes the generations of topological defects when a systemundergoes a second-order phase transition via quenches. We study the holographic KZ scaling usingholographic superconductors. The scaling can be understood analytically from a scaling analysis ofthe bulk action. The argument is reminiscent of the scaling analysis of the mean-field theory but ismore subtle and is not entirely obvious. This is because the scaling is not the one of the originalbulk theory but is an emergent one that appears only at the critical point. The analysis is alsouseful to determine the dynamic critical exponent z . I. INTRODUCTION
The AdS/CFT duality [1–4] has been useful to study“real-world” which is rather difficult to analyze viaconventional methods. The duality has been appliedto QCD, condensed-matter physics, and nonequilibriumphysics (see., e.g. , Refs. [5–8] for textbooks). In this pa-per, we analytically study the holographic Kibble-Zurek(KZ) mechanism; The KZ mechanism describes the gen-erations of topological defects when a system undergoesa second-order phase transition via quenches (cooling)[9–12].For a system with a second-order phase transition, thecorrelation length ξ diverges, and as a result various phys-ical quantities diverge. The divergences are parametrizedby critical exponents, and those exponents define the uni-versality class of the system. In the static case, ther-modynamic quantities diverge but in the dynamic case,the relaxation time of the order parameter also diverges,which is known as the critical slowing down [13–15]. Therelaxation time τ behaves as τ ∝ ξ z , where z is a dy-namic critical exponent. The study of dynamic criticalphenomena in the AdS/CFT duality was pioneered byRefs. [16, 17].When the system undergoes the phase transitionthrough a quench, topological defects are generated spon-taneously. As an example, for superconductors, the de-fects generated are vortices. By adding the quench, thesymmetry is broken but spatially separated regions canselect different states. The typical size of the correlatedregions (domains) and the density of defects show power-law behaviors on the quench time scale τ Q . This KZmechanism is closely related to the critical slowing down. ∗ [email protected]; Also at Department of Particle andNuclear Physics, SOKENDAI (The Graduate University for Ad-vanced Studies), 1-1 Oho, Tsukuba, Ibaraki, 305-0801, Japan;Department of Physics Engineering, Mie University, Tsu, 514-8507, Japan. † [email protected] Recently, the holographic KZ mechanism has beenstudied numerically [18, 19], and the KZ scaling has beenconfirmed. These works study holographic supercon-ductors [20–22] which undergo second-order phase tran-sitions. Typically, a holographic superconductor is anEintein-Maxwell-complex scalar system. For
T > T c , thesolution is a standard black hole without scalar, but for T < T c , the solution becomes unstable and is replaced bya solution with scalar “hair.” According to Refs. [18, 19],after t = τ KZ , where τ KZ is called the “freeze-out time,”one starts to have “droplets,” which are localized solu-tions of nonzero condensate. Droplets subsequently grow,and eventually droplets merge into | Ψ | ≃ (constant) so-lution leaving isolated regions with Ψ = 0. They arevortices with winding number. Static holographic vortexsolutions are obtained in Refs. [23–26].The holographic KZ scaling has been confirmed nu-merically. On the other hand, the KZ scaling can beunderstood from a scaling analysis of the mean-field the-ory such as the time-dependent Ginzburg-Landau theory(see, e.g. , Ref. [27]). Similarly, we would like to under-stand the holographic KZ scaling analytically. Our re-sults are summarized as follows:1. The holographic KZ scaling can be understood an-alytically from a scaling analysis of the bulk ac-tion/equation of motion. The argument is reminis-cent of the scaling analysis of the KZ mechanism.2. While the scaling analysis of the KZ mechanism,which is reviewed in Sec. II, is straightforward, theholographic scaling analysis is more subtle and isnot entirely obvious. This is because the scalingis not the one of the original bulk theory but isan emergent one which appears only at the criticalpoint.Namely, the bulk equation of motion has the relativistic“ z = 1” scaling, which comes from the underlying AdSgeometry. However, the operator L Ψ (3.13c) of the com-plex scalar field, which represents the time-independenthomogeneous part of the equation of motion, has a zeroeigenvalue at the critical point. In this case, the “ z = 2”scaling, which acts on the AdS radial coordinate triv-ially, is allowed. This “ z = 2” scaling gives rise to theKZ scaling.The emergent scaling is by no means surprising. Recallthat the scaling in critical phenomena itself is an emer-gent one and is not transparent to see it from the under-lying microscopic theory, e.g. , from the Ising model. Sim-ilarly, a holographic analysis is a “first-principle” compu-tation in principle. The holographic scaling is emergentand is not transparent just like the scaling from the Isingmodel.As the result of the first-principle computation, a holo-graphic analysis is usually not as easy as a mean-fieldanalysis. A bulk system often has several fields and solv-ing the equations of motion is not very easy. Thus, oneoften needs numerical analysis even to obtain mean-fieldresults. However, a lesson of our analysis is that onedoes not have to solve the bulk system in order to obtainsome qualitative information such as critical exponentsand KZ scalings. The trick is that the eigenmode of thezero eigenvalue plays the role of the mean-field. There isa trade-off for analytic arguments however. We cannotreally address the details of dynamical evolutions. Theyrequire numerical computations such as Refs. [18, 19].The plan of this paper is as follows. In Sec. II, wereview the scaling argument to determine critical expo-nents and the KZ scaling. In Sec. III, we set up ourconventions for holographic superconductors especially inthe Eddington-Finkelstein coordinates. The holographiccounterpart of the scaling argument is given in Sec. IV.We first use our scaling argument to rederive critical ex-ponents ( ν, z ) = (1 / , II. KZ MECHANISM AND SCALING
Consider the standard Ginzburg-Landau theory, or the φ mean-field universality class. The pseudofree energy I [ φ ] is given by I [ φ ] = Z d x (cid:26) | ∇ φ | + ǫ | φ | + g | φ | (cid:27) , (2.1)where ǫ := ( T − T c ) /T c . Phenomenologically, the relax-ation of a system is well-described by the time-dependentGinzburg-Landau (TDGL) equation: ∂ t φ ( t, x ) = − Z d y Γ (cid:0) | x − y | (cid:1) δI [ φ ] δφ ( t, y ) + ζ ( t, x ) , (2.2)where Γ (cid:0) | x − y | (cid:1) is a transport coefficient, and ζ is a ran-dom Gaussian variable with h ζ ( t, x ) ζ ( t ′ , x ′ ) i = 2 T Γ( x − x ′ ) δ ( t − t ′ ). The existence of ζ is crucial to produce fluc-tuations, but it does not play an important role below,so we henceforth ignore the term. The details of dynamic universality classes partly de-pend on Γ. We shall focus on “Model A” dynamic uni-versality class [13] because this is the dynamic univer-sality class of holographic superconductors [17]. This isthe case where the order parameter is not a conservedcharge. (On the other hand, the conserved charge caseis the “Model B” universality class.) In this case, Γ is aconstant, and the TDGL equation becomes ∂ t φ = − Γ (cid:2) − ∇ φ + ǫ φ + gφ | φ | (cid:3) . (2.3) A. Critical exponents from scaling
We first determine critical exponents via a scaling ar-gument. Consider the scaling ˜ t = at , ˜ x = b x , ˜ φ = φ/b . (2.4)Under the scaling, the TDGL equation becomes ∂ ˜ t ˜ φ = − Γ (cid:20) − b a ˜ ∇ ˜ φ + ǫa ˜ φ + b a g ˜ φ | ˜ φ | (cid:21) . (2.5)The ( T − T c )-dependence can be eliminated by choosing a = ǫ ∝ ( T − T c ) , (2.6a) b = a / ∝ ( T − T c ) / . (2.6b)Since the ( T − T c )-dependence is eliminated, the systemis away from the critical point in rescaled variables. Sup-pose the correlation length ˜ ξ and the relaxation time ˜ τ in rescaled variables are ˜ ξ ∼ O (1) and ˜ τ ∼ O (1). Then,in original variables, τ = a − ˜ τ ∝ ( T − T c ) − , (2.7a) ξ = b − ˜ ξ ∝ ( T − T c ) − / . (2.7b)The static exponent ν and the dynamic exponent z aredefined by τ ∝ ξ z ∝ | T − T c | − νz , (2.8a) ξ ∝ | T − T c | − ν . (2.8b)Thus, ( ν, z ) = (1 / ,
2) for Model A.
B. KZ scaling
We now consider the quench from high-temperaturephase to low-temperature phase. One typically considersthe linear “quench protocol” ǫ = − tτ Q . (2.9) The scaling of φ is chosen to be consistent with Eq. (2.6). The quench is added for the initial temperature T i > T c to the final temperature T f < T c according to Eq. (2.9)so that the system crosses the critical point at t = 0.When the temperature change is slow enough com-pared with the relaxation time, the order parameter canadjust to the change, and the evolution is adiabatic.However, as we saw, the relaxation time of the orderparameter diverges because of the critical slowing down.The evolution of the order parameter is slow, and the sys-tem cannot adjust to the change any more no matter howslow the quench is. The order parameter cannot adjustto the change globally and can adjust to the change onlylocally. As a result, defects form. This happens when t < | τ KZ | , where τ KZ is called the “freeze-out time.” Thesize of the typical domain is called the “freeze-out length”and is denoted as ξ KZ .To determine τ KZ and ξ KZ , again consider the scaling(2.4). In this case, the “mass term” [the second term ofthe right-hand side of Eq. (2.5)] takes the form − ˜ ta τ Q ˜ φ , (2.10)so the τ Q -dependence can be eliminated by choosing a = τ − / Q , (2.11a) b = a / = τ − / Q . (2.11b)In rescaled variables, the relaxation time ˜ τ KZ and thecorrelation length ˜ ξ KZ do not depend on τ Q . Then, inoriginal variables, τ KZ ∝ a − = τ / Q , (2.12a) ξ KZ ∝ b − = τ / Q . (2.12b)This agrees with the KZ prediction τ KZ ∝ τ νz/ (1+ νz ) Q , (2.13a) ξ KZ ∝ τ ν/ (1+ νz ) Q , (2.13b)for Model A [10].It is easy to generalize the scaling argument to thepolynomial quench of the form ǫ = − (cid:12)(cid:12)(cid:12)(cid:12) tτ Q (cid:12)(cid:12)(cid:12)(cid:12) n sgn( t ) . (2.14)In this case, τ KZ ∝ τ n/ ( n +1) Q , (2.15a) ξ KZ ∝ τ n/ n +1) Q . (2.15b)To summarize, scaling arguments eliminate the ǫ -dependence in the previous subsection and the τ Q -dependence in this subsection, which determines criti-cal exponents ( ν, z ) and the KZ exponents. We essen-tially repeat similar scaling arguments holographically inSec. IV. III. HOLOGRAPHIC SUPERCONDUCTORA. Preliminaries
We consider the ( p +2)-dimensional s -wave holographicsuperconductors given by S = Z d p +2 x √− g (cid:20) R − − e (cid:26) F MN + | D Ψ | + V (cid:0) | Ψ | (cid:1)(cid:27) (cid:21) , (3.1)where F MN = 2 ∂ [ M A N ] , D M := ∇ M − iA M , (3.2)Λ = − p ( p + 1)2 L , V = m | Ψ | . (3.3)We use capital Latin indices M, N, . . . for the ( p + 2)-dimensional bulk spacetime coordinates and use Greekindices µ, ν, . . . for the ( p + 1)-dimensional boundary co-ordinates.Below, we take the probe limit e ≫
1, where the back-reaction of the matter fields onto the geometry is ignored.Then, in the static case, the background metric is givenby the Schwarzschild-AdS black hole: ds p +2 = (cid:18) Lu (cid:19) (cid:18) − f ( u ) dt + d~x p + du f ( u ) (cid:19) , (3.4)where f ( u ) := 1 − (cid:18) uu h (cid:19) p +1 . (3.5)Here, the boundary coordinates are written as x µ =( t, ~x ) = ( t, x i ), and u = 0 at the AdS boundary. TheHawking temperature is given by T = ( p + 1) / (4 πu h ).We set L = e = 1 below.In the A u = 0 gauge, the asymptotic behavior of mat-ter fields is given byΨ( x, u ) ∼ Ψ ( − ) ( x ) u ∆ − + Ψ (+) ( x ) u ∆ + , (3.6a)∆ ± := p + 12 ± s(cid:18) p + 12 (cid:19) + m , (3.6b) A µ ( x, u ) ∼ A (0) µ ( x ) + A (1) µ ( x ) u p − . (3.6c)Ψ (+) represents the order parameter expectation value hOi , and Ψ ( − ) represents the external source for O . Sincewe are interested in the spontaneous condensate, we setΨ ( − ) = 0 . Similarly, the fast falloff A (1) µ represents the For simplicity, we do not consider the “alternative quantization,”where the role of Ψ (+) and Ψ ( − ) is exchanged [32]. boundary current h J µ i , and the slow falloff A (0) µ repre-sents its source (the external chemical potential µ andvector potential).On the horizon, we impose the regularity condi-tion for a time-independent problem, and we imposethe “incoming-wave” boundary condition for a time-dependent problem. The boundary condition is discussedmore in next subsection. B. Eddington-Finkelstein coordinates
It is convenient to introduce the tortoise coordinate u ∗ and the ingoing Eddington-Finkelstein (EF) coordinatesystem ( v, z ), where du ∗ := − duf , (3.7) v := t + u ∗ , z = u . (3.8)The horizon is located at u ∗ → −∞ . The metric becomes ds p +2 = 1 z (cid:0) − f ( z ) dv − dvdz + d~x p (cid:1) . (3.9)The inverse metric is g vz = − z and g zz = z f .The Maxwell field components in the EF coordinatesystem are related to the ones in the Schwarzschild-likecoordinate system in Eq. (3.4) as A v = A t , (3.10a) A z = A u + A t f , (3.10b)since A t dt + A u du = A v dv + A z dz .The Maxwell-scalar system admits a static solution A v = µ ϕ ( z/z h ) , ϕ ( x ) := 1 − x p − , (3.11a) A z = A i = 0 , (3.11b) Ψ = 0 , (3.11c)where boldface letters indicate background values. How-ever, at the critical point, the Ψ = 0 solution becomesunstable and is replaced by a Ψ = 0 solution. For p = 2, T c ≈ . µ .We approach the critical point from high temperature.When Ψ = 0, the scalar perturbation decouples fromthe Maxwell perturbations, and it is enough to solve thescalar perturbation only. The scalar action in the EF A z = 0 is our gauge choice, and it is different from choosing A u = 0 in the Schwarzschild-like coordinates from Eq. (3.10). coordinate system is given by S Ψ = Z d p +2 x L , (3.12a) L = 1 z p h (cid:0) ∂ v Ψ (cid:1) † ∂ z Ψ + (cid:0) ∂ z Ψ (cid:1) † ∂ v Ψ − δ ij ( ∂ i Ψ) † ∂ j Ψ i − z p " f (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂ z + iA v f (cid:19) Ψ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) m z − A v f (cid:19) | Ψ | . (3.12b)For brevity, we write A M as A M . After integrating byparts, S Ψ = Z d p +2 x L , (3.13a) L = 1 z p h (cid:0) ∂ v Ψ (cid:1) † ∂ z Ψ + (cid:0) ∂ z Ψ (cid:1) † ∂ v Ψ − δ ij ( ∂ i Ψ) † ∂ j Ψ i − Ψ † L Ψ Ψ (3.13b) L Ψ := − (cid:18) ∂ z + iA v f (cid:19) fz p (cid:18) ∂ z + iA v f (cid:19) + 1 z p (cid:18) m z − A v f (cid:19) . (3.13c)The operator L Ψ plays the important role below. Itrepresents the time-independent homogeneous part ofthe equation of motion. Thus, at the critical point,there must exist a nontrivial solution of L Ψ Ψ = 0 withΨ ( − ) = 0 .The EF coordinate system is convenient because theboundary condition at the horizon is simple. One of-ten uses the Schwarschild-like coordinates and imposesthe “incoming-wave” boundary condition at the horizon.In the incoming EF-coordinates, the boundary conditionreduces to the regularity condition. In the near-horizonlimit z → z h , the Lagrangian becomes L ∝ − ( ∂ v Ψ) † ∂ ∗ Ψ − ( ∂ ∗ Ψ) † ∂ v Ψ + Ψ † ∂ ∗ Ψ , (3.14)and the equation of motion becomes ∂ ∗ (2 ∂ v + ∂ ∗ )Ψ ∼ . (3.15)There are two solutions. The solution of ∂ ∗ Ψ ∼
0, namelyΨ = Ψ( v ) is the incoming-wave, and the other one is theoutgoing-wave.In the EF-coordinates, the boundary condition be-comes simple. This allows us to implement the holo-graphic scaling analysis directly at the action level. Inthe Schwarzschild-like coordinates, this structure is man-ifest only after one imposes the “incoming-wave” bound-ary condition explicitly.The action (3.13) allows the obvious “ z = 1” scaling v → av , x i → ax i , z → az . (3.16) Actually, the iA v -dependence in L Ψ can be gauged away, but wekeep this form (until Sec. IV D) so that the A v -dependence inthe action (3.13) is contained only in L Ψ . Under the scaling, the horizon radius and the chemicalpotential scale as z h → az h , µ → µ/a , (3.17)but this does not change physics since the systemis parametrized by the dimensionless parameter ¯ µ := z h µ ∝ µ/T . The scaling comes from the underlying AdSblack hole geometry (3.9). Then, one would naively ex-pect z = 1, i.e. , τ ∼ ξ , but at the critical point, oneactually has z = 2 as we see below. This is because ofthe emergent scaling at the critical point. At the criticalpoint, L Ψ Ψ = 0. This allows the scaling v → av , x i → a / x i , z → z . (3.18) IV. HOLOGRAPHIC SCALING ARGUMENT
Following the spirit of the TDGL scaling argument inSec. II, consider the scaling˜ v = a v , ˜ x i = b x i , ˜ z = c z , (4.1)in the scalar action Eq. (3.13). • In general, one would allow a Ψ-scaling like theTDGL analysis (2.4), but our scalar action isquadratic in Ψ, so the Ψ-scaling is irrelevant in thediscussion below. • The scaling c can be set to a desired value withoutloss of generality, and we choose convenient onesbelow. • Only constant scalings are considered to keep a sim-ple scaling property of the “kinetic term” [the firstline of Eq. (3.13b)].
A. Assumptions
In the TDGL analysis, one can determine the ( T − T c )-dependence. A parallel discussion becomes possible byutilizing some generic properties of the eigenvalue prob-lem L Ψ Ψ = λ Ψ . (4.2)The eigenvalue λ (¯ µ ) depends on ¯ µ := z h µ ∝ µ/T . Weimpose the boundary conditions where Ψ ∼ z ∆ + asymp-totically and is regular at the horizon. We impose thefollowing assumptions on this eigenvalue problem:1. Under our boundary conditions, we assume that L Ψ has a discrete and positive spectrum above T c (or below ¯ µ c ).2. Denote the lowest eigenvalue of L Ψ as λ (¯ µ ). Asone lowers T , a nontrivial source-free solution of L Ψ Ψ = 0 first appears at the critical point ¯ µ = ¯ µ c ,so λ (¯ µ c ) = 0. 3. The dynamics of Ψ is governed by the λ -eigenmode near the critical point .For the moment, we take these assumptions for granted,but we justify Assumptions 1 and 2 later. The (gauge-equivalent) operator is written as L g Ψ = L Ψ (¯ µ = 0) − A v z p f . (4.3)We will show that L Ψ (¯ µ = 0) has positive-definite eigen-values if m satisfies the Breitenlohner-Freedman (BF)bound [33]. But the Maxwell field contribution is neg-ative: this decreases the L Ψ -eigenvalues when one in-creases ¯ µ , and L Ψ develops a zero eigenvalue at the crit-ical point. B. Critical exponents from scaling
In this subsection, we determine critical exponents ν and z . We first choose c . To determine critical exponents,it is enough to consider the static background wherethe temperature ( ∼ /z h ) is constant, and we choose c = 1 /z h . Then, the coordinate ˜ z is dimensionless. Thehorizon is located at ˜ z h = 1, and f = 1 − ˜ z p +1 .The scaled action then becomes S Ψ = 1( z h b ) p Z d p +2 ˜ x ˜ L , (4.4a)˜ L = 1˜ z p h (cid:0) ∂ ˜ v Ψ (cid:1) † ∂ ˜ z Ψ + (cid:0) ∂ ˜ z Ψ (cid:1) † ∂ ˜ v Ψ − z h b a δ ij (cid:0) ˜ ∂ i Ψ (cid:1) † ˜ ∂ j Ψ (cid:21) − Ψ † (cid:18) ˜ L Ψ z h a (cid:19) Ψ , (4.4b)˜ L Ψ = − (cid:18) ∂ ˜ z + i ¯ µ ϕ (˜ z ) f (˜ z ) (cid:19) f ˜ z p (cid:18) ∂ ˜ z + i ¯ µ ϕ (˜ z ) f (˜ z ) (cid:19) + 1˜ z p (cid:26) m ˜ z − ¯ µ ϕ (˜ z ) f (˜ z ) (cid:27) . (4.4c)The scaled action reduces to the original action if onechooses z h a = ( z h b ) , (4.5a) z h a = 1 . (4.5b)Note that the second condition comes from the ˜ L Ψ -term.These two conditions give a = b = 1 /z h , but it is just theobvious scaling (3.16).However, ˜ L Ψ has a zero eigenvalue at the critical point, i.e. , ˜ L Ψ Ψ = 0. Then, the story is different, and theanisotropic “ z = 2” scaling a ∝ b is allowed since onedoes not have to impose Eq. (4.5b). Reference [28] also emphasizes the importance of the zero eigen-value mode when one discusses scaling properties in a quenchproblem.
Instead, from our Assumption 3,Ψ † ˜ L Ψ z h a Ψ ∼ λ (¯ µ ) z h a Ψ † Ψ , (4.6)near the critical point, where Ψ is the λ -eigenmode.So, one can choose z h a = λ (¯ µ ) . (4.7)In rescaled variables, the system is away from the criticalpoint. Let the correlation length ˜ ξ and the relaxationtime ˜ τ in rescaled variables be ˜ ξ ∼ O (1) and ˜ τ ∼ O (1).Then, in original variables, τ = a − ˜ τ ∝ λ − (¯ µ ) , (4.8a) ξ = b − ˜ ξ ∝ a − / ∝ λ − / (¯ µ ) , (4.8b) → τ ∝ ξ . (4.8c)The correlation length and the relaxation time diverge atthe critical point as expected, and we obtain z = 2.Note that the lowest eigenvalue λ plays the role of ǫ ∝ T − T c in the TDGL scaling argument in Sec. II.In both arguments, we eliminate the dependence of thedeviation from the critical point by the scaling, and thescaling determines the critical exponents.Denote the deviation from the critical point as ǫ µ := 1 − ¯ µ ¯ µ c . (4.9)The static exponent ν defined by ξ ∝ | ǫ µ | − ν can be de-termined as follows. Expand ˜ L Ψ around ¯ µ c in the ǫ µ -expansion. The operator becomes L Ψ (¯ µ c ) at the leadingorder, so it vanishes for Ψ . At next order, the operatoris proportional to ǫ µ ( c.f. , see the next subsection. Wewill carry out such an expansion.) Thus, λ (¯ µ ) ∝ | ǫ µ | ,and ξ ∝ λ − / ∝ | ǫ µ | − / , (4.10)namely ν = 1 /
2. Put differently, the ¯ µ -dependence in L Ψ is regular around ¯ µ c , which allows the ǫ µ -expansion.This is the essential reason why ν = 1 / C. Holographic KZ scaling
Now, consider the quench case. The system isparametrized by µ/T , but a time-dependent chemical po-tential is physically meaningless, so we consider the time-dependent black hole temperature. Consider the quenchprotocol as z h ( v ) = z h,c (cid:26) v ) (cid:18) | v | τ Q (cid:19) n (cid:27) (4.11)= z h,c { − ǫ µ ( v ) } , (4.12) where z h,c is the horizon radius at the critical point. ǫ µ ( v )represents the deviation from the critical point: ǫ µ ( v ) := 1 − z h ( v ) z h,c . (4.13)One should actually consider a dynamical black holeand would change the black hole temperature T , or thehorizon radius z h dynamically. Instead, we keep using thebackground (3.9) with z h = z h ( v ). The background is nolonger an exact solution; the Einstein equation gets cor-rections of O ( ∂ v z h ). In principle, one can construct a dy-namical solution in the expansion of ∂ v z h . But our back-ground remains a good approximation when the quenchis slow enough. This is because ∂ v z h ∝ z h,c /τ Q (4.14)for our protocol. A similar remark also applies to theMaxwell solution (3.11). Alternatively, for the Maxwellsolution, one can use an exact solution in the background(3.9) with z h = z h ( v ) .In previous subsection, we chose c = 1 /z h . In thequench case, z h = z h ( v ), and we should not choose c =1 /z h ( v ); this would change the form of the kinetic term(3.13b). In the action (3.13), the horizon radius z h ( v )appears only in f and ϕ ; it appears in the form of zz h ( v ) = zz h,c − ǫ µ ( v ) = ˜ zc z h,c − ǫ µ ( v ) . (4.15)So, in this case, it is convenient to choose c = 1 /z h,c .Again choose the scaling z h,c a = ( z h,c b ) and rewrite thescaled action in the ǫ µ -expansion. Then, we eliminate the τ Q -dependence (in ǫ µ ) by choosing a appropriately.Then, the scaled action becomes S Ψ = 1( z h,c b ) p Z d p +2 ˜ x ˜ L , (4.16a)˜ L = 1˜ z p h (cid:0) ∂ ˜ v Ψ (cid:1) † ∂ ˜ z Ψ + (cid:0) ∂ ˜ z Ψ (cid:1) † ∂ ˜ v Ψ − δ ij (cid:0) ˜ ∂ i Ψ (cid:1) † ˜ ∂ j Ψ i − Ψ † (cid:18) ˜ L Ψ z h,c a (cid:19) Ψ , (4.16b) Consider a time-dependent chemical potential µ ( v ) instead ofconstant µ . This is not physical, and we just use a gauge degreeof freedom. Then, A v = cz h ( v )(1 − z/z h ( v )) is a solution. Inthis case, µ ( v ) ∝ ¯ µ c { − ǫ µ ( v ) } , and one has to take into accountthe ǫ µ -correction in the following discussion. The scaled action(4.16c) takes a slightly different form, but Eq. (4.16d) remainsvalid. ˜ L Ψ = − (cid:18) ∂ ˜ z + i ¯ µ c ϕ ( z/z h ) f ( z/z h ) (cid:19) f ˜ z p (cid:18) ∂ ˜ z + i ¯ µ c ϕ ( z/z h ) f ( z/z h ) (cid:19) + 1˜ z p (cid:26) m ˜ z − ¯ µ c ϕ ( z/z h ) f ( z/z h ) (cid:27) (4.16c)= − (cid:18) ∂ ˜ z + i ¯ µ c ϕ (˜ z ) f (˜ z ) (cid:19) f (˜ z )˜ z p (cid:18) ∂ ˜ z + i ¯ µ c ϕ (˜ z ) f (˜ z ) (cid:19) + 1˜ z p (cid:26) m ˜ z − ¯ µ c ϕ (˜ z ) f (˜ z ) (cid:27) + O ( ǫ µ ( v )) , (4.16d)where we expand f and ϕ in ǫ µ in Eq. (4.16d). The firstand the second lines of Eq. (4.16d) are ˜ L Ψ at the criticalpoint, so it vanishes for Ψ . Thus, near the critical point,˜ L Ψ z h,c a Ψ ∼ ǫ µ ( v ) z h,c a Ψ = 1 z h,c a (cid:18) | v | τ Q (cid:19) n Ψ (4.17)= z nh,c ( z h,c a ) n +1 (cid:18) | ˜ v | τ Q (cid:19) n Ψ . (4.18)The τ Q -dependence can be eliminated by choosing a as z h,c a = (cid:18) z h,c τ Q (cid:19) n/ ( n +1) . (4.19)In rescaled variables, the relaxation time ˜ τ KZ and thecorrelation length ˜ ξ KZ do not depend on τ Q . Then, inoriginal variables, τ KZ ∝ a − ∝ τ n/ ( n +1) Q , (4.20a) ξ KZ ∝ b − ∝ a − / ∝ τ n/ n +1) Q . (4.20b)This agrees with the TDGL analysis (2.15).To be precise, in the above discussion, Ψ is the λ -eigenmode at v = 0. We impose the regularity boundarycondition at the dynamical horizon z = z h ( v ). Thus, theboundary condition is imposed at a different position fora different v . See Appendix A for more details. D. Checking assumptions
Let us go back to Assumptions 1 and 2 in Sec. IV Aand justify them. We consider the eigenvalue problem L Ψ (¯ µ )Ψ = λ (¯ µ )Ψ , (4.21a) L Ψ := − (cid:18) ∂ z + iA v f (cid:19) fz p (cid:18) ∂ z + i A v f (cid:19) + V Ψ z p , (4.21b) V Ψ := m z − A v f . (4.21c)What one can show is that1. λ N := λ (¯ µ = 0) > m satisfies the BF bound.2. The lowest eigenvalue λ (¯ µ = 0) is less than λ N := λ (¯ µ = 0) because of the A v -term in V Ψ . First, the operator L Ψ is hermitian with respect to Ψwhich satisfy our boundary conditions, so the eigenval-ues are real. The spectrum is also discrete as a two-boundary-value problem.It is easier to reduce to an equivalent eigenvalue prob-lem. First, the iA v -dependence in L Ψ can be gaugedaway by Ψ = U Ψ g where U := exp( − i R dz A v /f ): L g Ψ Ψ g := ( U † L Ψ U )Ψ g = λ Ψ g , (4.22) L g Ψ = − ∂ z fz p ∂ z + V Ψ z p . (4.23)Further define a new variable ψ as Ψ g = G ( z ) ψ . Thefunction G is chosen below. Then, the problem reducesto L ψ ψ := ( G L g Ψ G ) ψ = λG ψ . (4.24)Showing λ N > λ = R dz ψ † L ψ ψ R dz G | ψ | (4.25)is positive-definite for ¯ µ = 0. In the variable Ψ, thepotential V Ψ has the m < µ = 0. But in the new variable ψ ,the positivity can be seen explicitly. For simplicity, ψ isnormalized as R dz G | ψ | = 1 . By integrating by parts, Z dz ψ † L ψ ψ = Z dzz p (cid:20) f G | ∂ z ψ | + V ψ G | ψ | (cid:21) + (surface term) . (4.26)The kinetic term is positive-definite. The surface termmakes no contribution under our boundary conditions.The new potential term is given by V ψ := m z − z p G (cid:18) f G ′ z p (cid:19) ′ − A v f . (4.27)Choose a polynomial form of G ( z ) = z α/ with constant α . Then, V ψ = 1 z (cid:26) − ( α − p − (cid:18) p + 12 (cid:19) + m + α (cid:18) zz h (cid:19) p +1 (cid:27) − A v f . (4.28)By choosing α = p + 1, V ψ (¯ µ = 0) is positive-definiteif m > − ( p + 1) / i.e. , if m satisfies the BF bound.Thus, the integral (4.26) is positive-definite for ¯ µ = 0,and λ N > µ = 0 problem. Since V ψ < V ψ (¯ µ =0), λ < λ N . For convenience, we repeat the argumentin Ref. [34]. Let the lowest eigenfunctions of L ψ (¯ µ ) and L ψ (¯ µ = 0) be ψ and ψ N . Then, λ N = Z dz ψ † N L ψ (¯ µ = 0) ψ N > Z dz ψ † N L ψ (¯ µ ) ψ N ≥ Z dz ψ † L ψ (¯ µ ) ψ = λ (¯ µ ) , (4.29)where the variational argument is used in the last line.Although λ N > λ (¯ µ ) may not be because of theMaxwell field contribution. It is then natural to expectthat λ (¯ µ ) remains positive for a small ¯ µ , but λ (¯ µ ) tendsto decrease as one increases ¯ µ , and λ = 0 for a largeenough ¯ µ c , which is the critical point. V. RELATION TO PREVIOUS WORKS
Our work is related to various previous works, and it isworthwhile to mention them and to compare with thembriefly.Reference [17] obtained critical exponents for holo-graphic superconductors analytically. The analysis is car-ried out in the Schwarzschild-like coordinates and in themomentum space. But, in this paper, we are interested indefect formations, or solutions which are inhomogeneousin boundary spatial directions, so it is more appropriateto work in the real space. In any case, the main pointsof the paper are (i) the existence of a nontrivial solutionwith Ψ ( − ) = 0 at the critical point, or L Ψ Ψ = 0, and (ii)after imposing the boundary condition at the horizon,the scalar equation of motion has O ( ω ) and O ( q ) terms.More explicitly, the paper considers the solution of theform Ψ = Ψ ω,q ( u ) e − iωt + iqx . In order to implement the“incoming-wave” boundary condition, write the solutionas Ψ ω,q ( u ) =: (1 − u ) − iω/ (4 πT ) ϕ ω,q ( u ) . (5.1)Then, the ϕ equation has O ( ω ) and O ( q ) terms. Thisallows us to expand ϕ as ϕ ω,q = ϕ + ωϕ (1 , + q ϕ (0 , + · · · . The main objet computed in the paper is the “orderparameter response function” χ ω,q given by χ ω,q ∝ Ψ (+) ω,q Ψ ( − ) ω,q (5.2a) ∝ ϕ (+)0 + ωϕ (+)(1 , + q ϕ (+)(0 , + · · · ϕ ( − )0 + ωϕ ( − )(1 , + q ϕ ( − )(0 , + · · · (5.2b) ∼ ϕ (+)0 ϕ ( − )(0 , − i Γ ω + q + ξ , (5.2c) where
1Γ := i ϕ ( − )(1 , ϕ ( − )(0 , , ξ := ϕ ( − )0 ϕ ( − )(0 , . (5.3)From Eq. (5.2c), one obtains ( γ, ν, η, z ) = (1 , / , , , where the point (i), namely ϕ ( − )0 | T c = 0, ϕ (+)0 | T c = 0 isessential.The quench in this paper is cooling, the standardquench discussed in the context of the KZ mechanism.Such a quench is called a “thermal quench.” This quenchis added by the time-dependent black hole temperature T ( t ). But a different type of quench is discussed in theliterature. They typically consider the time-dependentsource Ψ ( − ) ( t ) for the order parameter. Such a quench iscalled a “source quench.”For example, Das and his collaborators consider sourcequenches at T = 0 [28–30]. Their analysis does not ad-dress defect formations for two reasons. First, the bound-ary theory is spatially homogeneous. Second, a sourcequench drives a spontaneously symmetry breaking sys-tem to an explicit symmetry breaking system. We con-sider the second-order phase transition and defect forma-tions, so it is not appropriate to consider a source quench.Although their problem is not a defect formation prob-lem, it is fine as a quench problem in a broad sense, andthey use a similar scaling argument as ours.A source quench for holographic superconductors isanalyzed numerically in Ref. [31]. Their quench pro-tocol has a Gaussian profile, so it drives the systemto an explicit symmetry breaking one only in a limitedtime. They start from the ordered phase, add the sourcequench, and follow the time-evolution of hO ( t ) i . Thesystem ends up with the disordered phase if the sourcequench is strong enough. ACKNOWLEDGMENTS
We would like to thank Takeshi Morita for useful dis-cussions. This research was supported in part by a Grant-in-Aid for Scientific Research (23540326 and 17K05427)from the Ministry of Education, Culture, Sports, Scienceand Technology, Japan. According to an explicit numerical computation, Γ is complex,so the order parameter is not purely diffusive [17].
Appendix A: Time-dependent boundary conditionand its effect
We impose the boundary condition at the dynamicalhorizon ˜ z = z h ( v ) /z h,c = 1 − ǫ µ ( v ). The boundary condi-tion thus has the O ( ǫ µ )-dependence. Consequently, ˜ L Ψ -eigenfunctions differ for a different v . The λ -eigenmodesget only O ( ǫ µ )-corrections, however. Namely, Ψ ( v ) ∼ Ψ ( v = 0) + O ( ǫ µ ), and Eq. (4.17) remains valid.Alternatively, one can modify the argument so that theboundary condition has no explicit ǫ µ -dependence. Then,the ǫ µ -dependence is contained entirely in the operator˜ L Ψ , and the eigenfunctions remain the same for all v . Todo so, introduce a new variable σ = z/z h ( v ), and rewrite˜ L Ψ in terms of σ . Then, the boundary condition is alwaysimposed at σ = 1.Introducing σ is not a coordinate transformation. A v -dependent coordinate transformation would spoil theform of the kinetic term (3.13b). Writing in terms of σ is just a convenient way to shift the effect of the time-dependent boundary condition to the operator ˜ L Ψ . Asthe ˜ L Ψ -eigenvalue problem, one can regard v just as anexternal parameter.Then, the effect of the time-dependent boundary con- dition is incorporated in the scaled action ˜ L Ψ as˜ L Ψ = (cid:18) z h,c z h (cid:19) p +2 " − (cid:26) ∂ σ + i (cid:18) ¯ µ c z h z h,c (cid:19) ϕ ( σ ) f ( σ ) (cid:27) f ( σ ) σ p (cid:26) ∂ σ + i (cid:18) ¯ µ c z h z h,c (cid:19) ϕ ( σ ) f ( σ ) (cid:27) + 1 σ p ( m σ − (cid:18) ¯ µ c z h z h,c (cid:19) ϕ ( σ ) f ( σ ) ) (A1a)= (cid:18) z h,c z h (cid:19) p +2 " − (cid:26) ∂ σ + i ¯ µ c ϕ ( σ ) f ( σ ) (cid:27) f ( σ ) σ p (cid:26) ∂ σ + i ¯ µ c ϕ ( σ ) f ( σ ) (cid:27) + 1 σ p (cid:26) m σ − ¯ µ c ϕ ( σ ) f ( σ ) (cid:27) + O ( ǫ µ ( v )) , (A1b)where we expand ˜ L Ψ in ǫ µ as in Eq. (4.16d). The prefac-tor ( z h,c /z h ) p +2 can also be expanded in ǫ µ ( v ). It is notnecessary however since the first term vanishes for Ψ atthe critical point.The rest of the discussion remains the same as the text,and we get Eq. (4.17). [1] J. M. Maldacena, “The Large N limit of superconfor-mal field theories and supergravity,” Int. J. Theor. Phys. (1999) 1113 [Adv. Theor. Math. Phys. (1998) 231][hep-th/9711200].[2] E. Witten, “Anti-de Sitter space and holography,” Adv.Theor. Math. Phys. (1998) 253 [hep-th/9802150].[3] E. Witten, “Anti-de Sitter space, thermal phase transi-tion, and confinement in gauge theories,” Adv. Theor.Math. Phys. (1998) 505 [hep-th/9803131].[4] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gaugetheory correlators from noncritical string theory,” Phys.Lett. B (1998) 105 [hep-th/9802109].[5] J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopaland U. A. Wiedemann, Gauge/String Duality, Hot QCDand Heavy Ion Collisions (Cambridge Univ. Press, 2014)[arXiv:1101.0618 [hep-th]].[6] M. Natsuume,
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