Far-from-equilibrium coarsening , defect formation , and holography

Abstract We study a thermal quench across a continuous transition from the disordered phase to the ordered phase. We develop a theoretical framework for early-time coarsening based on the spectrum of unstable modes after the quench. Our analysis unveils a novel period of non-adiabatic evolution after the system passes through the phase transition, where a parametrically large amount of coarsening occurs before a well-defined condensate forms. Our formalism predicts a rate of defect formation parametrically smaller than the Kibble-Zurek prediction and yields a criterion for the break-down of Kibble-Zurek scaling for sufficiently fast quenches. We numerically test our formalism for a thermal quench in a 2 + 1 dimensional holographic superfluid.

Let us briefly review the KZM [28][29][30]. Consider a system with a second order phase transition at temperature T c , below which a symmetry is spontaneously broken and an order parameter ψ develops a condensate. In equilibrium at temperature T > T c the correlation length ξ eq and relaxation time τ eq are related to the reduced temperature ≡ 1 − T Tc by ξ eq = ξ s | | −ν , τ eq = τ s | | −νz , (1.1) for some scales ξ s , τ s and critical exponents ν, z. Consider a quench from T i > T c to T f < T c with quench protocol (t) = t/τ Q , t ∈ (t i , t f ), (1.2) where t i = (1 − T i /T c )τ Q < 0 and t f = (1 − T f /T c )τ Q > 0. The system can respond adiabatically to the change in temperature until τ eq (t) ∼ |t|. This condition defines the freeze-out time and length scale During the interval t ∈ (−t freeze , t freeze ) the evolution of the system is essentially frozen. 1 The correlation length ξ freeze then imprints itself on the state at t = +t freeze . 1 Strictly speaking one should distinguish t > freeze for T > T c and t < freeze for T < T c as they can differ by an O(1) constant. We will suppress such differences for notational simplicities.
The density of topological defects generated across the phase transition can then be estimated as (1.4) where d is the number of the spatial dimensions and D is the number of dimensions of a defect. While the KZM is only supposed to determine the density of defects up to an O (1) factor, it often significantly overestimates the real density of defects observed in numerical calculations: one needs a "fudge" factor f multiplying ξ freeze with f = O(10) [30]. See also [31] for a recent discussion.
One motivation of this paper is to develop a formalism for describing the growth and coarsening of the order parameter after t freeze in a general critical system. Our analysis stresses a period of non-adiabatic evolution, before a well-defined condensate forms, where the system coarsens and the correlation length grows parametrically larger than ξ freeze . In particular, we show that in many systems, including conventional superconductors and superfluid 4 He, there could be a large logarithmic hierarchy between t freeze and the time scale we refer to t eq when one can sensibly measure the density of defects. Thus our analysis reconciles the need for a "fudge" factor f . Moreover, our analysis yields a new criterion for the breaking of the KZ scaling (1.4). Our discussion can also be applied without essential changes to quantum phase transitions. For definiteness, we will restrict discussion to thermal phase transitions throughout the paper.
A second motivation of this paper is to test the scaling predicted by the KZM and its refinement in strongly coupled systems using holographic duality. Holography equates certain systems of quantum matter without gravity to classical gravitational systems in one higher spatial dimension [32][33][34]. Hence complicated many-body physics can be mapped onto a solvable numerical gravity problem. Some examples include [35][36][37][38][39][40][41][42][43][44] (see also [45,46] for a discussion of KZM for a holographic quantum quench). In this paper we study the KZM in a holographic superfluid in 2 + 1 spacetime dimensions. Our gravity calculation will provide a first check of KZ scaling in a strongly coupled system without quasiparticles and will verify key features of the coarsening physics discussed in the next section.
Note: Independently, Sonner, del Campo and Zurek [47] have found universal scaling behavior in the dynamics of strongly-coupled superconductors with a holographic dual.

A. Unstable critical modes
We now develop a formalism to describe a period of non-adiabatic growth of the order parameter ψ after t freeze . The seeds for condensate growth come from thermal and quantum fluctuations, whose effects on the macroscopic evolution of ψ can be described in terms of a stochastic source ϕ for ψ. In the IR the statistics of the fluctuations in ϕ read where ζ is a (weakly) temperature dependent constant.
Let ψ(t, q) and ϕ(t, q) be the Fourier transformed order parameter and noise respectively.
At early times ψ(t, q) is small and can be described by linear response, where G R (t, t , q) is the retarded ψ correlator. Statistical homogeneity and isotropy imply G R only depends on q = |q|. The regime of validity for the linear response will be discussed below. To elucidate the growth of ψ and to extract the time evolution of the correlation length after t freeze , we study the evolution of the correlation function Averaging over the noise (2.1) we find As the dynamics is essentially frozen between (−t freeze , t freeze ), at t ∼ t freeze , the system is in a supercooled state for which the leading time dependence of G R can be obtained by analytically continuing to below T c the equilibrium retarded correlator G eq above T c . Close to T c the time dependence of G eq (t, q) should be dominated by the leading pole w 0 (q) (the critical mode) of G eq (ω, q) in the complex frequency plane, i.e.
G eq (t, q) = θ(t)H(q)e −iw 0 ( ,q)t , w 0 ( , q) = zν h(q −ν ), (2.5) where H(q) is some function which depends weakly on q. h(x) is a universal scaling function which is analytic in x 2 for small x. For T > T c , w 0 (q, T ) lies in the lower half ω-plane, and its imaginary part at q = 0 gives the inverse of the relaxation time. 2 When continued to a supercooled state at T < T c , w 0 moves to the upper half frequency plane for q smaller than a certain q max , and for such q's (2.5) grows exponentially with time. More explicitly, for positive we can expand Im w 0 in small q as where a and b are positive constants. Hence Im w 0 > 0 until q ∼ q max with For slow quenches in which (t) → 0 the retarded correlator must be given by The earliest time when (2.8) can be applied is when |∂ t log w 0 | < |w 0 | and thus t > t freeze .
Substituting (2.8) into (2.4) we then secure The · · · in (2.9) denotes the contributions in (2.4) coming from the integration domain t < t freeze , which will be neglected in our discussion below as the first term in (2.9) grows exponentially with time and will soon dominate. 3 Let us consider the behavior of the above integral for t parametrically large compared to t freeze assuming for the moment that the linear analysis holds. For this purpose it is convenient to introducē (2.10) In the regimet 1, we find for qξ freeze 1 (see Appendix A for details), In the language of the dual gravitational description discussed below, w 0 is the lowest quasinormal mode frequency of a dual black hole. 3 In addition to the exponential suppression in time, when Fourier transformed to real space the omitted terms in (2.9) also fall off parametrically faster with distance than the first term. and a 1 , a 2 , a 3 are O(τ 0 Q ) constants. Fourier transforming q to coordinate space we find (2.15) Equations (2.11)-(2.15) are our main results of this section. We now proceed to discuss their physical meaning and physical implications.
Introducing a scale t eq by requiring we expect the linear analysis to break down for t ∼ t eq . In particular, for t t eq , we expect the condensate growth to transition from the exponential growth of (2.13) to the adiabatic growth governed by (2.16) with in (2.16) given by the time-dependent reduced temperature (1.2). Moreover, the system does not contain a well-defined number of topological defects until a well defined condensate forms which necessarily lies outside the domain of linear response. Thus t eq is also the natural time scale to measure the density of topological defects.
To estimate t eq , it is convenient to introduce the ratio In this case there is no hierarchy of scales between t freeze and t eq and the condensate begins to grow adiabatically after t freeze . In other words, in this case our analysis reduces to the standard story of the KZM and the density of topological defects is given by (1.4). When R 1 there is, however, a hierarchy between t eq and t freeze , and (2.13) applies over a parametrically large interval of time during which the condensate grows with time exponentially, and the coarsening length co (t), which controls the typical size of a condensate droplet, grows with time as a power. In particular in the limit R → ∞, from (2.13), (2.17) and (2.12) we see and Thus for R 1 a parametrically large amount of coarsening occurs before a well-defined condensate even forms. The density of topological defects of dimension D is then (using (2.12)) As a result of early-time coarsening, the defect density ρ is parametrically much smaller than Kibble-Zurek prediction ρ KZ and the standard KZ scaling is corrected by a logarithmic prefactor. Possible systems with R 1 will be further discussed in the conclusion section.
We stress that the time dependence of (2.13) differs from the scaling behavior of standard coarsening physics [48], which applies only after the magnitude |ψ| has achieved its equilibrium value. The possible importance of early-time coarsening physics in the KZM has recently also been discussed in [31], but it assumed the scaling behavior of standard coarsening physics and thus is not compatible with our result.

C. Rapid quenches
By decreasing τ Q (while keeping T i , T f fixed), eventually the scaling (2.21) for the defect density must break down. In standard KZ discussions, this should happen when t f t freeze .
Here we point out that for systems with t eq t freeze , the scaling (2.21) breaks down for t f t eq , and can happen even for t f parametrically much larger than t freeze . This is easy to understand; for t eq t f t freeze , since the system stays at T f after t f , the growth of the condensate will largely be controlled by the unstable modes at T f , and the defect density will be determined by T f rather than τ Q . We now generalize the above discussion of far-from-equilibrium coarsening to such a case, where equation (2.9) should be modified to where as commented below (2.9), · · · denotes contributions from earlier times which can be results in a simple e t growth for any ν, z (compare with (2.11)). Fourier transforming the above expression, then C(t, r) can be written in a scaling form (see Appendix A for details) for some scaling function f . For νz f (t − t freeze ) 1 and r ν f 1 (assuming linear response still applies), f can be obtained explicitly and one finds Note that in comparing with (2.11)-(2.12), we see that both the logarithm of the condensate square and the coarsening length square grow linearly with time.
Parallel to the earlier discussion, we postulate that the linear response analysis breaks down when the condensate squared obtained from (2.24) becomes comparable to |ψ| 2 eq . To estimate the time scale t eq when this happens, it is again convenient to introduce 26) and the criterion for linear response to apply for νz f (t − t freeze ) 1 is again R f 1. In particular, the equilibrium time t eq and the density of defects should be given by Clearly ρ is independent of τ Q .
For very fast quenches, i.e. t f t freeze , the whole quench from T i to T f will be nonadiabatic. In such a case, at the end of quench, the system will have correlation length G N of the bulk gravity such that G N ∼ 1 N 2 ; the classical gravity approximation in the bulk thus corresponds to the large N limit in the boundary theory. Finite temperature in the boundary system is described on the gravity side by a black hole. In the large N limit, thermal and quantum fluctuations are suppressed by 1/N 2 and on the gravity side are encoded in quantum gravity effects induced from the black hole's Hawking radiation.
In this paper we consider a holographic superfluid phase transition in two spatial dimension with relevant topological defects being point-like vortices. In the large N limit, the phase transition has mean field critical exponents with ν = 1 2 , z = 2, β = 1 2 , and ζ in (2.1) of order O( 1 N 2 ). For such a system, the predictions from the KZM for density of superfluid vortices read Strictly speaking, the above discussion applies to unstable q modes with q −1 < ξ i will be averaged out and only those modes with q −1 > ξ i can grow.
Applying the discussion of last section to such large N theories, we can make the following predictions: 1. For slow quenches, i.e. quenches with t f t eq , with d = 2 and mean field exponents, equation (2.13) becomes (t = t/t freeze ) Furthermore, from (2.18) we find Λ = −1 and thus (3.3) In the large N limit, we always have R 1 and from (2.19) and (2.21) f is always much greater than 1 in the large N limit, and we have from (2.27) Note that both quantities above are independent of τ Q .

B. Numerical results
We have performed numerical simulations of thermal quenches across a second order phase transition of a 2 + 1 dimensional holographic superfluid. We employ the linear quench (1.2) which in the gravity context translates into a black hole with a time dependent temperature.
Instead of directly computing fluctuations from Hawking radiation (see e.g. [49,50]), we model fluctuations from quantum gravity effects as a random noise that enters as a nontrivial boundary condition in the gravity equation of motion. In such a formulation, ζ can taken as an adjustable parameter which we take to be numerically small so as to imitate the O(1/N 2 )  We begin our analysis by studying the normalized average order parameter The sum over i represents an ensemble average over M configurations at fixed τ Q . The time evolution of A(t) for various τ Q are given in Fig. 1. The left and middle plots correspond to slow quenches where we see all curves experience a period of rapid growth after t freeze followed by a period of approximate linear growth. We operationally define t freeze as the time at which . The rapid growth can be identified with the regime described by (2.13) and (3.2) as indicated by the middle plot. The linear growth can be identified as the regime of adiabatic condensate growth. To see this note that for mean field |ψ| 2 eq ∼ (t) = t τ Q implying A(t) ∼ t/τ Q for adiabatic growth. This conclusion is supported by the slope of the linear growth, the observation that the termination of the linear condensate growth coincides with the end of the quench, and that when extrapolated to t = 0 the linear curves have A = 0.
The crossover from exponential to linear growth corresponds to the equilibration time t eq (2.17), which we operationally defined as the time in which A (t eq ) < 0.1 max {A (t)}.
A key feature of the middle plot of Fig. 1 is that curves of different τ Q all lie top of one another when we plot them in terms of scaling variablest 2 = (t/t freeze ) 2 . In particular, the lineart 2 growth in the logarithmic plot agrees very well with the prediction of (3.2). The right plot describes fast quenches discussed in Sec. II C, with all the qualitative features of (3.6) confirmed numerically, namely, e t growth as compared with the e t 2 growth of slow quenches, and all curves of different τ Q lying on top of one another when plotted v.s. t−t freeze .
For such a "rapid" quench, the growth of condensate and the resulting defect density are dictated by T f and are independent of τ Q . This expectation is also borne out in Fig. 2 where we plot the freeze-out time t freeze and equilibration time t eq as a function of τ Q . While for large τ Q , their behavior is consistent with √ τ Q scaling, for rapid quenches, t eq approaches a constant. The right panel of Fig. 2 also shows that our numerical results are consistent with the presence of a logarithmic hierarchy between the two time scales as predicted in (3.4).
In Fig. 3 we plot the time evolution of |ψ(t, x)| 2 /|ψ(t = ∞, x)| 2 for two values of τ Q at various times up to t = t eq . These plots help to visualize the key point that before t eq when a relative uniform |ψ| 2 has not formed one cannot sensibly count defects. Moreover, it is evident that the defect density is higher for the faster quench.
To quantify the time evolution of coarsening and smoothing of the condensate we numerically compute the correlation function C(t, r) by computing the average in (2.3) over an ensemble of solutions at fixed τ Q . The results are in Fig. 4, where we also present the full width half max ξ FWHM (t) of C(t, r). Before t freeze , ξ FWHM is dominated by the fluctuations (4) and is constant. After t ≈ t freeze , ξ FWHM experiences a period of rapid growth which is consistent with our prediction (3.2) including the scaling behavior. Note ξ FWHM (t eq ) is significantly larger than ξ FWHM (t freeze ), which highlights the importance of the "fudge" factor needed to account for the correct defect density. This is in line with our expectation from Eqs. (2.12) and (3.4).
Finally, in the left panel of Fig. 5 we show that for slow quenches our numerical results reproduce the KZ scaling of the number of vortices N vortices . For τ Q < 200 our numerics are consistent with N vortices = const. This is the expected behavior from our discussion of the breakdown the KZ scaling in the preceding section: the density of defects should asymptote to a constant in the limit of sudden quenches. For such rapid quenches, the right plot (lower) at times t = t freeze , t = 0.7 t eq , t = 0.85t eq and t = t eq . The key message is that we can sensibly talk about defect density only after t eq . At t = t freeze the order parameter is very small and dominated by fluctuations. These fluctuations seed droplets of condensate, whose subsequent causal connection can be seen at time t = 0.7t eq . At such a time, the droplets are still separated by large regions where there is no condensate. Subsequently, the droplets expand and grow in amplitude and the system becomes smoother and smoother. By time t eq the droplets have merged into a comparatively uniform condensate with isolated regions where ψ = 0. The non-uniformities -the localized blue "dots" -are superfluid vortices with winding number ±1.
confirms the scaling of the defect density with f as predicted in (2.27) and (3.8). For both situations, our statistics are not enough to resolve the logarithms predicted in (3.5) and (3.8).
Also included in the left panel of Fig. 5 is a plot of (LB/ξ FWHM (t eq )) 2 where B ≈ 1.92. The fantastic agreement between (LB/ξ FWHM (t eq )) 2 and N vortices for all τ Q bolsters the notion that the vortex density is a measure of the correlation length. Moreover, the observation that B = O(1) and ξ FWHM (t eq ) ξ FWHM (t freeze ) is consistent with our argument that coarsening during the early stages of the evolution can dramatically increases the correlation length the red stars correspond to ξ FWHM (t eq ). At t ≈ t freeze , ξ FWHM starts a period of growth. Note . The collapse of all curves between t freeze and t eq is consistent with the scaling behavior of (3.2).
and decreases the expected density of defects from the KZ prediction (1.4).

IV. CONCLUSION AND DISCUSSION
To summarize, we elucidated a novel period of non-adiabatic evolution after a system passes through a second order phase transition, where a parametrically large amount of coarsening occurs before a well-defined condensate forms. The physical origin of the coarsening can be traced to the fact that when the system passes through the phase transition, IR modes of the order parameter become unstable and exponentially grow. We showed that such a far-from-equilibrium coarsening regime could have important consequences for defect formation. We also numerically simulated thermal quenches in a 2 + 1 dimensional holographic superfluid, which provided strong support for our analytic results.
For slow quenches a key quantity we introduced is R of (2.18) which we copy here for N vortices = const., which is consistent with our expectation that the density of defects should asymptote to a constant in the limit of sudden quenches. Also included is a plot of (LB/ξ FWHM (t eq )) 2 where L is our box size and B ≈ 1.92. Our statistics are not sufficient to resolve the logarithmic prefactor in (3.5). Right: N vortices verses f = (T f ) for sufficiently small τ Q . The results are consistent with (3.8) with N vortices ∼ f . Our statistics are again not sufficient to resolve the logarithmic prefactor in (3.8).
For R 1, there is large hierarchy between t freeze and t eq , and the density of defects can be significantly lower than that predicted by KZ.  [51]. Recall that the Van Hove theory of critical slowing down predicted exactly z = 2 − η. Renormalization group analysis give z slightly greater than this value [52], which means that generically for model A, Λ is only slightly negative, and thus (4.6) essentially translates into ζ 1. As an explicit example, conventional superconductors have a very small ζ and thus we expect them to have a large hierarchy between t freeze and t eq .
For fast quenches the analogous quantity is R f defined in (2.26 Here Greek indices run over boundary spacetime coordinates and r is the AdS radial coordinate with r = ∞ the AdS boundary. With our choice of coordinates lines of constant t represent infalling null radial geodesics affinely parameterized by r. In addition we choose to work in the gauge A r = 0. For simplicity we choose to work in the probe limit e → ∞ where gravitational dynamics decouple from the dynamics of the gauge and scalar fields. The equations of motion following from (B1) are then simply Since the boundary of AdS is time-like, the equations of motion (B4) require boundary conditions to be imposed there. As the boundary geometry of AdS corresponds to the geometry the dual quantum theory lives in, we demand that the boundary geometry be that of flat 2+1 dimensional Minkowski space. This is accomplished by setting lim r→∞ g µν = η µν .
The near-boundary behavior of the gauge and scalar fields can easily be worked by from Eqs. (B4b) and (B4c) and read On the gauge field we impose the boundary condition where µ is a constant. In the dual QFT µ is interpreted as a chemical potential for the conserved U (1) charge. As a final boundary condition we set with ϕ random variable satisfying statistics (2.1). The stochastic driving of the scalar field mimics the effect of quantum and thermal fluctuations induced by the black brane's Hawking radiation. In the dual quantum theory the boundary condition (B8) amounts to deforming the Hamiltonian Note ϕ has mass dimension one and ψ has mass dimension two. In terms of the asymptotic behavior of the scalar field (B6) the boundary order parameter reads Let us first discuss static equilibrium solutions to the set of equations of motion (B4).
Translationally invariant equilibrium solutions to Einstein's equations consist of black branes, where The Hawking temperature T of the black brane is related to the horizon radius r h by and corresponds to the temperature of the dual quantum theory.
Static equilibrium solutions to the scalar-gauge field system (B4c) and (B4b) were first explored in [53]. One static solution to (B4c) and (B4b) (with ε = 0 and hence no stochastic driving) is simply However, for sufficiently low temperatures this solution is unstable and not thermodynamically preferred. For T < T c , where the thermodynamically preferred solution has Φ = 0. Hence the bulk U (1) gauge redundancy is spontaneously broken at low temperatures and the black brane develops a charged scalar atmosphere. Likewise, via (B10) the boundary order parameter is non-zero and the global U (1) symmetry on the boundary is spontaneously broken. The gravitational and boundary systems have a second order phase transition at T = T c with mean-field critical exponents.
To study the Kibble-Zurek mechanism gravitationally we drive the system stochastically with the boundary condition (B8) and choose to dynamically cool the black brane geometry through T c . When the geometry cools through T c the aforementioned instability will result in the scalar field Φ growing and the black brane developing a scalar atmosphere. Likewise, as this happens the boundary QFT condensate (B10) will grow in amplitude.
Instead of solving Einstein's equations (B4a) for a black brane with dynamic temperature, we chose to fix the geometry to be the equilibrium geometry (B11) but with a time dependent temperature T (t) equal to the boundary quench protocol temperature (1.2), which we control. The metric will therefore no longer satisfy Einstein's equations. Why is it reasonable to employ a geometry that does not satisfy Einstein's equations? To answer this question we note that to cool the system through T c we can couple it to an external thermal bath at controllable temperature T ext (t). This can be done by, for example, putting our system in a box of size L and putting the surface of the box in contact with the thermal reservoir. As we are ultimately interested in slow quenches where T ext (t) is parametrically small, we expect thermal equilibration and T (t) ≈ T ext (t). In this limit Einstein's equations can be solved with the gradient expansion of fluid/gravity [54]. At leading order in gradients the solution is simply (B11), but with the time dependent temperature T (t).
Our numerical methods used to solve the scalar/gauge field system (B4c) and (B4b) are outlined in [35]. We use pseudospectral methods and discretize the AdS radial coordinate using 20 Chebyshev polynomials. In the spatial directions we work in a periodic spatial box and discretize using a basis of 201 plane waves. We chose box size LT c = 30.8 and measure all other dimensionfull quantities in units of T c . We choose noise amplitude ζT c = 1.5×10 −3 .
As our quench protocol (1.2) starts off at temperatures T > T c , in the infinite past we choose initial conditions (B14).