Holographic thermal relaxation in superfluid turbulence

Journal of High Energy Physics, Dec 2015

Holographic duality provides a first-principles approach to investigate real time processes in quantum many-body systems, in particular at finite temperature and far-from-equilibrium. We use this approach to study the dynamical evolution of vortex number in a two-dimensional (2D) turbulent superfluid through numerically solving its gravity dual. We find that the temporal evolution of the vortex number can be well fit statistically by two-body decay due to the vortex pair annihilation featured relaxation process, thus confirm the previous suspicion based on the experimental data for turbulent superfluid in highly oblate Bose-Einstein condensates. Furthermore, the decay rate near the critical temperature is in good agreement with the recently developed effective theory of 2D superfluid turbulence.

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Holographic thermal relaxation in superfluid turbulence

HJE Holographic thermal relaxation in super uid turbulence Yiqiang Du 0 1 3 7 8 9 Chao Niu 0 1 3 5 8 9 Yu Tian 0 1 3 6 7 8 9 Hongbao Zhang 0 1 2 3 4 8 9 0 Beijing 100875 , China 1 Beijing 100190 , China 2 Department of Physics, Beijing Normal University 3 Beijing 100049 , China 4 Theoretische Natuurkunde, Vrije Universiteit Brussel and The International Solvay Institutes 5 Institute of High Energy Physics, Chinese Academy of Sciences 6 State Key Laboratory of Theoretical Physics 7 School of Physics, University of Chinese Academy of Sciences 8 Pleinlaan 2 , B-1050 Brussels , Belgium 9 Institute of Theoretical Physics, Chinese Academy of Sciences Holographic duality provides a time processes in quantum many-body systems, in particular at far-from-equilibrium. We use this approach to study the dynamical evolution of vortex number in a two-dimensional (2D) turbulent super uid through numerically solving its Holography and condensed matter physics (AdS/CMT); Black Holes - gravity dual. We nd that the temporal evolution of the vortex number can be well t statistically by two-body decay due to the vortex pair annihilation featured relaxation process, thus con rm the previous suspicion based on the experimental data for turbulent super uid in highly oblate Bose-Einstein condensates. Furthermore, the decay rate near the critical temperature is in good agreement with the recently developed e ective theory of 2D super uid turbulence. 1 Introduction 2 3 4 More on our numerical scheme real experimented sample by hand [3]. Compared to these signi cant experimental developments, our theoretical understanding of dynamics of these quantized vortices is still limited because at nite temperature the e ective dissipative hydrodynamical description for normal uids does not work in the presence of quantized vortices and all the conventional approaches rely on some phenomenological models, which nevertheless have signi cant shortcomings. With this in mind, any ab initio theoretical framework would be greatly desirable. Gratefully, holographic duality provides us with such a satisfactory theoretical framework, in which a complete description of a strongly coupled quantum many-body system, valid at all scales, can be encoded in a classical gravitational system with one extra dimension. In particular, the super uid at nite temperature is dual to a hairy black hole in the bulk and the dissipation mechanism is naturally built in the bulk in terms of excitations absorbed by the hairy black hole. Thus it allows a rst-principles investigation of vortex dynamics by using the dual gravity description of super uid phase. It is recently shown by holography in the seminal work [ 4 ] that although the 2D turbulent super uid kinetic energy spectrum obeys Kolmogorov 5=3 scaling law as it does for turbulent ows in normal uids, the super uid turbulence demonstrates a direct energy cascade towards a short-distance scale set by the vortex core size, in stark contrast to the hydrodynamical argument for the inverse energy cascade of 2D normal { 1 { uid turbulence due to the conservation of enstrophy, which is violated in a super uid by vortex pair annihilation anyhow. Inspired by the above sharp results derived from the holographic principle as well as the aforementioned experimental investigation of thermal relaxation for super uid turbulence, in this paper we shall make such a holographic duality contact closer with the experimental data by initiating a quantitative investigation of temporal evolution of vortex number during the thermal relaxation of super uid turbulence through numerically solving the coupled nonlinear equations of motion of its gravity dual. Remarkably, not only does our holographic result con rm the suspected vortex pair annihilation induced two-body decay rate, but also the decay rate near the critical temperature is in good agreement with the recently developed e ective eld theory description of 2D super uid turbulence in [5].1 We also have reliable results at moderate temperature for the decay rate of vortex number, but more detailed comparison between our holographic simulation and experiments may need more experimental data and new technology due to the drift-out e ect and the deviation of oblate Bose-Einstein condensates from truly 2D systems. 2 Holographic setup The simple holographic model for 2D super uid consists of gravity in asymptotically AdS4 spacetime coupled to a U(1) gauge eld A and a complex scalar eld with charge q and mass m. The corresponding bulk action is given by [6, 7] 1 is required such that classical gravity is reliable, which corresponds to the large N limit of the dual eld systems such as ABJM theory. In addition, we shall work in the probe limit, namely the matter elds decouple from gravity, which can be achieved by taking the large q limit.2 One thus can put the matter elds on top of Schwarzschild black brane background, which can be written in the infalling Eddington coordinates as L2 z2 ds2 = ( f (z)dt2 2dtdz + dx2); where the blackening factor f (z) = 1 ( zzh )3 with z = zh the location of horizon and z = 0 the AdS boundary. The behavior of matter elds is controlled by the equations of motion in the bulk as DaDa m2 = 0; raF ab = i( Db Db ): 1For the seemingly disparity between our holographic result and the result by e ective eld theory, please check the conclusion section for a detailed discussion. 2For notational convenience we shall set the overall factor 16 Gq2 = 1 later on. { 2 { By the holographic dictionary, the dual boundary system is placed at a nite temperature given by T = 3 4 zh ; where a conserved current operator J is sourced by the boundary value of the bulk gauge eld A and a scalar operator O of conformal dimension by the near boundary data of scalar eld . For simplicity but without loss of generality, we shall focus only on the case of m2L2 = 2 in the axial gauge Az = 0, in which the asymptotic solution of A and can be expanded near the AdS boundary as q 94 + m2L2 is sourced A = a + b z + o(z); = variation of renormalized bulk on-shell action with respect to the sources, i.e., HJEP12(05)8 hJ i = hOi = j j2, and the dot denotes the time derivative [8]. hOi = de ned as [ 4 ] When this scalar operator O develops a nonzero expectation value spontaneously in the situation where the source is switched o , the system is driven into a super uid phase with characterizing the super uid condensate.3 Generically such a super uid phase has gapped vortex excitations with the circulation quantized. With the super uid velocity u = ; j = the winding number w of a vortex is determined by (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) where denotes a counterclockwise oriented path surrounding a single vortex. In particular, close to the core of a single vortex with winding number w, the condensate for w > 0 and / (z z0) w for w < 0 with z the complex coordinate and z0 the location / (z z0)w of the core. Thus not only does the magnitude of condensate apparently vanish at the core of a vortex but also the corresponding phase shift around the vortex is given precisely by 2 w. This is the characteristic property of a vortex and will be used as an e cient way to identify vortices in our later vortex counting. To address the vortex pair annihilation in a turbulent super uid by holography, we would rst like to impose the following boundary conditions onto the bulk elds, i.e., > c with c the critical chemical potential for the onset of a homogeneous super uid phase, given by c 4:07. Explicit gravity solutions dual to a static vortex of arbitrary winding number have been numerically constructed in [9{11]. But we are required to prepare an initial bulk con guration for at the Eddington time t = 0 such that the dual initial boundary state includes 300 vortex-antivortex pairs in 100 square box with periodic boundary conditions, where the vortices (each with winding number w = 1) and antivortices (each with winding number w = 1) are randomly placed, mimicking the experimental setup [3]. The detailed construction of the initial bulk con guration for is basically similar to that in the appendix of ref. [ 4 ], with the main di erence that our vortices are placed randomly instead of on a lattice. Due to the asymptotic behavior (2.6), we de ne as usual for our numerical convenience, and the homogeneous equilibrium con guration eq(z) can be easily obtained as in the ordinary holographic super uid. For the single vortex con guration w = g(r) eq(z)eiw (w = 1) in polar coordinates (r; ), which is used to compose the initial state here, the concrete form of the pro le function g(r) is unimportant. Actually, besides g(r) ! 1 for r ! 1, the only requirement is g(r) / r for r ! 0, which guarantees the smoothness of w(r ! 0). In practice, g(r) = tanh(r=c) with the constant c O(1) just works, and we have checked that there is no observable e ect on our results by di erent choices of c or di erent forms of g(r). := z Then the initial data of At can be determined by the constraint equation once A is given at t = 0 supplemented by the second boundary condition @zAtjz=0 = with the boundary charge density, in addition to at = Atjz=0 = . For convenience but without loss of generality, we shall let the initial value A = 0. Importantly, the initial charge density is taken to be that for the homogeneous and isotropic super uid phase corresponding to the chemical potential , the reason of which is explained in detail in the appendix. With the above initial data and boundary conditions, the later time behavior of bulk elds can be obtained by the following evolution equations 1 2 iA)2 z ]; 2At (2.13) (2.14) (2.15) We numerically solve these non-trivial evolution equations by employing pseudo-spectral methods plus Runge-Kutta method. Namely, we expand all the involved bulk elds in a basis of Chebyshev polynomials in the z direction as well as Fourier series in the x direction, and plug such expansions into the above (3+1)D partial di erential equations { 4 { to make them boil down into a set of 1D ordinary di erential equations, which is well amenable to the time evolution with the fourth order Runge-Kutta scheme. We also use the constraint equation (2.12) to check the validity of our resultant numerical solution. The above numerical evolution scheme has the drawback that the numerical violation of the constraint equation (2.12) is accumulative in time, so in principle it is not suitable for long time evolution, though we have carefully made the violation under control by appropriately choosing the numbers of the Chebyshev and Fourier expansion modes as well as the time step of the Runge-Kutta method. Alternatively, we can use another numerical evolution scheme as described in the appendix, which has a constraint violation accumulated in the spatial direction z instead of time and is believed to be more suitable for long time evolution than the scheme described above. In fact, both these schemes have been used in our study to achieve double con dence. Finally, the vortex dynamics can be decoded by extracting the near boundary behavior of according to (2.6), (2.8) and (2.11), i.e. the condensate hOi = is just conjugate of the boundary (z = 0) value of . For identifying vortices, we calculate winding numbers (2.10) by computing the total phase di erence of around each plaquette formed by the nearest neighboring collocation points in the pseudo-spectral Fourier collocation on the z = 0 surface. 3 Numerical results We now describe the typical behaviors in the holographic turbulent super uid constructed above by numerically solving the bulk equations of motion for a variety of random initial conditions at each chemical potential we choose. As time passes, we never see the merging of vortices with the same circulation. Instead, we observe that the coalescence of vortex and antivortex cores is followed by formation of a crescent-shaped gray soliton when the size of the vortex dipole becomes smaller than a certain threshold value d. Such a crescent-shaped gray solitons originates in the fact that the coalesced vortex and antivortex cores generically march forward leading to a perpendicular linear momentum of the vortex dipole to the vortex dipole direction, and converts eventually into a shock wave, dissipatively propagating in the super uid. With such vortex pair annihilation featured process, the vortex number decreases and eventually the turbulent condensate relaxes into a homogeneous and isotropic equilibrium state. For the purpose of demonstration, we plot one early time and one late time con gurations of turbulent super uid at the chemical potential = 6:25 respectively in gure 1, where 30 vortex-antivortex pairs are prepared in a 30 30 periodic square box as the initial state.4 It is noted that the crescent shape of the gray solitons in our simulation is remarkably similar to that of the density-depleted regions observed in the experiment (see gure 3 in [3]). Instead of the energy spectrum investigated in [ 4 ], we shall focus on the quantitative behavior for the temporal evolution of vortex number in the above relaxation process because this behavior has already been accessible experimentally [3]. Note that it takes N vortices to nd N antivortices, thus it is reasonable for one to expect that the annihilation 4The videos are available at the http URL as http://people.ucas.ac.cn/ ytian?language=en#171556. { 5 { rate should be proportional to N decay takes the following form whereby one can obtain N = N 2. In terms of the number density, the vortex dn(t) dt = n(t) = n(t)2; 1 t + n10 ; at the position where the condensate vanishes. where the decay rate is suggested by the kinetic theory to be proportional to the product of the velocity and cross section of vortices, namely = v2d with v the velocity of vortices if the vortices can be regarded as a gas of particles. Thus the decay rate is supposed to be uniquely determined by the chemical potential through v and d. On the other hand, it is important to focus on the statistical laws because the driven turbulent ow is chaotically sensitive to our randomly prepared initial conditions. Therefore we run 12 groups of data for each chemical potential and extract the corresponding decay rate by tting the temporal evolution of the averaged number density with the statistical error by the formula (3.2). As a demonstration, we only plot the relevant result for the case of chemical potential = 6:25 in gure 2. Obviously, the decrease of vortex number density is well captured by vortex pair annihilation induced two-body decay as (3.2) from a very early time on. The similar results are also obtained for other chemical potentials. It is noteworthy that although we here focus on the early time evolution such a decay pattern is believed to persist towards a { 6 { (3.1) (3.2) 0.05 0.04 1 n t    100 t 0 50 150 200 0 50 150 200 over 12 groups of data with randomly prepared initial conditions at the chemical potential which is well described by the formula (3.2), as implied by vortex pair annihilation mechanism. very late time [12].5 The upshot is that because the drift-out e ect is favorably absent from our holographic super uid due to the periodic boundary conditions imposed on the square box, the above results inarguably con rms the suspected two-body decay mechanism by vortex pair annihilation in [3]. We further plot the variation of decay rate with respect to the temperature in gure 3 by the scaling symmetry of our theory. As illustrated, the decay rate is increased with the temperature within the error bars. In particular, the decay rate is expected to be divergent when one approaches the critical point. Inspired by the e ective description of 2D super uid turbulence in which the decay rate is expected to be proportional to the inverse of super uid density ns [5],6 we further t the near critical point data by the following formula (T )Tc = 1 a T Tc (3.3) with Tc the critical temperature because ns = jhOij2 / 1 TTc near the critical point, which is typical of second order phase transitions [6]. As one can see, the data roughly con rm the above ansatz for the temperature dependence of decay rate in (3.3). We would like to end this section by mentioning the near zero temperature behavior of decay rate. Actually the above e ective description of 2D super uid turbulence is further indicative of the low temperature decay rate / T 2 by using the very fact that the force on a moving vortex at low temperature can be expressed in terms of Kubo formulas of defect CFT operators because at low energies the vortex in a holographic super uid can be 5We are grateful to Andreas Samberg for his private communication on this issue. 6We are grateful to Paul Chesler for his private communication on this issue. { 7 { HJEP12(05)8 scaling symmetry of our theory from the data for those equally spaced chemical potentials from = 4:75 to = 7:0. The decay rate is further t by the formula (3.3) for the six data points nearest to the critical temperature (the rightmost points). viewed as a conformal defect, with a CFT1 living on it [13]. Although as shown in [7], the probe limit we are working with can capture accurately the essential physics all the way down to zero temperature, the numerical simulation turns out to be too time consuming to be worked out by our limited computational resources for the vortex dynamics at low temperature. But we hope to report it elsewhere in the future. 4 Conclusion and discussion The super uid dynamics at zero temperature is generally described with the GrossPitaevskii equation. But in order to address the nite temperature super uid dynamics, the dissipative terms are usually introduced in a purely phenomenological way. On the contrary, holographic duality, as a new laboratory and powerful tool, o ers a rst-principles method to study vortex dynamics in the turbulent super uid, where the super uid at nite temperature is dual to a hairy black hole in the bulk and the dissipation mechanism is universally captured by excitations absorbed by the hairy black hole. We use this gravitational description to numerically construct turbulent noncounter ows with the initial vortices and antivortices placed randomly in the 2D nite temperature holographic super uid. We nd that in the thermal relaxation process the decrease of the vortex number obeys the intrinsic two-body decay due to the vortex pair annihilation, thus con rm the recent experiment data based suspicion. Furthermore, near the critical temperature such a decay pattern is in good agreement with the recently developed e ective theory of 2D super uid turbulence. It is worthwhile mentioning the relevant result obtained by the e ective eld theory of super uid dynamics in [5], where the N 2 decay behavior is found for the laminar ow while 5 N 3 decay behavior for the turbulent ow. This result appears to be at odds with the N 2 decay behavior for our turbulent ow. But the point is that we are dealing with the two di erent regimes of super uid dynamics. To be more precise, ref. [5] is dealing with the { 8 { super uid at a very low temperature with a very tiny dissipative term and dilute vortices while we are dealing with the super uid at the temperature order of critical temperature where the dissipation is supposed to be very large and the vortices are not necessarily dilute. So there is no reason to claim that the super uid dynamics should exhibit the similar behavior at these two regimes. Actually the inverse energy cascade is obeyed by the turbulent ow in [5] while as shown in [ 4 ] our super uid turbulence exhibits a direct energy cascade. Therefore it is highly interesting to see whether there is a critical temperature for our super uid to transition from the direct energy cascade N 2 behavior to the inverse 5 energy cascade N 3 behavior, albeit beyond the scope of our paper. In addition, in order for more detailed comparison between the holographic simulation and the experiments, more works should be done on both sides. As mentioned previously, at very low temperature the current numerical approach tends to break down even in the probe limit, since both the equilibrium con guration in the z direction and the vortex con guration in the radial direction (with respect to the vortex center) tend to be nonanalytic, so a very large number of expansion modes should be taken in order to maintain precision, which challenges any computational resources. Possibly novel approach should be used to overcome this di culty. On the experimental side, after all, the highly oblate BoseEinstein condensate is not a genuine 2D system [3]. It is also di cult (if ever possible) to properly subtract the drift-out e ect, which seems inevitable in experimental setups. This fact makes the existing experimental data insu cient to determine the intrinsic physics of vortex pair annihilation. In addition, the experimental data at very low temperature or near criticality is also lacking. We hope that signi cant progresses will be achieved on both the theoretical (numerical) and experimental sides in the near future. Acknowledgments We thank Yong-il Shin for his stimulating talk and later helpful discussion on vortex pair annihilation on the focus conference \Precision Tests of Many-Body Physics with Ultracold Quantum Gases" during the long program \Precision Many-Body Physics of Strong Correlated Quantum Matter" at KITPC. We would like to express a special thanks to Hong Liu for his insightful comments and valuable suggestions throughout this whole project. We acknowledge the organizers of another long term program \Quantum Gravity, Black Holes and Strings" at KITPC for the fantastic infrastructure they provide and the generous nancial support they o er. H.Z. is grateful to the Mainz Institute for Theoretical Physics for its hospitality and its partial support during his attending the program \String Theory and its Applications", and CERN for its nancial support during his attending \CERN-CKC TH Institute on Numerical Holography", where he bene ts much from the discussions with Andreas Samberg. He also acknowledges the Erwin Schrodinger Institute for Mathematical Physics for the nancial support during his participation in the program \Topological Phases of Quantum Matter", where the relevant conversation with Michael Stone is much appreciated. Finally, H.Z. thanks the Galileo Galilei Institute for Theoretical Physics for the hospitality and the INFN for partial support during his attending the workshop \Holographic Methods for Strongly Coupled Systems", where the revision of this work is being conducted. Y.D. and Y.T. are partially supported by NSFC with Grant { 9 { No. 11475179. C.N. is partially supported by NSFC with Grant No. 11275208. H.Z. is supported in part by the Belgian Federal Science Policy O ce through the Interuniversity Attraction Pole P7/37, by FWO-Vlaanderen through the project G020714N, and by the Vrije Universiteit Brussel through the Strategic Research Program \High-Energy Physics". He is also an individual FWO fellow supported by 12G3515N. A More on our numerical scheme Besides the numerical scheme described in the main body of this paper, where the constraint equation (2.12) is imposed at t = 0, and used as a later monitor of the reliability of our numerical evolution by the equations (2.13), (2.14), and (2.15), we have an alternative scheme described as follows. In this scheme, we still evolve the scalar eld and the gauge eld A in time by (2.13) and (2.14), respectively, with the fourth order Runge-Kutta method. However, we no longer use the equation (2.15) to achieve the time evolution of At except at z = 0, where (2.15) boils down into which is nothing but the conservation equation for the boundary global current (2.7) in the boundary system (z = 0). Upon integration on the constant time surface (within the boundary z = 0), this current gives the conserved charge during the dynamical evolution process, which guarantees that the boundary system will eventually settle down to the desired chemical potential if we use the corresponding (equilibrium) charge density as the initial charge density. Then with the solution for @zAt(t)jz=0 as well as At(t)jz=0 = , we shall instead use the constraint equation (11) to solve At from and A not only at the initial time t = 0 but also at every step of the later time evolution. It is not hard to show that the equation (2.15) will automatically hold elsewhere by virtue of the equations (2.12), (2.13), and (2.14), if it does hold at a given constant z hypersurface. In both schemes, we take 28 Chebyshev modes in the z direction and 361 Fourier modes in both the x and y directions. The advantage of this alternative scheme is that it can undergo a very long time evolution because the main numerical error comes from the violation of the equation (2.15) deep into the bulk horizon, which is nevertheless not accumulated fast with the time evolution. For the chemical potential we have chosen, the violation of the equation (2.15) at the horizon z = 1 is of the order of 10 4, which we deem to be acceptable. Regarding the main results presented in this paper, both schemes are in good agreement with each other, which can be regarded as a double check of the reliability of our results. For the case of the smallest size of a vortex core ( = 7:0), there are roughly 20 grids within each vortex core, which is expected to be ne enough. For the time step t in the Runge-Kutta evolution, we have taken both t = 0:05 and t = 0:025, and nd no observable di erence for a given initial state on either the vortex number counting (to t = 200) or the constraint violation as described above. For t = 0:025, it takes about two weeks on a mainstream desktop computer for one run (to t = 200). Actually, we have 10 di erent chemical potentials and for each chemical potential we need 12 groups of data, which are accomplished on a parallel work station. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Dynamics of Single Vortex Lines and Vortex Dipoles in a Bose-Einstein Condensate, Science 329 (2010) 1182. 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Yiqiang Du, Chao Niu, Yu Tian, Hongbao Zhang. Holographic thermal relaxation in superfluid turbulence, Journal of High Energy Physics, 2015, 18, DOI: 10.1007/JHEP12(2015)018