Holographic Thermal Relaxation in Superfluid Turbulence

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.

Introduction.-Quantized vortices are topological objects in a 2D superfluid, where vortex-antivortex pairs play a crucial role in various superfluid phenomena. Amazingly, vortex dipole dynamics can nowadays be experimented in a controllable way in atomic Bose-Einstein condensate systems [1,2], where in particular not only has the thermal activation of vortex pairs been observed in quasi-2D Bose gases [3,4], but also most recently the vortex pair annihilation has been investigated experimentally during the relaxation process of turbulent superflow [5].
Compared to these fascinating experimental developments, our theoretical understanding of dynamics of these quantized vortices is still limited because the effective dissipative hydrodynamical description for normal fluids does not work in the presence of quantized vortices and all the conventional approaches rely on some phenomenological modes, which nevertheless have significant 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. Thus it allows a firstprinciples investigation of vortex dynamics by using the dual gravity description of superfluid phase. In particular, it is recently shown by holography in the seminal work [6] that although the 2D turbulent superfluid kinetic energy spectrum obeys Kolmogorov −5/3 scaling law as it does for turbulent flows in normal fluids, the superfluid 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 fluid turbulence due to the conservation of enstrophy, which is violated in a superfluid by vortex-antivortex pair annihilation anyhow.
Inspired by the above sharp results derived from the holographic principle as well as the aforementioned experimental investigation of vortex pair annihilation, in this Letter we shall make such a holographic duality contact closer with experimental data by initiating a systematic investigation of vortex pair annihilation in 2D turbulent superfluid through numerically solving the equations of motion of its gravity dual.
Holographic setup.-The simple holographic model for 2D superfluid consists of gravity in asymptotically AdS 4 spacetime coupled to a U (1) gauge field A and a complex scalar field Ψ with charge q and mass m. The corresponding bulk action is given by [7,8] where G is the Newton's constant, L is the radius of curvature of AdS, and the matter Lagrangian reads with D = ∇ − iA. We shall work in the probe limit, namely the matter fields decouple from gravity, which can be achieved by taking the large q limit. One thus can put the matter fields on top of Schwarzschild black arXiv:1412.8417v2 [hep-th] 19 Jan 2015 brane background, which can be written in the infalling Eddington coordinates as where the blackening factor f (z) = 1 − ( z z h ) 3 with z = z h the location of horizon and z = 0 the AdS boundary. The behavior of matter fields is controlled by the equations of motion in the bulk as By the holographic dictionary, the dual boundary system is placed at a finite temperature given by where a conserved current operator J is sourced by the boundary value of the bulk gauge field A and a scalar operator O of conformal dimension ∆ = 3 2 ± 9 4 + m 2 L 2 is sourced by the near boundary data of scalar field Ψ. For simplicity but without loss of generality, we shall focus only on the case of m 2 L 2 = −2 in the axial gauge A z = 0, in which the asymptotic solution of A and Ψ can be expanded near the AdS boundary as Then the expectation value of J and O can be explicitly obtained by holography as the variation of renormalized bulk on-shell action with respect to the sources, i.e., where the renormalized action is obtained by adding a counter term to the original action to make it finite as S ren = S − 1 √ −γ|Ψ| 2 , and the dot denotes the time derivative [9]. When this scalar operator O develops nonzero expectation value spontaneously in the situation where the source is switched off, the system is driven into a superfluid phase with O =ψ characterizing the superfluid condensate [10]. Generically such a superfluid phase has gapped vortex excitations with the circulation quantized. With the superfluid velocity defined as [6] the winding number w of a vortex is determined by where γ denotes a counterclockwise oriented path surrounding a single vortex. In particular, close to the core of a single vortex with winding number w, the condensateψ ∝ (z − z 0 ) w for w > 0 and ψ ∝ (z − z 0 ) −w for w < 0 with z the complex coordinate and z 0 the location 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 2πw . This is the characteristic property of a vortex and will be used as an efficient way to identify vortices in our later vortex counting.
To address the vortex pair annihilation in turbulent superfluid by holography, in particular to mimic the experimental situation in [5] to some extent, we would first like to impose the following boundary conditions onto the bulk fields, i.e., where we set the chemical potential µ > µ c with µ c the critical chemical potential for the onset of a homogeneous superfluid phase, given by µ c ≈ 4.07. Explicit gravity solutions dual to a static vortex of arbitrary winding number have been numerically constructed in [11][12][13]. But we are required to prepare an initial bulk configuration for Ψ at the Eddington time t = 0 such that the dual initial boundary state includes 300 vortex-antivortex pairs in a 100 × 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. Then the initial data of A t can be determined by the constraint equation once A is given at t = 0. For convenience but without loss of generality, we shall set the initial value A = 0. With the above initial data and boundary conditions, the later time behavior of bulk fields can be obtained by the following evolution equations We numerically solve these non-trivial evolution equations by employing pseudo-spectral methods plus Runge-Kutta method. Namely we expand all the involved bulk fields in a basis of Chebyshev polynomials in the z direction as well as Fourier series in x direction, and plug such expansions into the above 3+1D partial differential equations to make them boil down into a set of 1D ordinary differential equations, which is well amenable to the time evolution with the fourth order Runge-Kutta scheme. We also use the constraint equation (11)  the validity of our resultant numerical solution. Finally the vortex dynamics can be decoded by extracting the near boundary behavior of Ψ according to (6) and (7). Numerical results.-We now describe the typical behaviors in the holographic turbulent superfluid 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 crescentshaped gray soliton when the size of the vortex dipole The temporal evolution of averaged vortex number density in the turbulent superfluid over 12 groups of data with randomly prepared initial conditions at the chemical potential µ = 6.25, which is well described by the formula (16), as implied by vortex pair annihilation mechanism.
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 superfluid. 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 configurations of turbulent superfluid at the chemical potential µ = 6.25 respectively in FIG.1, where 30 vortex-antivortex pairs are prepared in a 30×30 periodic square box as the initial state [14]. Instead of the energy spectrum investigated in [6], 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 [5]. Note that it takes N vortices to find N antivortices, thus it is reasonable for one to expect that the annihilation rate should be proportional to N × N = N 2 . In terms of the number density, the vortex decay takes the following form whereby one can obtain 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 Γ = vd 2 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 flow 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 fitting the temporal evolution of the averaged number density with the statistical error by the formula (16). As a demonstration, we only plot the relevant result for the case of chemical potential µ = 6.25 in FIG.2. Obviously, the decay of vortex number density is well captured by vortex-antivortex pair annihilation as (16) 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 very late time [15]. This universal result is remarkable because all other known universal behaviors generically show up only in the intermediate time regime or late time regime for the holographic superfluid turbulence [6,16].
We further plot the variation of decay rate with respect to the chemical potential in FIG.3. As illustrated, the decay rate is decreased with the chemical potential within the error bars. This is reasonable because it amounts to meaning that the decay rate is increased with the temperature due to the scaling symmetry of our theory. In addition, the decay rate is expected to be divergent when one approaches the critical point. In particular, inspired by the effective description of 2D superfluid turbulence in which the decay rate is expected to be proportional to the inverse of superfluid density n s [17,18], we fit the resultant data by the following formula with R = µc µ = T Tc because n s = | O | 2 ∝ 1 − R near the critical point, which is typical of second order phase transitions [8]. This fit seems to work well even for the regime far away from the critical point and indicates that vortex pair annihilation persists all the way down to zero temperature, which is believed to be a universal feature for holographic superfluid because there always exist gapless modes associated with the presence of the bulk horizon. In particular, if we subtract the decay rate by such zero temperature offset, our fit is suggestive of the low temperature decay rate proportional to T µ , which is apparently in tension with the aforementioned experimental result, where Γ ∝ T 2 µ [5]. We would like to conclude this section with speculation on the possible reasons for such a distinction. First, on top of the vortex pair annihilation, the experimented sample also undergoes the drift-out process, which is absent in our holographic superfluid because we impose the periodic boundary conditions. There may be manyvortex effects on this drift-out process such that the measured two-body decay rate will not be precisely the same as the intrinsic one from the vortex pair annihilation. Second, as evident from the fact that the measured twobody decay rate is not a function of T µ , the experimented system is not truly 2D as ours, but quasi-2D superfluid, where the finite length of the vortex line might also play a role in the vortex pair annihilation. Third, the experiment is carried out with inhomogeneous samples trapped in a harmonic potential, which may also modify the intrinsic vortex-pair annihilation. Fourth, the experimented sample may not be as strongly coupled as our holographic superfluid, which may lead to the total different universal behavior for the vortex pair annihilation. Last but not least, the probe limit we are working with will become unreliable when one approaches zero temperature, where the decay rate may be modified by taking into account the back reaction to the metric. To sharpen and resolve this seemingly conflict, one is required to make the simulated and experimented sample conditions closer to each other, which turns out to be highly nontrivial for both numerics and experiment. We hope to address this challenging issue from the side of numerical holography in the future.
Discussions.-The superfluid dynamics at zero temperature is generally described with the Gross-Pitaevskii equation. But in order to address the finite temperature superfluid dynamics, the dissipative terms are usually introduced in a phenomenological way. On the contrary, holographic duality, as a new laboratory and powerful tool, offers a first principles method to study vortex dynamics in the turbulent superfluid, where the superfluid at finite 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.
We use this gravitational description to numerically construct turbulent non-counterflows with the initial vortices and antivortices placed randomly in the 2D finite temperature holographic superfluid. We find that in the thermal relaxation process the decrease of the vortex number obeys the intrinsic two-body decay due to the vortex pair annihilation remarkably from a very early time on and the decay rate is decreased (increased) with the chemical potential (temperature). In particular, our result is consistent with the effective theory of 2D superfluid turbulence and signals a seemingly distinct decay pattern from that demonstrated in the recent real experimental data for superfluid turbulence in highly oblate Bose-Einstein condensates.
Acknowledgements.-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 financial support they offer. 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 financial sup-port during his attending "CERN-CKC TH Institute on Numerical Holography", where he benefits much from the discussions with Andreas Samberg. He also acknowledges the Erwin Schrödinger Institute for Mathematical Physics for the financial support during his participation in the program "Topological Phases of Quantum Matter", where the relevant conversation with Michael Stone is much appreciated. Y.D. and Y.T. are partially supported by NSFC with Grant 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 Office 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.