Real-Time dynamics and phase separation in a holographic first order phase transition

Romuald A. Janik,1, ∗ Jakub Jankowski,2, † and Hesam Soltanpanahi3, ‡

1Institute of Physics, Jagiellonian University, Lojasiewicza 11, 30-348 Krakow, Poland2Faculty of Physics, University of Warsaw, ulica Pasteura 5, 02-093 Warsaw, Poland

3School of Physics, Institute for Research in Fundamental Sciences (IPM),P.O.Box 19395-5531, Teheran 19538-33511, Iran

We study the fully nonlinear time evolution of a holographic system possessing a first order phasetransition. The initial state is chosen in the spinodal region of the phase diagram, and includesan inhomogeneous perturbation in one of the field theory directions. The final state of the timeevolution shows a clear phase separation in the form of domain formation. The results indicate theexistence of a very rich class of inhomogeneous black hole solutions.

Introduction. The concept of a phase transition is in-herently an equilibrium one and it is a theoretical chal-lenge to formulate a framework in which it can be quan-titatively studied in cases involving real time dynamics.Such a framework is offered by gauge-gravity duality putforward about twenty years ago [1]. Gauge-gravity du-ality, also referred to as holography, is immanently re-lated to field theory systems at strong coupling and thusit is very useful to study nonperturbative physics espe-cially as within this framework one can work directly inMinkowski signature and study real time dynamics. Re-cently, it has been demonstrated that well known holo-graphic 1st order phase transitions [2] are accompaniedby an unstable spinodal region [3, 4] in accordance withstandard predictions [5]. The analysis was based on lin-ear response theory, however the true power of the dualgravitational approach is the possibility of investigatingfully nonlinear dynamic evolution of the system. Simul-taneously with the present work, this particular directionhas been recently undertaken in Ref. [6] where spinodalregion was studied in the holographic model of a phasetransition. The final state of the time evolution was aninhomogeneous configuration approached at late timeswithin the hydrodynamic approximation. A different set-up of a homogeneous evolution has been undertaken inRef. [7]. Inhomogeneous, static configurations appearingin the context of a holographic first order phase transitionwere also recently studied in Ref. [8].

A natural question which arises is whether the finalstate will exhibit domains of the two coexisting phaseswith the same values of the free energy. The main resultof the present letter is to demonstrate for the first timethat in the case of a three-dimensional nonconformal sys-tem with a holographic dual such a phase separation willarise dynamically through a real time evolution from aperturbation in the spinoidal region. The respective en-ergy densities of the two components of the final state arevery close to the corresponding energy densities deter-mined at the critical temperature. This implies that thesystem undergoes a dynamical transition during whichdifferent regions of space become occupied with differentphases of matter.

The holographic framework. The holographic modelwe use is a bottom-up construction containing Einsteingravity coupled to a real, self-interacting scalar field. Theaction takes the standard form

S =1

2κ24

∫d4x√−g[R− 1

2(∂φ)

2 − V (φ)

]+SGH+Sct ,

(1)where is κ4 related to the four dimensional Newton’s con-stant κ4 =

√8πG4, and the self interaction potential

V (φ) is given below. We include the boundary terms inthe form of Gibbons-Hawking SGH [9] and holographiccounterterms Sct [10, 11] contributions. We chose towork in 3+1-dimensional bulk spacetime, which is dual to2+1-dimensional field theory, for the absence of confor-mal anomaly in odd dimensions. This, in turn, makes theexpansions near the conformal boundary free from loga-rithms, which allows for the usage of Chebyshev spectralmethods for numerical integration. The V (φ) potentialis constructed so that the dual field theory undergoesan equilibrium first order phase transition. The specificchoice that we use is

V (φ) = −6 cosh

(φ√3

)+ b4φ

4 , (2)

where b4 = −0.2. This functional form is dual to a rel-evant deformation of the boundary conformal field the-ory with an operator of conformal dimension ∆ = 2.When b4 = 0 the potential is that of N = 2 supergrav-ity in D = 4 after dimensional reduction from D = 11[12]. The physical scale, breaking conformal invariance,is set by the source of the operator, and is chosen tobe Λ = 1. The equilibrium structure of this model isdescribed in terms of dual black hole geometries charac-terized by specifying the horizon value of the scalar field,i.e., φ(z = 1) = φH . The entropy and the free energy ofthe system are, in turn, given by the Bekenstein-Hawkingformula, and the on-shell value of the action (1) respec-tively. The corresponding thermodynamics reveals theappearance of a first order phase transition between dif-ferent branches of black hole geometries as determined bythe difference of free energies. The order of the transitionis established by a discontinuity of the first derivative of

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[he

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] 1

5 Ja

n 20

18

2

the free energy of the system. This effect is illustrated inthe lower panel of Fig. 1. The value of the critical tem-perature is Tc ' 0.246 (in Λ = 1 units). This transitionis of similar nature as the Hawking-Page transition inthe case of anti-de Sitter (AdS) space [13, 14] (an impor-tant difference, however, is that in the current setup allphases are of the black hole type). The equation of state(EOS) is displayed in the upper panel of Fig. 1 as a tem-perature dependence of the energy density. This EOS issimilar to the five-dimensional case studied in [3, 4] andthe detailed analysis of the linearized dynamics proved,in accordance with the general lore, the existence of aspinodal region separating stable configurations [3, 4].

FIG. 1. Upper panel: The temperature dependence of theholographic energy density. Vertical green line represents thetransition temperature. Orange points represent sample cho-sen initial configurations for time evolution. Horizontal linesshow the domain energies in the final state (solid - cosine per-turbation, dashed - Gauss perturbation). Lower panel: freeenergy as a function of temperature.

Time dependent configurations. To study the timeevolution of the system we adopt the following metricansatz in the Eddington-Finkelstein (EF) coordinates

ds2 = −Adt2−2 dt dz

z2−2B dt dx+S2

(Gdx2 +G−1 dy2

),

(3)where all functions are x, t, z dependent. We can take

the initial state to lie in the spinodal region of the phasediagram and add an x-dependent perturbation to the Sfunction. By the proper choice of the perturbing function

δS(t, x, z) = S0 z2 (1− z)3 cos (kx) , (4)

we can turn on a particular unstable mode or add a mix-ture of all the modes

δS(t, x, z) = S0 z2 (1− z)3 exp

[−w0 cos

(kx)2]

, (5)

with different widths. By solving the time dependentEinstein-matter equations of motion with proper AdSboundary conditions at z = 0 1 we determine the non-linear evolution of the system. Using the procedure ofholographic renormalization we then read-off the relevantobservables like the energy density of the the boundarytheory from subleading terms in the near-boundary ex-pansion [10, 11].

The system is essentially studied in the microcanon-ical ensemble as the total energy density of the systemis fixed throughout the evolution. The gravitational for-mulation of the problem is now given by a coupled set ofnon-linear partial differential equations. We solve thatproblem numerically using the characteristic formulationof General Relativity [15] along with spectral methods[16]. In the relevant spatial direction we use periodicboundary conditions with spectral Fourier discretization.The remaining spatial direction is uncompactified.Results. Since for this system the energy density inequilibrium uniquely determines the temperature, wemay deduce that the final state of evolution starting fromthe unstable spinoidal branch necessarily has to be inho-mogeneous. The physical expectation that the final statewill consist of well separated phases at the phase tran-sition temperature T = Tc would manifest itself in theexistence of spatial domains characterized by very flatenergy density, the values of which should coincide withthe energy densities of the two stable phases at T = Tc.The fact that such a configuration is attained dynam-ically even when starting from points on the spinoidalbranch with temperatures differing from Tc is far fromtrivial. This is the main result of the present work.

To illustrate the effect of the appearance of differentphases during the time evolution we run the simulationsfor a number of initial configurations, covering a rep-resentative region of interest. Some of the configura-tions are marked with orange dots in Fig. 1. As it wasexplained in the previous section we use two differentshapes of perturbing function given in Eq. (4) and (5)with different values for parameters. Particularly trans-parent results appear for the value of the momentum

1 In the EF coordinates A ∼ 1/z2 + O(1), S ∼ 1/z + O(1), G ∼O(z), B ∼ O(z) for z → 0.

3

equal to k = 1/6 and k = 1/12, with w0 = 10, and wehave chosen to present those in this letter. The ampli-tudes of the perturbations are in the range S0 = 0.1−0.5.

The first point of interest is a large black hole configu-ration with temperature below Tc, but still on the stablebranch e.g. with φH = 1. The linear analysis showsno instability of that configuration. However, one couldstill expect a non-linear instability due to overcooling. Inour simulations we found, however, no evidence of thatwithin the considered framework.

The second considered point, with φH = 2, is placeddeep in the unstable region. The temporal and spatialdependence of the energy density is shown in the upperpanel of Fig. 2. Initially small Gaussian perturbationgrows with time, and after around 400 units of simula-tion time starts settling down to an inhomogeneous finalstate. The maximum and the minimum energy of thisstate, marked as horizontal solid lines in Fig. 1, approachwithin less than 1% the energy densities determined bythe transition temperature. Pronounced flat regions ofconstant energy density are apparent at late stages ofthe evolution. It is clearly visible that in the final statedifferent parts of the system are occupied by the differentphases, joined by a domain wall.

The third considered configuration, with φH = 3, liesclose to the end of the spinodal region. The time depen-dence of the energy density in this case is displayed inthe lower panel of Fig. 2. The perturbation added is asingle cosine mode. In this case, after about 150 unitsof simulation time, the configuration settles down to aninhomogeneous final state. The maximum and the mini-mum energy densities of this final state are marked withhorizontal, dashed lines in Fig. 1. Similarly to the pre-vious configuration the extrema of energy density reachthe corresponding densities determined at the transitiontemperature T = Tc. However, due to the fact that theinitial energy density is smaller than the energy densityof a configuration with φH = 2, we observe a smallerregion of the high-energy phase in the final state.

To summarize the above results we display the finalsate energy density as a function of x in Fig. 3 for bothunstable initial configurations. The Hawking tempera-ture of the final state geometry is constant and equal tothe critical temperature Tc. From the field theory per-spective this is a clear demonstration of the coexistencephenomenon, where inhomogeneous state has constanttemperature. Different regions of space are occupied bydifferent phases connected by domain walls. Despite thefact that in both cases the final state is rather universalthe time evolution is substantially different. The greatnovelty of our approach is that details of dynamical for-mation of domains of different phases can be studiedquantitatively. In order to do so it is convenient to intro-

FIG. 2. The energy density as a function of time for the initialconfiguration in spinodal region: φH = 2 Gaussian perturba-tion (upper panel), φH = 3 cosine perturbation (lower panel).

duce the following observable

Aε =1

12π

∫ε>ε0

ε(t, x)dx , (6)

where ε0 is the mean energy of the system at t = t0. Theabove quantity essentially measures the amount of energyabove the mean energy stored in the system at t = t0. Asis clearly seen in Fig. 4 in both considered cases the de-tails of dynamics are different2. For the configurationwith φH = 2 the initial perturbation develops into twobubbles which subsequently move away from the centerand then violently coalesce and merge into one final do-main. Because of that, the final state is approached withlarge, damped oscillations. In contrast, the configura-tion with φH = 3 displays less violent evolution. We canidentify three stages of evolution in that case. First isan exponential growth of the instability, that takes placeroughly until the maximum energy reaches the equilib-rium energy density. The second stage displays linear

2 These differences stem from the different forms of the perturba-tion employed by us at φH = 2 and φH = 3.

4

FIG. 3. The final state energy density for different initialconfigurations: φH = 2 (red) and φH = 3 (blue). The hori-zontal green lines represent the energy densities at the criticaltemperature in isotropic solutions.

increase of the bubbles width with fixed height to forman extended region. The system finally saturates in thethird stage with small oscillations for late times. In bothcases the complicated dynamics is a consequence of non-linear nature of dual Einstein’s equations, and would beextremely difficult to study using conventional field the-ory techniques. In future work we intend to analyze inmore detail both of these phenomena: the collision offully formed domains of equilibrium phases as well as thedetails of the dynamics of bubble growth.Discussion. In this letter we demonstrated for the firsttime the existence of a phase separation effect at thetransition temperature of a first order phase transitionin the context of holographic models. The chosen set-upwas a bottom-up holographic construction designed toexhibit an equilibrium first order phase transition.

The full nonlinear time evolution of a perturbed con-figuration with a spinodal instability ends in an inho-mogeneous state composed of domains of different sta-ble phases as determined at the transition temperaturefixed by the equality of free energies. The dual gravi-tational configuration is a black hole with an inhomoge-neous horizon. However in contrast to the results of [6],the geometries obtained here correspond to domains ofspecific thermodynamic phases with very mild spatial de-pendence separated by relatively narrow domain walls.On the gravitational side this means that the bulk geom-etry consists of two distinct types of black holes, charac-terized e.g. by the value of the scalar field on the horizonsmoothly connected by interpolating domain walls (seeFig. 5 for an example configuration of the scalar field inthe bulk). Despite the inhomogeneity, the Hawking tem-perature is constant on the horizon. From the field the-ory interpretation we expect to have an immense modulispace of geometries which correspond to different con-figurations of phase domains coming from different seed

0 100 200 300 400 500

0.00

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t

Aϵ

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0.00

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FIG. 4. The time evolution of the observable Aε defined inEq. 6. Upper panel: configuration starting from φH = 2.Lower panel: configuration starting from φH = 3. The insetsshow profiles of the energy density at given instants of time.See discussion in text.

FIG. 5. The final configuration of the scalar field φ(x, z)/zextending to the inhomogeneous horizon at z = 1 for thesolution starting from φH = 3.

perturbations. Indeed, we found also solutions with mul-tiple potential domains which, however, were of size ofthe order of the domain wall width and thus were moresimilar to the solutions of [6]. It is clear that with anappropriately large transverse spatial extent of the simu-lation one can explicitly construct solutions with multipledomains. We intend to investigate this in the future.

5

We have also demonstrated that large black holes withtemperatures below the critical value are stable againstthe perturbations that we tried. This indicates that inthe holographic description, we may naturally expect anovercooled phase which only starts to nucleate once ithas dynamically moved into the spinoidal branch.

There are numerous directions for further research.Apart from studying the space of domain geometriesmentioned earlier, it would be very interesting to inves-tigate in detail the various temporal regimes which canbe seen in Fig. 4 and study the dynamics of the phasedomains. Naturally it would be good to investigate ex-tensions to other dimensions as well as ultimately relaxany symmetry assumptions. An additional bonus of thepresent setup is the appearance configurations breakingtranslation invariance without specifying explicit inho-mogeneous sources. In other words, the final state solu-tions spontaneously break translational invariance. Thisopens a possibility of applications in the context of con-densed matter physics [17, 18].

Acknowledgments. We would like to thank Z. Ba-jnok for extensive discussions and collaboration in theinitial stage of the project. RJ was supported by theNCN grant 2012/06/A/ST2/00396. JJ was supportedby the NCN grant No. UMO-2016/23/D/ST2/03125. JJand HS would like to thank theory division at CERN,Wigner institute and Jagiellonian University where a partof this research was carried out. HS would like to thankto Witwatersrand university where part of this researchwas done. RJ would like to thank the Galileo GalileiInstitute for Theoretical Physics for hospitality and theINFN for partial support during the completion of thiswork.

∗ romuald@th.if.uj.edu.pl† jjankowski@fuw.edu.pl‡ hsoltan@ipm.ir

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