IFT-UAM/CSIC-17-002 Asimpleholographicscenarioforgappedquenches Esperanza Lopez and Guillermo Milans del Bosch Instituto de F´ısica Teo´rica UAM/CSIC, C/ Nicola´s Cabrera 13-15, Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain Weconstructgravitationalbackgroundsdualtoafamilyoffieldtheoriesparameterizedbyarelevantcoupling. Theycombineanon-trivialscalarfieldprofilewithanakedsingularity. Thenakedsingularityisnecessaryto preserveLorentzinvariancealongtheboundarydirections. Thesingularityishoweverexcisedbyintroducing aninfraredcutoffinthegeometry.Theholographicdictionaryassociatedtotheinfraredboundaryisdeveloped. Weimplementquenchesbetweentwodifferentvaluesofthecoupling.Thisrequiresconsideringtimedependent boundaryconditionsforthescalarfieldbothattheAdSboundaryandtheinfraredwall. Introduction.Modelingquantumquenchesinaholographic AdSwithaninfraredhardwallisawellknownroughholo- setuphasattractedconsiderableattentioninthelastyears. A graphicmodelforconfiningtheories[7]. Thenewingredient 7 1 remarkablesuccesshasbeenachievedinreproducingimpor- inthispaperistoconsiderthehardwallasaregularizingele- 0 tant aspects of the universal dynamics of quenches [1]-[5]. ment,whiletheinfraredphysicswillbelinkedtothestrength 2 Howevermostmodelslacksomeofthedefiningcharacteris- ofthenakedsingularity. n ticsofquenches.Notably,theysimulateaninjectionofenergy a inthesystemwithoutrealchangeinthehamiltonian. Staticbackgrounds. Wewillexploretheproposedscenario J Inthisnotewewanttopresentasimpleholographicmodel with a vanishing scalar potential. Equations (2) can then be 0 ofaquenchmodifyingtheinfraredphysics.Withthisaim,we solvedanalytically,withtheresult 1 searchforthegravitationaldualtoafamilyofd-dimensional QFT’sparameterizedbyarelevantcoupling. Asamaininput, A(z)=1+α2z2d , (3) h] thegroundstateforanyvalueofthecouplingisrequiredtobe φ(z)=β+d−1/2arcsinh(αzd), (4) t Lorentzinvariant. Wepursuetheminimalscenario,involving p- Einsteingravitycoupledtoarealscalarfield. Thepossibility whereα andβ arearbitraryconstants. β representsaglobal shiftinthevalueofthescalar,whichisofnophysicalconse- e ofextracompactifieddimensionsisexcluded. h Weusethefollowingansatzforthegroundstatemetrics quencewhenV(φ)=0. α inducesanon-trivialscalarprofile [ 1 (cid:18) dz2 (cid:19) α =z−dsinh(√d∆φ). (5) 1 ds2= −dt2+d(cid:126)x2 . (1) 0 v z2 A(z) d−1 1 with z0 denoting the radial position of the wall, and ∆φ = 7 Setting8πG=d−1,theequationsofmotionare φ −φ the variation of the scalar field between the wall and 0 ∞ 6 theAdSboundary. Ifextendedbeyondtheinfraredcutoff,all z 2 A(cid:48)=d(A−1)+2V(φ), A(cid:48)=2zAφ(cid:48)2. (2) backgroundswith∆φ(cid:54)=0haveanakedsingularity. 0 2 . 1 These equations have two integration constants, which can ω z 0 be related to the coefficients of the two independent scalar 0 0 ωn/ω0 5.0 7 5.0 modes. Askingforregularityofthegeometrylinkstheirval- 4.8 1 4.5 : ues, allowing to interpret one of them as a QFT coupling 4.6 4.0 v and the other as the expectation value of the sourced oper- 4.4 3.5 Xi ator. Regular solutions of (2), whose existence depends on 4.2 3.0 the scalar potential, describe RG flows into an infrared fixed 4.0 2.5 ar point independent of the integration constants. All other so- 3.8 Δϕ Δϕ lutions run into naked singularities. Considering naked sin- 0.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 gularities raises a number of serious problems. There is no conditionthatrelatesthetwointegrationconstants,challeng- Figure1. Left: Frequencyofthefundamentalscalarmode. Right: ingtheusualholographicdictionary. Relatedtothis,Lorentz Ratioofthenextthreenormalfrequenciestothefundamentalone. invariantmetricsarenotminimalenergysolutionswhenonly one integration constant is fixed at the AdS boundary. Actu- Holographyinterpretstheharmonicmodesofbulkfieldsas allytherearesolutionsofarbitrarynegativeenergy. excitationsinthedualQFT.InFig.1wehaveplottedthefre- Theseissuesadmitasimple,albeitcrudesolution,byintro- quencyofthelowerscalarmodesalongthefamily(3)-(4)for ducinganinfraredcutoffinthegeometry. Thiscreatesanew d=2. Whentheradialvariationofscalarprofileissmall,the boundaryandrendersnaturaltointerpretbothintegrationcon- spectrumisdeterminedbyz . Thespectrumbecomesinstead 0 stantsfrom(2)ascouplings. Fixingthetwocouplingssolves ruledby∆φ forlargervaluesofthisparameter. Fig.1ashows the vacuum stability problem [6]. Moreover regions of high that the mass gap, holographically given by ω , grows with 0 curvature,whichbringoutsidetheregimeofvalidityofclas- ∆φ and implies that this is a relevant coupling. Interestingly sicalgravity,areexcised. theratioofhighernormalfrequenciestothefundamentalone 2 showsanapproximatelineargrowth,seeFig.1b.Hencethein- theinfraredboundary. Unlessthetimevariationisadiabatic, fraredphysicsassociatedtothefamily(3)-(4)doesnotdiffer thispulseinducesanexcitedstateinthefinalQFT.Thevalue byamererescaling. Thelowestexcitationbecomesincreas- ofA willthenadjustsuchthatthetotalenergyisconserved. 0 inglyseparatedfromtherestthelargeris∆φ. Namely, if the initial theory belongs to the Lorentz invariant Modelling a quantum quench. We want to model a global subset, the final one will have negative ground state energy. quenchbetweenQFT’swhosegroundstatesareinthefamily Inordertomodelaquenchbetweentheoriesin(10),thewall (3)-(4). Aconvenientansatzfortheassociatedmetricis profile φ0(t) needs to be combined with an energy injection into the system. This can only happen at the AdS boundary, 1 (cid:18) dz2 (cid:19) inducedbyanon-trivialφ (t). ds2= −A(t,z)e−2δ(t,z)dt2+ +d(cid:126)x2 . (6) ∞ z2 A(t,z) d−1 A0 z02Mc Forvanishingscalarpotential,theequationsofmotionare 7 6 ����=-� 1.6 Tc/ω0 (cid:16) (cid:17)(cid:48) (cid:16) (cid:17)(cid:48) Φ˙ = Ae−δΠ , Π˙ =zd−1 z1−dAe−δΦ , (7) 5 -�/� 1.4 0.05 4 � 1.2 δ(cid:48)=z(Φ2+Π2), A(cid:48)=zA(Φ2+Π2)+d(A−1), (8) 3 �/� 1.0 0.04 Δϕ z 2 1 2 0.8 1 with Φ=φ(cid:48) and Π=A−1eδφ˙ encoding the radial and time 0 Δϕ 0.6 Δϕ scalar derivatives. Solving the equations of motion requires 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.5 1.0 1.5 2.0 giving a set of initial data together with boundary data at Figure2. Left: Shadowedinblue,couplingsadmittingstaticsolu- asymptotic AdS and the infrared wall. As boundary data, tionswithouthorizonsford=2. Equalmasscurvesarehighlighted. wewillallowfortimedependentprofilesφ (t)=φ(t,0)and ∞ Right:Thresholdmassforcollapseofpulses(12).Inset:Ratioofthe φ (t)=φ(t,z ). Dynamical processes triggered by φ (t) in 0 0 ∞ temperatureoftheblackholeatthresholdtothemassgap. the hard wall setup were studied in [8]. The possibility of imposingatimedependentscalarprofileatthewallhasbeen Theinitialstatewillbetakentohavevanishingφ ,φ =φ¯ ∞ 0 consideredin[9]. andA satisfying(10). Weshallchoosethewallprofile 0 The family (3)-(4) does not exhaust the set of static solu- tions to our gravity system. In general they break Lorentz φ (t)=φ¯+1η(cid:16)1+tanht(cid:17). (11) 0 invariance and are described by the ansatz (6). Their en- 2 a ergy density can be read from the asymptotic expansion A= This models a quench with a finite time span controlled by 1−2Mzd+.... It is then clear that all solutions (3)-(4) have the parameter a. After the quench φ =φ¯+η, while A will 0 0 zeromass. Forstaticsolutions befixedbytheconservationofenergyatthewall. Anyφ (t) ∞ M= 21z−0d(1−A0)+12(cid:90) z0z1−dAΦ2dz, (9) whahviechLfourlefinltlzs(i1n0v)araitalnattectoiumpelsin,gesn.suWreestdhoatnthoet wfinaanltQhFowTewvielrl 0 thatthequenchfollowsanarbitrarypathinthecouplingspace with A =A(z ). The first term represents the contribution ofFig.2a. Weaimtoonlyactonthecombinedcouplingthat 0 0 to the total energy from the geometry hidden by the infrared moves along the M=0 line. This involves a tuned variation cutoff. When this part encloses a naked singularity it can be of the scalar field at the wall and the AdS boundary, which arbitrarily negative. On the contrary, the second term is al- theabsenceoftime-likeKillingvectorrendersunclearhowto wayspositiveforsolutionswithouthorizons.Thereforenaked implement. Inthefollowingwewillassumethatthediagonal singularitiesareacrucialingredientforobtainingLorentzin- timecoordinatein(6)providesareasonablewaytoprojectthe variantbackgroundswithnon-trivialscalarprofiles. valueofφ ontothedualQFT.Hencewerequire(10)tohold 0 Uptoatrivialglobalshiftinthescalar,staticsolutionsare ateachconstanttimeslice. parameterizedby∆φ andA . Thebackgrounds(3)-(4)define 0 Numericalresults. Thecentralcharacteristicofthedynam- thecodimensiononesubset ics generated by (10)-(11), is whether or not it will generate √ A =cosh2( d∆φ), (10) ahorizon. Intheaffirmativecase,theendpointoftheevolu- 0 tionisaSchwarzchildblackholetrappingthetotalmass.This seeredlineinFig.2. SinceA isaboundarydata,itisnatural represents a unitary process in the dual QFT leading to ther- 0 toalsointerpretitasaQFTcoupling. Theuniquestaticsolu- malization[1][10]. Thosethatdonotformahorizonresultin tion without horizons for ∆φ and A in the shaded region of ascalarpulsethatbouncesforeverbetweenAdSboundaryand 0 Fig.2a,representsthegroundstateoftheassociatedQFT[9]. wall[8]. Bouncinggeometriesprovidetheholographiccoun- WeconsiderthattheQFTbeforethequenchisintheground terparttoperiodicreconstructionsofquantumcorrelationsin state for chosen couplings in the subset (10). Acting on the thedualfieldtheory[11],knownasquantumrevivals[12]. boundaryvaluessuchthatφ changeswhileφ remainscon- Theonlytopologicalobstructiontotheformationofahori- ∞ 0 stant,clearlybringsoutside(10). Thesameactuallyhappens zoninoursetupisthepresenceofthewall,enforcingMz2> 0 intheoppositecase. Whenφ iskeptconstant,theequations 1/2. Afirstquestiontheniswhetherthetypicalscaletrigger- ∞ of motion ensure the conservation of total mass. A time de- ingfastthermalizationissetbyz ordependsonthe∆φ. Be- 0 pendentφ generatesascalarpulsethatentersthegeometryat forestudyingquenches,weanalyzetheinfallofascalarshell 0 3 modellinganenergyinjectionwithoutvariationofthehamil- inthemassiveSchwingermodelafteraquench[13]. Theim- tonian. We restrict in the following to d=2 for numerics. portantdifferenceinourcaseistheirenergydensity. Itcanbe Sincetheshapeofthepulseinfluencestheevolution,wecon- muchlargerthanthemassgap,properinholographicmodels sideratypicalshell,radiallylocalizedandofgaussianform ofaconfiningphase,ranginguptoO(1/G),closetothetyp- ical values in the plasma phase. In spite of that, the physics Π(t=0)∝z2e−σ12tan2(2πzz0), (12) drivingthermalizationdoesnotrefertoω0. Thisisillustrated intheinsetofFig.2b. Thetemperatureoftheblackholeatthe withσ=0.1andΦ(t=0)inthefamily(3)-(4). Thethreshold collapsethresholdforthegaussianpulses(12),iswellbelow mass for gravitational collapse without bounces is plotted in themassgap. Fig.2b. Itstronglygrowswith∆φ, confirmingthesecondary Traveling pulses exhibit radial localization and displace- roleoftheinfraredwall. Usingthehardwallasanauxiliary ment. They represent in general partial revivals. They have element,weareactuallyobtainingabasicmodelofasoftwall. largermasses,andtheassociatedQFTstatesarethusexpected tocontainhigherenergyexcitationsandnon-zeromomentum z02M 〈O∞〉 modes. The former should be connected with the projection 1.4 a=0.05 6 �=��� �=��� ofnarrowpulsesonhigherharmonicmodes. Theradialinfall 1.2 a=0.15 �=���� ofanarrowshellhasbeenrelatedtotheevolutionofthesepa- a=0.1 rationbetweenentangledexcitationsafteraquench[1][3][5], 1.0 a=0.2 4 0.8 a=0.25 theso-calledhorizoneffect[14].Inthissense,radialdisplace- 0.6 a=0.3 tω0 mentindicatesthepresenceofnon-zeromomentummodesin 0.4 a=0.4 -1 1 2 2π thedualfieldtheorystate. Sincethequenchwearemodelling 0.2 isglobal,finitemomentummodescanonlybecreatedinpairs. a=0.6 0.00.0 0.2 0.4 0.6 0.8 η 0 Fig.1bshowsthat2ω0≈ω1foralargerangeofcouplings,ex- plaining why radial localization and displacement appear at Figure3. Left: Energydensitygeneratedby(10)-(11)forφ¯=0.7 similarenergies. andd=2.Right:(cid:104)O∞(cid:105)forthreequencheswithdifferenttimespans. Contrary to standing waves, traveling configurations gen- erated by (10)-(11) are composed of two distinct sub-pulses, one entering from the AdS boundary and the other from the Weexplorenowtheevolutionsafteraquenchmodelledby wall. This is clearly appreciated in the one-point functions. (10)-(11)ind=2. ThequenchwillbeappliedtotheLorentz invariantbackground∆φ=φ¯=0.7. Atthisvaluetheinfrared Fig.3bshowsthevevoftheoperatorsourcedbyφ∞ forthree examples from Fig.3a. We use a rescaled time such that the physics starts to be dominated by the hidden singularity in- fundamentalfrequencyforthefinalcouplingsis2π. Theos- steadofthewallposition,seeFig.1. Wefocusonη>0,and cillationsof(cid:104)O (cid:105)areplottedinblueforaslowquench,with hencethequenchwillincreasethemassgap.Fig.3ashowsthe ∞ a=0.6, resulting in a standing wave. A traveling configura- final energy density as a function η for several values of the tionswithtwosub-pulsesproducingsignalsofsimilarmagni- timespana.Itsgrowthwithηismorepronouncedthesmaller tude is obtained for a=0.15 and shown ingreen. The effect isa. Wehaveshadedinbluetheparametersthatleadtoblack ofbothsub-pulsessuperposes,givingrisetooscillationswith hole formation. Processes where a horizon is generated af- roughlytwicethefundamentalfrequency.Inmagentawehave tersomebouncingcyclesoccupyjustasmallwindowonthe anintermediumconfiguration,withasmallboundarycompo- boundaryoftheblueregion. Otherwiseweobtaingeometries nent. Itisworthmentioningaslightincreaseintheperiodof that keep bouncing as far as our simulation could go. Only oscillations between the a=0.6 and a=0.15 pulses. This is sufficiently fast quenches, those with a<0.25 in the exam- duetotheirdifferentfinalenergies: M=0.002andM=0.02 ple of Fig.3a, can generate enough energy density to trigger respectively. The increase of the period with the energy is thermalization. generic in holographic quenches, finding some analogues in Bouncinggeometriescanberoughlydividedintwotypes: thecondensedmatterliterature[11]. standingandtravelingwaves. Standingwavesprojectmainly Thedistinctionbetweenfastandslowquenchesshouldre- on the fundamental harmonic of the static background asso- fer to the characteristic scale of the infrared physics. Slow ciated to the final couplings. It is convenient to restore the naturalmassunits,M→ d−1M,withGextremelysmall. Ac- quenches can be unambiguously defined as those producing 8πG standingorquasi-standingwaves. Weconsidernowquenches cordingtotheholographicdictionary, 1/Gisproportionalto with fixed amplitude η and time span a, but different initial thenumberofelementarydegreesoffreedominthedualQFT. coupling φ¯. Fig.1a shows that the mass gap grows with the HenceMtranslatesintoanenergydensityperspeciesinfield coupling. Thereforethe quench shouldresult in acollapsing theory terms. Although the mass of standing waves is much shell,abouncingpulseorastandingwaveaswechooselarger smallerthanthatrequiredforcollapse,itcanbeparametrically values of φ¯. Alternatively, the energy density in units of the larger than G. Indeed quenches in Fig.3a generate standing final mass gap must be a monotonically decreasing function waveswhena≥0.6,havingmassesuptoMz2≈0.1. 0 ofφ¯. ThisquantityisplottedinFig.4aforη=0.2andseveral Standing waves oscillate with the frequency of the mass smallvaluesofa,confirmingtheexpectedbehavior. gap, ω . It is then natural to holographically identify them 0 withcoherentstatesof(cid:126)k=0modesofthelowestQFTexcita- Onepointfunctionsatthewall. Wehaveassumedthatthe tion.Revivalswiththesameinterpretationappearforexample boundaryvaluesφ andA relatetocouplingswithawellde- 0 0 4 atthewall. Usingtheaboveproposedvaluefor(cid:104)O (cid:105),(15)at M/ω2 0 0 thewallcanberewrittenas 0.35 �=���� 15 0.3 00..2350 ������� 10 〈�∞〉 00..12 〈��〉 φ˙0(cid:104)O0(cid:105)+12A˙0z−0de−δ0 =0. (16) 0.20 0 t 0.15 ���� 5 1 2 3 ThereforetheexpectationvalueoftheoperatorO sourcedby A 0.10 0.05 1 2 3 4 5 t A0, isgivenbytheexpressionmultiplyingitstimederivative 〈� 〉 inthepreviousequation. ϕ � 0.0 0.5 1.0 1.5 -5 A check on the consistency of these assignments is how they behave when a horizon forms. Thermalization after a Figure4. Left:Energydensitynormalizedbythesquareofthefinal global quench in an infinite system only happens at the lo- mass gap in quenches with different initial coupling and η=0.2. cal level. Namely, for any late but finite time there are suf- Right: Evolutionofone-pointfunctionsafteraquenchwitha=0.3 ficiently large regions where non-local observables have not andη=1,leadingtothermalization. yet achieved thermal values. Such observables, as for exam- ple the entanglement entropy, require information from be- fined,localprojectiononthefieldtheorytimecoordinate.The hindtheapparenthorizonfortheirholographicdetermination sameasφ ,theyshouldsourcelocaloperators. Weaimtode- ∞ [1][4]. One-pointfunctionsarelocalobservables,whichthus terminetheirexpectationvalues. shouldonlyimplythegeometryoutsideit. Wehaveusedcon- Symmetryunderglobalshiftsofthescalarfieldimpliesthat stantt slicestotranslatewallboundaryvaluesintoQFTcou- onlythedifference∆φ=φ0−φ∞isphysicallyrelevant. Hence plings. Constantt slices only approach the apparent horizon the ground state expectation values of the operators O and 0 asymptotically at late times, in the region where it has prac- O cannotbeindependent. Whilethelatterisdictatedbythe ∞ tically achieved its final value z . They depart again from BH asymptoticexpansionsattheAdSboundary,theformerhasto itatz>z , andfinallyreachthewall. Thisimpliesthatin- BH depend on quantities evaluated at the wall. The scalar equa- deed, (cid:104)O (cid:105)and(cid:104)O (cid:105)donotrequireinformationfrombehind tionforstaticsolutionsreduceto(z1−dAe−δΦ)(cid:48)=0,implying 0 A theapparenthorizonatanyinstanceoftheirevolution. Theonlynonvanishingone-pointfunctionassociatedtoa lim(z1−dΦ)=z10−dA0e−δ0Φ0, (13) Schwarzchildgeometryisthatofthestresstensor. Thusother z→0 expectationvaluesshouldtendtozerointheprocessofgravi- wherewehavegaugefixedt tobethepropertimeattheAdS tationalcollapse. Whenahorizonemerges,thepartofthege- boundary,i.e.δ∞=0.Thelhsisprecisely(cid:104)O∞(cid:105)[15].Defining ometrywithz>zBH getsfrozenforobserversusingtheproper (cid:104)O (cid:105)asminustherhs,weobtainarelationofthedesiredform time at the AdS boundary. This is implemented by the ex- 0 ponential vanishing of e−δ in that region. According to the (cid:104)O∞(cid:105)+(cid:104)O0(cid:105)=0. (14) previousassignmentsboth(cid:104)O0(cid:105)and(cid:104)OA(cid:105)areproportionalto e−δ0,whichinsuresthatindeedtheytendtozeroasahorizon The sign has been chosen such that the operator sourced by forms. Clearly so does (cid:104)O (cid:105). The evolution of the three ob- ∞ φ∞+φ0 has a vanishing vev in the ground state. Notice that servablesafteraquenchgeneratingahorizon,orequivalently (13)wouldnotholdwithV(φ)(cid:54)=0,whenneitheraglobalshift leadingtothermalization,isshowninFig.4b. onthescalarisasymmetryofthesystem. We have proposed a simple holographic scenario, easily ThemetricfunctionAsatisfiestheevolutionequation accessible to numerics, modeling quenches where a relevant A˙=2zAΦφ˙. (15) coupling changes. A number of checks have been success- fullyperformed. Wehopethatthiscanhelpplacinghologra- ThezdcoefficientintheasymptoticexpansionofAdetermines phyamongthestandardtoolsforstudyingoutofequilibrium physics. thedualQFTenergydensity[15]. Thepreviousequationim- plies M˙+φ˙∞(cid:104)O∞(cid:105)=0. However the field theory Ward iden- Acknowledgements. We thank M. Garcia-Perez, R. Em- titiesdictateasumoverallcouplings, M˙+∑λ˙i(cid:104)Oi(cid:105)=0[15]. paran and D. Mateos for discussions. This work was sup- Itisthennecessarythatthecontributionsfromφ andA ex- ported by project FPA2015-65480-P and Centro de Excelen- 0 0 actlycancel,whichistherequirementofenergyconservation ciaSeveroOchoaProgrammeundergrantSEV-2012-0249. [1] J.Abajo-Arrastia,J.AparicioandE.Lopez,JHEP1011(2010) [7] J.Erlich,E.Katz,D.T.SonandM.A.Stephanov,Phys.Rev. 149. Lett. 95 (2005) 261602; L. Da Rold and A. Pomarol, Nucl. [2] T.AlbashandC.V.Johnson,NewJ.Phys.13(2011)045017. Phys.B721(2005)79. [3] V.Balasubramanianetal.,Phys.Rev.Lett.106(2011)191601. [8] B. Craps, E. Kiritsis, C. Rosen, A. Taliotis, J. Vanhoof and [4] T.HartmanandJ.Maldacena,JHEP1305(2013)014. 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