A simple holographic scenario for gapped quenches

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by  Esperanza Lopez

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.
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