EffectiveFieldTheoryofDissipativeFluids

HongLiu

MichaelCrossley

arXiv:1511.03646

PaoloGlorioso

ConservedquantitiesConsideralong wavelengthdisturbanceofasysteminthermalequilibrium

conserved quantities:cannot relaxlocally,onlyviatransports

Conserved quantitiesGaplessandlong-livedmodes

(only onesinamedium)

Thereshouldexistauniversal lowenergyeffectivetheory.

non-conserved quantities:relaxlocally,

Hydrodynamics

Thermalequilibrium:

Promotethesequantitiestodynamicalvariables:(localequilibrium)

slowlyvaryingfunctionsofspacetime

Expressexpectationvaluesofthestresstensorandconservedcurrent intermsofderivativeexpansionofthesevariables:constitutiverelations.

Equationsofmotion:

d+1variables,d+1equations

Despitethelongandglorioushistoryofhydrodynamics

Itdoesnot capturefluctuations.

Fluctuations

Therearealwaysstatistical fluctuations…..

transports,

Importantinmanycontexts:

Atlowtemperatures,quantum fluctuationscanalsobeimportant.

Longtimetail

dynamicalaspectsofphasetransitions,

non-equilibriumstates,turbulence,

finitesizesystems….

Phenomenologicallevel:stochastic hydro(Landau,Lifshitz)

:noiseswithlocalGaussiandistribution

3.fluctuationsofdynamicalvariablesthemselves

Expect:

1.interactionsamongnoises

2.interactionsbetweendynamicalvariablesandnoises

Untilnownotknownhowtotreatsuchnonlineareffectssystematically.Notevenclearitisagoodquestion.

particularlyimportantfornon-equilibriumsituations.

Constraints

Constitutiverelations:notenough tojustwritedownthemostgeneralderivativeexpansionconsistentwithsymmetries.

1.Entropycondition

2.Onsagerrelations:linearresponsematrixmustbesymmetric

Phenomenologicalconstraints:solutionsshouldsatisfy:

Arethesecomplete?

Currentformulationofhydrodynamicsis awkward.

awkward:usesolutionstoconstrainequationsofmotion

Microscopicderivation?

EffectivetheoryapproachmayalsomakeiteasiertogeneralizehydrodynamicsEOMtolessfamiliarsituations,saywithmomentumdissipations,anomalies.....

develophydrodynamicsasabonafidelowenergyeffectivefieldtheoryofageneralmany-bodysystematfinitetemperature

1.givesafullinteractingtheory ofnoises.

2.Microscopicoriginandcompletenessofphenomenologicalconstraints

3.Newconstraints(nonlinearOnsagerrelations)

Actionprinciplewhichincorporatesbothdissipationsandnoises

Shouldbedistinguished fromanactionwhichjustreproducesstandardeoms (whichmaynotcapturefluctuationscorrectly)

Searchingforanactionprincipleforhydrodynamicshasbeenalongstandingopenproblem,datingbackatleasttoG.Herglotz in1911….....

Allresultsatnon-dissipativelevel….

Manyactivitiessince70’stounderstandhydrodynamicfluctuations….....

Results

1.Hydrodynamicswithclassicalstatisticalfluctuations

isdescribedbyasupersymmetric quantum fieldtheory

2.Hydrodynamicswithquantumfluctuationsalsoincorporated

isdescribedbya“quantum-deformed”(supersymmetric)quantumfieldtheory.

SeealsoHaehl,Loganayagam,Rangamani

Approach:putarelativisticquantummany-bodyssystem inacurvedspacetime

PartII:formulation

Transitionamplitudesv.s.expectationvalues

Weareinterestedinaneffectivetheorydescribingnonlineardynamicsaroundastate.

ShouldbecontrastedwithEFTdescribingtransitionamplitudes,

Closedtimepath(CTP)orSchwinger-Keldysh contour

Shoulddouble alldegreesoffreedom

Hydroeffectivefieldtheory

hydrodynamicmodes

EFTapproach:

1.Whatare? donotwork

2.Whatarethesymmetriesof?

3.Integrationmeasure?

Atlongdistancesandlargetimes:

Allcorrelationfunctionsofthestresstensorandconservedcurrentsinthermalequilibrium

Dynamicalvariables:integratinginToyexample:asingleconservedcurrent

1.Currentconservation:

2.Wmustbenonlocal:Non-localitysolely duetointegratingouthydromodes

Integratein hydromodes:

(a):local (b):Ensure1issatisfied

(c):EOMsmustbeequivalenttocurrentconservations

Proposal:(usetheusualStueckelberger trick)

isalocal action. :hydromodes

Satisfythefollowingconsistencyrequirements:

1.

2.Eoms ofareequivalenttocurrentconservations.

Dynamicalvariables(II)Forstresstensor,weputthesysteminacurvedspacetime

Conservationofstresstensor:

Integrateinhydromodes: Promotespacetime coordinatestodynamicalfields

1.2.Xeoms areequivalenttoconservationofstresstensor

anemergent spacetime withcoordinates

Interpretationof: labelindividualfluidelements, internaltime

:motionofafluidelementinphysicalspacetime

SowejustrecoveredtheLagrangedescriptionofafluid!

Asastartingpoint,wecouldsimplydoublethedegreesoffreedomintheLagrangedescription.

Abithistory:

NickelandSonshowedthecovariantversionarisesnaturallyfromholography(arXiv:1103.2137).

DoubledcopiesappearedinHaehl,Loganayagam,RangamaniarXiv:1502.00636, andCrossley,Glorioso,HL,WangarXiv:1504.07611.

UsingasinglecopyofasdynamicalvariableforanidealfluidactiondatedbacktoG.Herglotz in1911.

CovariantwasusedbyTaub in1954.

Rediscoveredin2005byDubovsky,Gregoire,Nicolis andRattazziinhep-th/0512260andfurtherdevelopedbyDubovsky,Hui,Nicolis and Son inarXiv:1107.0731 ,......

Standardhydrovariables(whicharenowderivedquantities)

Asignificantchallenge: ensuretheeoms fromtheactionofXandcanbesolelyexpressedintermsofthesevelocitytypeofvariables.(e.g.solids v.s.fluids)

Symmetries(I)Nowneedtospecifythesymmetriesof

Notethatitisdefinedinfluidspacetime

Requiretheactiontobeinvariantunder:

Interpretationof: labelindividualfluidelements, internaltime

definewhatisafluid!

Itturnsoutthesesymmetriesindeeddomagicforyou:

atthelevelofequationsofmotion,theyensurealldependenceondynamicalvariablescanbeexpressedin

Recoverstandardformulationofhydrodynamics(modulo phenomenological constraints)

Thiswouldbethefullthestoryinausualsituation.

Fullnon-linearfluidfluctuatingdynamicsencodedinnon-trivialdifferentialgeometry:

Symmetries(II)WeareconsideringEFTforasystemdefinedwithCTP:

Thegeneratingfunctionalhasthefollowingproperties:

• KMSconditionplusPTimplyaZ2 symmetryonW:

• Reflectivitycondition:

• Unitaritycondition:

Fullbosonic theoryReflectivityconditioncanbeeasilyimposed,leadingtoacomplex action.

ImposingKMSconditionisverytricky.

AlltheconstraintsfromentropycurrentconditionandlinearOnsagerrelations

NewconstraintsonequationsofmotionfromnonlinearOnsagerrelations.

proposal:localKMScondition,aZ2symmetryontheaction

Imaginarypartoftheactionnon-negative

FermionsandSupersymmetry

isa“topological”conditiononthemeasureofpathintegrals

Introducefermionic partners(“ghost”fieds)fordynamicalvariablesandrequiretheactiontohaveaBRSTtypesymmetry.

SeealsoHaehl etalarXiv:1510.024941511.07809Unitarity condition:

Ataquadratic levelindynamicalfields,onefindsthatlocalKMSconditionleadstoanemergentfermionic symmetry.

Butnotclearhowtowritedownanonlinearactionwithsuchanalgebra.

Requiresa“quantum-deformed”SUSY

Classicallimit:

becomestandardsupersymmetry intimedirection.

Inthislimitonecanwritedownasupersymmetric completionofthefullbosonic hydrodynamicaction.

Notethatintheclassicallimit,pathintegralremains,capturingstatisticalfluctuations.

Example:nonlinearstochasticdiffusion

Considerthetheoryforasingleconservedcurrent,wheretherelevantphysicsisdiffusion.

Dynamicalvariables: (or)

Roughly,:standarddiffusionmode,:thenoise.

Ifignoringinteractionsofnoise

AvariationofKardar-Parisi-Zhang equation

Summary

Fermionic excitationsandEmergentsupersymmetry.

AnEFTforgeneraldissipativefluids.

Recoversthestandardhydrodynamicsasequationsofmotion,constitutiverelations,constraints.

Encodesquantumandthermalfluctuationssystematicallyinapathintegralexpansion.

Fullnon-linearfluidfluctuatingdynamicsencodedinnon-trivialdifferentialgeometry.

FuturedirectionsFormalism:

Non-relativisticlimit,superfluids,Anisotropic,inhomogeneous,“quantum-deformed”Supersymmetry

…....Applications:

Longtimetails,runningofviscosities,

Dynamicalaspectsofclassicalandquantumphasetransitions

ScalingbehaviorinhydrobehaviorviafixedpointsofQFTs,suchasKPZscaling,turbulence….

….........

Non-equilibriumsteadystates,dynamicalflowsofQGP

ThankYou