Adiabatic Hydrodynamics and the Eightfold Way to Dissipation

Transcription

1 Adiabatic Hydrodynamics and the Eightfold Way to Dissipation Felix Haehl (Durham University & PI) Perimeter Institute 12 November 2015 FH, R. Loganayagam, M. Rangamani [ ], [ ], (see also [ ], [ ], [1511.xxxxx],...)

2 Outline The hydrodynamic gradient expansion Classification of transport Example: conformal fluid Schwinger-Keldysh and emergent gauge symmetry Outlook and conclusion

3 Felix Haehl (Durham University), 1/27 The hydrodynamic gradient expansion Hydrodynamics: generic near-equilibrium eff. field theory for long wavelength fluctuations microscopic theory L l mfp macroscopic fluid variables: u µ (x), T (x), µ(x) (u 2 = 1) background sources: g µν (x), A µ (x) Constitutive relations: T µν = T µν (0) + T µν (1) +... J α = J α (0) + J α (1) +... Dynamics: ν T µν F µν J ν + T ν H α J α 0J H E.g.: T µν (0) = ε(t, µ) uµ u ν + p(t, µ) (g µν + u µ u ν ), J(0) α = q(t, µ) uα

4 Felix Haehl (Durham University), 1/27 The hydrodynamic gradient expansion Hydrodynamics: generic near-equilibrium eff. field theory for long wavelength fluctuations microscopic theory L l mfp macroscopic fluid variables: u µ (x), T (x), µ(x) (u 2 = 1) background sources: g µν (x), A µ (x) Constitutive relations: T µν = T µν (0) + T µν (1) +... J α = J α (0) + J α (1) +... Dynamics: ν T µν F µν J ν + T ν α J α J H J H T ν H (cov. anomalies) E.g.: T µν (0) = ε(t, µ) uµ u ν + p(t, µ) (g µν + u µ u ν ), J(0) α = q(t, µ) uα

5 Felix Haehl (Durham University), 2/27 The hydrodynamic gradient expansion Solving hydrodynamics in this talk does not mean to calculate transport coefficients (ε, p,...) for any particular microscopic system... to solve the fluid equations for {u µ (x), T (x), µ(x)} It rather means: to provide all symmetry-allowed constitutive relations order by order in µ which are consistent with the Second law constraint J µ S = s(t, µ) uµ + J µ S,(1) +... with µj µ S 0 (on-shell) Gives quite non-trivial constraints on physically allowed constitutive relations, e.g.: Neutral ideal fluid: ε + p = s T Neutral 1 st order: viscosities η, ζ 0 Neutral 2 nd order: 5 relations among 15 a-priori independent transport coefficients Anomaly induced transport completely fixed Bhattacharyya 12 Son-Surowka 09 Jensen-Loganayagam-Yarom 13 kkkkkk.

6 Felix Haehl (Durham University), 3/27 Plan This talk: the structure of hydrodynamics Understand constitutive relations allowed by Second Law: Classify hydrodynamic transport in a physically useful way Construct most general solution at all orders Suggest a unifying framework for adiabatic transport: New gauge symmetry that explains the 2nd law constraint Talk on Nov 20: the field theory behind hydrodynamics Use hydrodynamics as a tractable starting point to learn basic lessons about some important problems across physics: How to understand hydrodynamics as a Wilsonian field theorist? Ingredients: Schwinger-Keldysh, SUSY, topological σ-models,... Black holes via AdS/CFT: dissipation, entropy, unitarity,...

7 Outline The hydrodynamic gradient expansion Classification of transport Example: conformal fluid Schwinger-Keldysh and emergent gauge symmetry Outlook and conclusion

8 Felix Haehl (Durham University), 4/27 Disclaimer From now on I will only discuss neutral fluids. Adding an arbitrary number of abelian or non-abelian flavours is just a technical task without new conceptual ideas, see [ ].

9 Felix Haehl (Durham University), 5/27 Off-shell entropy production and adiabaticity Inequality constraint µ J µ S 0 is much more conveniently incorporated if we don t have to simplify it using equations of motion. Use Lagrange multiplier β µ and consider off-shell statement: µ J µ S + β µ ( ν T µν T µ H ) 0 Natural Lagrange multiplier: β µ = 1 T uµ (local thermal vector) Loganayagam 11 Task: solve for {J µ S, T µν } as functionals of {β µ, g µν } Ideally: find effective action which defines all solutions off-shell Marginal case = 0: adiabaticity equation Particularly rich structure! focus on this first

10 Felix Haehl (Durham University), 6/27 Aside: adiabaticity equation for free energy current µ J µ S + β µ ( ν T µν T µ H ) = 0 Can trade entropy current J µ S for free energy current Gµ : Gµ T J µ S (J µ S ) canonical with (J µ S ) canonical = β νt µν Grand-canonical version of adiabaticity equation: [ µ ( G µ T ) ] G H = 1 T 2 T µν β g µν Solve for {G µ, T µν } as functionals of {β µ, g µν }

11 Classification of hydrodynamic transport Felix Haehl (Durham University), 7/27

12 Classification of hydrodynamic transport Felix Haehl (Durham University), 7/27

13 Felix Haehl (Durham University), 8/27 Anomaly induced transport (Class A) [ σ ( G σ T ) ] G H = 1 T 2 T µν β g µν + First of all: let s get rid of anomalies G H = u νt ν H Can always split off from a solution {G σ, T µν } a particular solution {(G σ ) A, (T µν ) A } that takes care of anomalies with ( ) A = 0: [ ( ) ] (G σ ) A σ T G HT = 1 2 (T µν ) A β g µν Loganayagam 11 Jensen-Loganayagam-Yarom 13 Anomalous transport coefficients fixed in terms of anomaly polynomial finite class

14 Felix Haehl (Durham University), 9/27 Dissipative transport (Class D) ( G σ ) σ T 1 2 T µν β g µν 0 Now consider transport which does generically produce entropy ( > 0) Such terms appear in three varieties: 1 Sign-definite terms (inequalities from 2 nd law) These only show up at leading order! 2 Sign-indefinite terms which are dominated by sign-definite terms (no constraints from 2 nd law) 3 Sign-indefinite terms which are dominant in derivative expansion (forbidden by 2 nd law) Example: T µν (1) = ζ Θ (gµν + u µ u ν ) (Θ µ u µ ) gives = ζ 1 T Θ2 ζ 0 (type 1 ) At higher orders: any T µν (2) = γ [ ]µν s.t. = γ Θ 3 will be subdominant, hence unconstrained (type 2 ) Bhattacharyya

15 Felix Haehl (Durham University), 10/27 Hydrostatically forbidden terms (Class H F ) ( G σ ) σ T 1 2 T µν β g µν 0 Type 3 : sign-indefinite terms at dominant order in Need to be zero for consistency with 2 nd law! Example: Ideal fluid T µν (0) = ε uµ u ν + p (g µν + u µ u ν ), J µ S,(0) ( = s uµ Θ (T s ε p) }{{} T + T ds dt dε ) (u )T dt T =0 (!) }{{} =0 (!) A-priori: 3 parameters But second law enforces: 2 relations This is Class H F : combinations forbidden by 2 nd law

16 Felix Haehl (Durham University), 11/27 Dissipative transport (Class D) ( G σ ) σ T 1 2 T µν β g µν 0 There s a simple way to construct Class D explicitly: ( (T µν ) D = N (µν)(αβ) + N (αβ)(µν)) β g αβ (G σ ) D = 0 Every term in involves a total square of the form = N ((µν)(αβ)) ( β g µν)( β g αβ ) Hence 0 automatic as long as the leading order is sign-definite Easy task at any order in : find all tensor structures N (αβ)(µν) [β µ, g µν ]

17 Felix Haehl (Durham University), 12/27 Hydrostatics (Class H) Hydrostatic transport: time-independent equilibrium configurations timelike Killing vector K µ = β µ equil. : K g µν = 0 Spacetime manifold M: Euclidean structure Σ M S 1 K µ Σ M Σ M Transport captured by Euclidean path integral/partition function: [ˆ ( G σ ) ] W Hydrostatic = [total free energy] = d d 1 S σ T Σ M Decompose: G σ = S β σ + V σ This splits Class H into two subclasses: H = H S H V Variation w.r.t. gµν gives all hydrostatic T µν Hydrostatic

18 Felix Haehl (Durham University), 13/27 Lagrangian solutions (Class L) Consider Landau-Ginzburg action: Fields: fluid vector & background geometry = {β µ, g µν } Symmetries: diffeomorphism invariance ˆ g S eff = L[β µ, g µν ] Basic variation defines hydrodynamic currents: δs eff = [ g 1 2 T µν δg µν + T h σ δβ σ + µ ( ) µ ] }{{} surface term Further, define entropy current: [ ] J µ 1 S = s δs eff uµ with s g δt {u µ,g µν} fixed = h σ β σ Can show: {T µν, J µ S } solve adiabaticity equation (= off-shell version of 2 nd law constraint)

19 Lagrangian solutions (Class L) To get correct dynamics, formulate problem as a σ-model: X µ : worldvolume reference configuration space filling brane (physical fluid) g ab = Xµ σ a X ν σ b gµν[x(σ)], βa = σa X µ βµ [X(σ)] X µ (σ) {β µ, g µν} Vary pullback fields X µ, while holding the reference configuration β a fixed δs eff δx µ = 0 + diffeo Bianchi id. µt µν 0 σ a X µ {β a, g ab } Lesson: fluids are naturally σ-models with dynamical d.o.f. = pullback maps (c.f. formulation of non-dissipative fluids in terms of Goldstone modes Dubovsky-Hui-Nicolis-Son 11) Felix Haehl (Durham University), 14/27

20 Felix Haehl (Durham University), 15/27 What do we have so far? Class A: anomalies can be dealt with once and forever Class D: can genuinely produce entropy ( 0) Class H F : Constitutive relations forbidden by 2 nd law G σ = Sβ σ + V σ Class H = H S H V : Hydrostatic response to free energy density and flux Class L: Wilsonian action giving currents consistent with 2 nd law But Lagrangians being scalars, we only get: Class L = HS H S Some more situations that we re missing so far: Free energy current G σ could be zero or topological Non-hydrostatic free energy flux vectors (H V )

21 Felix Haehl (Durham University), 16/27 Berry-curvature type solutions (Class B) ( G σ ) σ T 1 2 T µν β g µν = 0 Consider the following currents: ( (T µν ) B N (µν)(αβ) N (αβ)(µν)) β g αβ (G σ ) B = 0 Trivially solve adiabaticity equation Manifestly non-hydrostatic ( β g µν = 0 in hydrostatics) Seemingly not captured by Lagrangians (Class L) Easy task at any order in : find all tensor structures N (αβ)(µν) built out of {β µ, g µν } Examples in d = 2 + 1: Hall conductivity, Hall viscosity

22 Felix Haehl (Durham University), 17/27 Transverse non-hydrostatic free energy (Class H V ) Remember splitting: G σ = S β σ + V σ with β σ V σ = 0 Consider solutions to adiabaticity equation with non-trivial and non-hydrostatic V σ Transport genuinely due to free energy flux These are in general parameterized as (T µν ) HV D ρc ρ(µν)(αβ) N β g αβ + 2 C ρ(µν)(αβ) N D ρ β g αβ Easy task at any order in : find all tensor structures C ρ(µν)(αβ) N built out of {β µ, g µν }

23 Felix Haehl (Durham University), 18/27 Conserved entropy current (Class C) Another trivial solution to adiabaticity equation: exactly conserved entropy current (J µ S ) C = J µ with D µ J µ 0, (T µν ) C = 0 If cohomologically non-trivial: describes topological states in the fluid (no energy/charge transport) Example: Euler current J σ in d = Euler ˆ D σ J σ 0, Euler γ (J σ u Euler σ) χ(σ M ) Σ M Golkar-Roberts-Son 14

24 Felix Haehl (Durham University), 19/27 Summary: Classification of hydrodynamic transport (finite) g L[β µ, g µν] equil. g L[β µ, g µν] (finite) C ρ(µν)(αβ) N [β µ, g µν] N ((µν)(αβ)) [β µ, g µν] N [(µν)(αβ)] [β µ, g µν] (finite) Theorem: The eightfold way of hydrodynamic transport FH-Loganayagam-Rangamani There are eight classes of {T µν, J µ S } consistent with µj µ S 0. All of them can be constructed easily at all orders in µ. Constitutive relations not produced by this algorithm, are forbidden by second law (Class H F ).

25 Outline The hydrodynamic gradient expansion Classification of transport Example: conformal fluid Schwinger-Keldysh and emergent gauge symmetry Outlook and conclusion

26 Felix Haehl (Durham University), 20/27 Example: neutral Weyl-invariant fluid at O( 2 ) Most general 2 nd order (neutral, Weyl-invariant) stress tensor: T µν (2) = (λ 1 κ) σ <µα σ α ν> + (λ 2 + 2τ 2κ) σ <µα ω α ν> + τ ( u α D W α σµν 2σ <µα ω α ν> ) + λ 3 ω <µα ν> ω α + κ (C ) µανβ u αu β + σ <µα σ ν> α + 2σ <µα ν> ω α

27 Felix Haehl (Durham University), 20/27 Example: neutral Weyl-invariant fluid at O( 2 ) Most general 2 nd order (neutral, Weyl-invariant) stress tensor: T µν (2) = (λ 1 κ) σ <µα σ α ν> + (λ 2 + 2τ 2κ) σ <µα ω α ν> + τ ( u α D W α σ µν 2σ <µα ω α ν> ) Class H S + λ 3 ω <µα ν> ω α ( ) + κ C µανβ u αu β + σ <µα σ ν> α + 2σ <µα ν> ω α Class H S Class H S τ, λ 3, κ Are all derivable from a Lagrangian (Class L) L W 2 = 1 [ 2 κ ] 4 (d 2) (W R) + 2 (κ τ) σ 2 + (λ 3 κ) ω 2 Note: λ 3 and κ are hydrostatic, τ is genuinely hydrodynamic

28 Felix Haehl (Durham University), 20/27 Example: neutral Weyl-invariant fluid at O( 2 ) Most general 2 nd order (neutral, Weyl-invariant) stress tensor: T µν (2) = (λ 1 κ) σ <µα σ α ν> + (λ 2 + 2τ 2κ) σ <µα ω α ν> Class D + τ ( u α D W α σµν 2σ <µα ω α ν> ) Class H S + λ 3 ω <µα ν> ω α ( ) + κ C µανβ u αu β + σ <µα σ ν> α + 2σ <µα ν> ω α Class H S Class H S (λ 1 κ) Leads to entropy production (λ 1 κ) 1 T σµ νσ ν ρσ ρ µ Dissipative ( not captured by L-G Lagrangian) (but unconstrained by second law, since σ 3 σ 2 )

29 Felix Haehl (Durham University), 20/27 Example: neutral Weyl-invariant fluid at O( 2 ) Most general 2 nd order (neutral, Weyl-invariant) stress tensor: T µν (2) = (λ 1 κ) σ <µα σ α ν> + (λ 2 + 2τ 2κ) σ <µα ω α ν> Class D Class B + τ ( u α D W α σµν 2σ <µα ω α ν> ) Class H S + λ 3 ω <µα ν> ω α ( ) + κ C µανβ u αu β + σ <µα σ ν> α + 2σ <µα ν> ω α Class H S Class H S (λ 2 + 2τ 2κ) Is of the form of a Class B constitutive relation (T µν ) B 1 4 (G σ ) B = 0 ( N (µν)(αβ) N (αβ)(µν)) β g αβ because of orthogonality: σ <µα ω α ν> β g µν = 0

30 Felix Haehl (Durham University), 21/27 Example: neutral Weyl-invariant fluid at O( 2 ) Out of 5 transport coefficients, 3 come from a Lagrangian: τ, λ 3 and κ For fluids described by L[β µ, g µν ], the other 2 combinations are zero: (λ 1 κ) = 0 and (λ 2 + 2τ 2κ) = 0 These relations have been observed in Einstein gravity Our simple Lagrangians seem to know about holography Derive L[β µ, g µν] from gravity directly? Nickel-Son 10 Haack-Yarom 08 de Boer et al. 15, Crossley et al. 15 First relation ensures no entropy production at subleading order (this is not required by second law!) Principle of minimum dissipation in holography? FH-Loganayagam-Rangamani 14

31 Outline The hydrodynamic gradient expansion Classification of transport Example: conformal fluid Schwinger-Keldysh and emergent gauge symmetry Outlook and conclusion

32 Felix Haehl (Durham University), 22/27 Schwinger-Keldysh doubling Non-equilibrium effective field theory in general described by Schwinger-Keldysh formalism Systems in mixed state: path integral evolves both and Double fields & symmetries: iε R copy O R Ô t H phys H R H L L copy O L i(ε β) Schwinger, Keldysh, Feynman-Vernon, 60s In principle many applications: off-equilibrium physics, dissipation, gravity with horizons, complementarity,... CFT L CFT R

33 Felix Haehl (Durham University), 23/27 Schwinger-Keldysh hydrodynamics? Can we upgrade Class L Lagrangians to describe all 8 classes, using Schwinger-Keldysh? Would like to double the sources: g µν {g µν, g µν }...and write something like ˆ g 1 S eff = 2 T µν [β µ, g µν] g µν 2 g δs eff δ g µν = T µν This would allow us to get any T µν from an action. Including Class H F! Problem: Just blindly doubling gives too much freedom. Can easily violate 2 nd law. Doubling is very powerful. But how to control it? Important obstacle for systematic understanding of eff. field theory of mixed states (dissipation, entanglement, horizons,...)

34 Felix Haehl (Durham University), 24/27 A new macroscopic symmetry We found a consistent Schwinger-Keldysh doubling for hydrodynamics. It is constrained in exactly the right way to reproduce the eightfold way. Danger of doubling controlled by introducing an emergent U(1) T symmetry.

35 Felix Haehl (Durham University), 24/27 A new macroscopic symmetry We found a consistent Schwinger-Keldysh doubling for hydrodynamics. It is constrained in exactly the right way to reproduce the eightfold way. Danger of doubling controlled by introducing an emergent U(1) T symmetry. Remember Class L free energy current: G µ = L u µ + T (bdy. term β ) µ N µ Gµ T is Noether current for diffeos along βµ Postulate a U(1) T gauge symmetry with gauge field A (T) µ coupling to N µ Proposed field content: Hydrodynamic field: β µ Background source: g µν SK copy of source: g µν U(1) T gauge field: A (T) µ

36 Felix Haehl (Durham University), 25/27 The eightfold master Lagrangian (Class L T ) Any constitutive relations {T µν, G σ } which satisfy adiabaticity equation can be obtained from a diffeo and U(1) T invariant Lagrangian: L T = 1 2 T µν [β µ, g µν ] g µν Gσ [β µ, g µν ] T A (T) σ FH-Loganayagam-Rangamani Bianchi identity for U(1)T invariance reduces to adiabaticity equation Equations of motion are: As in Class L: D νt µν 0 From A (T) σ: D µj µ S 0 Conversely: any diffeo and U(1) T invariant Lagrangian gives adiabatic constitutive relations That is: U(1) T invariance gives precisely the right constraint!

37 Outline The hydrodynamic gradient expansion Classification of transport Example: conformal fluid Schwinger-Keldysh and emergent gauge symmetry Outlook and conclusion

38 Felix Haehl (Durham University), 26/27 Outlook: Talk next Friday (Nov 20, string group meeting) Try to derive this scenario from first principles Study Schwinger-Keldysh path integrals in detail Find very interesting and universal structures: Hidden topological (BRST) supersymmetry behind every SK path integral (incl. relativistic fluids) Associated ghosts are crucial to retain unitarity Gauge theory of entropy: general field theory argument Formulate eff. action for all 8 classes Dissipation has to do with ghosts Via AdS/CFT: all these features should play some role in black holes lots of interesting conjectures to work on

39 Felix Haehl (Durham University), 27/27 Summary 8 classes of constitutive relations consistent with the Second Law I gave a recipe for constructing them at any order The classification is useful and physical: Computations become simpler in this framework (classification often tells what is the nicest basis to work in) Conjecture: long-wavelength near-horizon AdS dynamics can be usefully characterized using the Eightfold Way Minimum dissipation conjecture : Holographic fluids optimize entropy production. Schwinger-Keldysh doubling is powerful but dangerous Proposal: emergent U(1) T gauge symmetry controls the doubling in hydrodynamics Next week: systematic justification of U(1) T