Effective actions and hydrodynamic transport Mukund Rangamani Gauge/gravity duality 2013 Max Planck Institute for Physics, Munich August 2, 2013 J. Bhattacharya, S. Bhattacharyya, MR [1211.1020] F. Haehl, MR [1305.6968] F. Haehl, R. Loganayagam, MR (work in progress)
Non-dissipative fluids ๏ Introduction ๏ Effective actions ๏ Exhibit 1: Neutral fluids ๏ Exhibit 2: Hall transport ๏ Exhibit 3: Anomalous transport ๏ Summary
Hydrodynamics as an effective field theory ✦ Hydrodynamics describes low-energy, near-equilibrium behaviour fluctuations of an equilibrium density matrix on scales large compared to the characteristic mean free path. ✦ Organize data into conserved currents: T µ ν , J µ ✦ Dynamics: conservation laws for the currents (up to anomalies) r µ T µ ν = 0 , r µ J µ = 0 ✦ Summarize hydrodynamic data as constitutive relations for the currents in terms of operators built from the hydrodynamical variables T µ ν = ε u µ u ν + P P µ ν + Π µ ν J µ = q u µ + ν µ
Constraints on hydrodynamics ✦ Constitutive relations obtained in a gradient expansion with transport coefficients/thermodynamic response parameters determined by microscopics. Π µ ν = − η σ µ ν − ζ θ P µ ν + · · · ✦ The transport data are constrained macroscopically by demanding the second law of thermodynamics hold locally, eg., η , ζ ≥ 0 9 J µ r α J α ! S � 0 S ✦ Typically second law: ✴ implies inequalities for transport coefficients ✴ fixes thermodynamic response parameters (5 for neutral fluids at 2 ∂ ). S. Bhattacharyya ’12
Constraints on hydrodynamics ✦ Thermodynamic response are adiabatic data r µ J µ S = ( r µ J µ S ) diss + ( r µ J µ S ) adiabatic ✦ These can be captured from an equilibrium partition function Z [ g µ ν , A µ ] which is a functional of background sources. Banerjee et.al. ’12 Jensen et.al. ’12 ✦ Status quo: Jensen, Loganayagam, Yarom ’12 ✓ neutral/charged fluids to 2 ∂ ✓ Parity odd fluids to 1 ∂ ✓ superfluids to 1 ∂ ✓ anomalous transport
An autonomous theory of hydrodynamics? ✦ Are the constraints exhaustive? ✴ gradient expansion is systematic but not derived from usual principles for effective field theories ✦ First principles understanding of entropy current? ✦ Would ideally like to have an effective action for deriving the dynamics. ✴ dissipation introduces some difficulties. ✴ No obstruction if we switch off dissipation. ➡ Describe effective actions for non-dissipative fluids (NDF) postponing issues about physical reasonableness etc., till later.
Non-dissipative fluids: Definition ✦ Requirements of an effective action for NDF ✴ Dynamical eom = conservation equations r µ T µ ν = 0 δ S eff = 0 ) = ✴ Lack of dissipation conserved entropy current r α J α S = 0 ⇒ = ✦ Ideal fluids clearly comprise one such system. The surprise is that there are non-trivial non-linear examples which seem to suggest some interesting constraints on hydrodynamic transport. ✦ Formalism is quite old: Taub ’54, Carter ’73 ✦ Modern presentation: Dubovsky, Hui, Nicolis, Son ’11
Lagrangian fields & symmetries ✦ The fundamental fields for NDF are taken to be Lagrangian variables which are labels for the fluid elements: φ I , I = 1 , · · · , d − 1 ✦ NB: view fluid as a space filling D-brane. ✦ Field reparameterization invariance: require arbitrary volume preserving diffeomorphisms in configuration space Sdi ff ( M φ ) φ I ! ξ I ( φ ) , Jacobian( ξ , φ ) = 1 ✦ The diffeo invariance in configuration space guarantees that Euler-Lagrange equations are identical to energy momentum conservation. δ S eff 2 r µ T µ ν = 0 , T µ ν = � δ ϕ S eff = 0 p� g ( ) δ g µ ν
Entropy current ✦ Volume preserving symmetry conserved entropy current ⇒ = d − 1 1 J β = ( d � 1)! ✏ βα 1 ... α d − 1 ✏ I 1 ...I d − 1 Y @ α j � I j r α J α S = 0 j =1 ✦ Interpret this current as being the entropy current to all orders by passing to the entropy frame q J α = s u α � g αβ J α J β s = ✦ Operator dimensions as appropriate for a phase field: [ d φ ] = 0 ✦ Intuitively expect that all dissipative transport coefficients will be vanishing in the theory; borne out by explicit analysis.
Neutral fluids: 0 ∂ and 1 ∂ ✦ Zeroth order action reproducing ideal fluid behaviour Z d d x √− g f ( s ) S 0 ∝ T µ ν = ( s f 0 ( s ) − f ( s )) g µ ν + s f 0 ( s ) u µ u ν ✦ Basically the action is the energy density as a function of entropy density. ✦ 1 ∂ corrections: No corrections for parity-even fluid dynamics since only available term is a total derivative d d x p� g J α d d x p� g r α ( f 1 ( s ) J α Z Z s r α f 1 ( s ) = s ) S 1 / ✦ Exception: parity-odd fluids which we will visit in a bit.
Neutral fluids: 2 ∂ ✦ Obvious invariants at 2 ∂ . Use the decomposition Θ r µ u ν = � u µ a ν + σ µ ν + ω µ ν + d � 1 P µ ν linearly dependent a 2 , σ 2 , ω 2 , Θ 2 , R, R µ ν u µ u ν ⟹ 5 parameter family of neutral non-dissipative fluids 5 Z d d x √− g K m ( s ) O k X S 2 = m =1 ✦ 3 parameter family of scalar invariant NDF.
Neutral fluids: 2 ∂ ✦ Conserved entropy current for neutral fluids: standard analysis based on classifying independent tensor structures. 2 2 6 5 S = s u µ + J µ X δ k v µ X α l e µ X β m v µ X γ n j µ 1 ,k + 2 ,l + 2 ,m + 2 ,n m =1 n =1 k =1 l =1 2 13 2 T µ ν = T µ ν X η m t µ ν X X t µ ν ζ n ˆ ξ k t 2 ,k + ideal + 1 ,m + 2 ,n m =1 n =1 k =1 ✦ {2+13} pieces of data in entropy current: conservation implies {2+6} vanish. ✦ Of the remaining {0+7}, {0+2} are exactly conserved ⟹ {0+5} non-trivial. ✦ {2+15} pieces of data in stress tensor: {2+13} fixed by {0+5} entropy data and {0+2} are free. ⟹ 7 parameter family of neutral non-dissipative fluids!
Neutral fluids: Order 2 ✦ The unconstrained second order transport data is . { λ 2 , λ 0 − ξ 2 } Π µ ν = � η σ µ ν � ζ P µ ν Θ + T τ u α r α σ h µ ν i + κ 1 R h µ ν i + κ 2 F h µ ν i + λ 0 Θ σ µ ν � α σ αν i + λ 2 σ h µ α ω αν i + λ 3 ω h µ α ω αν i + λ 4 a h µ a ν i + λ 1 σ h µ � ζ 1 u α r α Θ + ζ 2 R + ζ 3 R 00 + ξ 1 Θ 2 + ξ 2 σ 2 + ξ 3 ω 2 + ξ 4 a 2 + T P µ ν = s u µ + r ν 2 A 1 u [ µ r ν ] T J µ + r ν ( A 2 T ω µ ν ) ⇥ ⇤ S = ✓ ◆ ✓ A 3 ◆ ⇥ R µ ν � 1 T + dA 3 Θ r µ T � P αβ r β u µ r α T 2 g µ ν R ⇤ + A 3 u ν + dT + ( B 1 ω 2 + B 2 Θ 2 + B 3 σ 2 ) u µ + B 4 [ r α s r α s u µ + 2 s Θ r µ s ] ✦ There is a four parameter family of scale invariant NDF. τ = 3 λ 0 , κ 2 = 2 κ 1 = κ , λ 1 , λ 2 , λ 3
Neutral fluids: 2 ∂ Comparison ✦ The NDF derived from the action is consistent with but more restrictive than the one obtained by demanding the existence of a conserved entropy current. ✦ All the transport data can be fixed in terms of the coupling functions K n . B 3 = � K 1 + K 2 A 3 = � K 5 B 1 = K 2 � K 1 T , , 2 T 2 T + s 2 ds � s 4 B 2 = � K 1 + K 2 dT dK 5 2 T K 3 6 T T 2 ds 2 T � 1 B 4 = K 4 dT dK 5 T 2 ds ds ✓ ◆ λ 0 = s dK 5 ds � 2 dK 5 dT + 1 s dK 2 ds � K 2 + s dK 1 ds � K 1 3 T T λ 2 = 2 K 2 + K 1 T
Adding charges ✦ Local label for Abelian charge with chemical shift symmetry ψ ψ → ψ + f ( φ ) ✦ Total action invariant under combination of Sdiff and chemical shift. ✦ Invariant: chemical potential (scalar) resulting from entropy current co- moving with fluid elements µ = u α D α ψ ✦ NB: Canonical coupling to metric and gauge fields ψ → ψ − λ ( x ) , A → A + d λ
Exhibit 2: Parity odd transport ✦ Charged fluid in 2+1 dimensions. Derive thermodynamics from Z d 3 x √− g f ( s, µ ) S 0 = ✦ Allowed parity odd data at 1 ∂ 2 parameter family of transport ⇒ = Z p� g h w ( s, µ ) ✏ ρσλ u ρ r σ u λ + b ( s, µ ) ✏ ρσλ u ρ r σ A λ i S 1 = ✦ Entropy analysis: 4 parameter family of transport ✴ 9 (4 even, 5 odd) vectors in entropy current Jensen et al. ’12 ✴ 6 (3 even, 3 odd) vectors in charge current ✴ 5 (2 even, 3 odd) tensors/scalars in stress tensor ✦ Surprise: Hall viscosity is forced to vanish and anomalous Hall conductivity fixed by . [Torsion doesn’t help]. b & w Nicolis, Son ’11
Exhibit 3: Abelian anomalies ✦ Anomaly induced transport is adiabatic and is recoverable from a straightforward entropy current analysis. Son, Surowka ’09 ✦ Equivalently, obtain the result for transport from equilibrium partition function. ✦ Adiabaticity of anomalous data suggests that NDF should be able to capture anomalous transport explicitly. One can in fact focus exclusively on the anomalous part of the effective action. ✦ Upshot: works like a charm in 2d. Dubovsky, Hui, Nicolis ’11 ✦ In higher dimensions: obtain the correct anomalous effective action, but not the correct dynamics (well naively, but....)
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