Effective actions and hydrodynamic transport Mukund Rangamani - - PowerPoint PPT Presentation

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: ✦ Dynamics: conservation laws for the currents (up to anomalies) ✦ Summarize hydrodynamic data as constitutive relations for the currents in

terms of operators built from the hydrodynamical variables T µν, Jµ rµT µν = 0 , rµJµ = 0 T µν = ε uµ uν + P P µν + Πµν Jµ = q uµ + νµ

Constraints on hydrodynamics

✦ Constitutive relations obtained in a gradient expansion with transport

coefficients/thermodynamic response parameters determined by microscopics.

✦ The transport data are constrained macroscopically by demanding the

second law of thermodynamics hold locally, eg.,

✦ Typically second law: ✴ implies inequalities for transport coefficients ✴ fixes thermodynamic response parameters (5 for neutral fluids at 2∂).

Πµν = −η σµν − ζ θ P µν + · · · η, ζ ≥ 0 9 Jµ

S

! rαJα

S 0

• S. Bhattacharyya ’12

Constraints on hydrodynamics

✦ Thermodynamic response are adiabatic data ✦ These can be captured from an equilibrium partition function

which is a functional of background sources.

✦ Status quo: ✓ neutral/charged fluids to 2∂ ✓ Parity odd fluids to 1∂ ✓ superfluids to 1∂ ✓ anomalous transport

rµJµ

S = (rµJµ S)diss + (rµJµ S)adiabatic

Z [gµν , Aµ]

Banerjee et.al. ’12 Jensen et.al. ’12 Jensen, Loganayagam, Yarom ’12

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 ✴ Lack of dissipation conserved entropy current

δSeff = 0 = ) rµT µν = 0 rαJα

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:

✦ NB: view fluid as a space filling D-brane. ✦ Field reparameterization invariance: require arbitrary volume preserving

diffeomorphisms in configuration space

✦ The diffeo invariance in configuration space guarantees that Euler-Lagrange

equations are identical to energy momentum conservation. φI ! ξI(φ) , Jacobian(ξ, φ) = 1 δϕSeff = 0 ( ) rµT µν = 0 , T µν = 2 pg δSeff δgµν φI , I = 1, · · · , d − 1 Sdiff(Mφ)

Entropy current

✦ Volume preserving symmetry conserved entropy current ✦ Interpret this current as being the entropy current to all orders by passing to

the entropy frame

✦ Operator dimensions as appropriate for a phase field: ✦ Intuitively expect that all dissipative transport coefficients will be vanishing in

the theory; borne out by explicit analysis.

Jβ = 1 (d 1)! ✏βα1...αd−1 ✏I1...Id−1

d−1

Y

j=1

@αjIj

rαJα

S = 0

Jα = s uα s = q gαβ Jα Jβ [dφ] = 0

= ⇒

Neutral fluids: 0∂ and 1∂

✦ Zeroth order action reproducing ideal fluid behaviour

T µν = (s f 0(s) − f(s)) gµν + s f 0(s) uµ uν S0 ∝ Z ddx √−g f(s)

✦ 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

S1 / Z ddx pg Jα

s rαf1(s) =

Z ddx pg rα (f1(s) Jα

s ) ✦ Exception: parity-odd fluids which we will visit in a bit.

Neutral fluids: 2∂

✦ Obvious invariants at 2∂. Use the decomposition

rµuν = uµ aν + σµν + ωµν + Θ d 1 Pµν

a2, σ2, ω2, Θ2, R, Rµν uµ uν

⟹ 5 parameter family of neutral non-dissipative fluids linearly dependent

S2 =

5

X

m=1

Z ddx √−g Km(s) Ok

✦ 3 parameter family of scalar invariant NDF.

Neutral fluids: 2∂

✦ Conserved entropy current for neutral fluids: standard analysis based on

classifying independent tensor structures.

T µν = T µν

ideal + 2

X

m=1

ηm tµν

1,m + 13

X

k=1

ξk t2,k +

2

X

n=1

ζnˆ tµν

2,n

Jµ

S = s uµ + 2

X

k=1

δk vµ

1,k + 2

X

l=1

αl eµ

2,l + 6

X

m=1

βm vµ

2,m + 5

X

n=1

γn jµ

2,n ✦ {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 .

{λ2, λ0 − ξ2}

Πµν = η σµν ζ PµνΘ + T  τ uαrασhµνi + κ1 Rhµνi + κ2 Fhµνi + λ0 Θ σµν + λ1 σhµ

α σανi + λ2 σhµ α ωανi + λ3 ωhµ α ωανi + λ4 ahµaνi

• + T Pµν

 ζ1 uαrαΘ + ζ2 R + ζ3 R00 + ξ1 Θ2 + ξ2 σ2 + ξ3 ω2 + ξ4 a2

• ✦ There is a four parameter family of scale invariant NDF.

τ = 3 λ0 , κ2 = 2 κ1 = κ , λ1 , λ2 , λ3

= s uµ + rν ⇥ 2 A1 u[µ rν]T ⇤ + rν(A2 T ωµν) + A3 ✓ Rµν 1 2gµνR ◆ uν + ✓A3 T + dA3 dT ◆ ⇥ Θ rµT P αβrβuµ rαT ⇤ + (B1 ω2 + B2 Θ2 + B3 σ2) uµ + B4 [rαs rαs uµ + 2 s Θ rµs]

Jµ

S =

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 Kn.

A3 = K5 T , B1 = K2 K1 2 T , B3 = K1 + K2 2 T B2 = K1 + K2 6 T + s2 T 2 dT ds dK5 ds s4 2 T K3 B4 = K4 2 T 1 T 2 dT ds dK5 ds λ0 = s T dK5 ds 2 3 dK5 dT + 1 T ✓ s dK2 ds K2 + s dK1 ds K1 ◆ λ2 = 2 K2 + K1 T

Adding charges

✦ Local label for Abelian charge with chemical shift symmetry

ψ µ = uα Dαψ ψ → ψ + f(φ) ψ → ψ − λ(x) , A → A + dλ

✦ Total action invariant under combination of Sdiff and chemical shift. ✦ Invariant: chemical potential (scalar) resulting from entropy current co-

moving with fluid elements

✦ NB: Canonical coupling to metric and gauge fields

Exhibit 2: Parity odd transport

✦ Charged fluid in 2+1 dimensions. Derive thermodynamics from

S0 = Z d3x √−g f(s, µ)

✦ Allowed parity odd data at 1∂ 2 parameter family of transport

S1 = Z pg h w(s, µ) ✏ρσλ uρrσuλ + b(s, µ) ✏ρσλ uρrσAλ i

✦ Entropy analysis: 4 parameter family of transport ✴ 9 (4 even, 5 odd) vectors in entropy current ✴ 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

Jensen et al. ’12 Nicolis, Son ’11

= ⇒

Exhibit 3: Abelian anomalies

✦ Anomaly induced transport is adiabatic and is recoverable from a

straightforward entropy current analysis.

✦ 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. ✦ In higher dimensions: obtain the correct anomalous effective action, but not

the correct dynamics (well naively, but....)

Dubovsky, Hui, Nicolis ’11 Son, Surowka ’09

Off-shell anomalous transport data

✦ Adiabaticity equation for anomalous transport:

rµT µν = F να (Jcov)α , rαJα

cov = C

2n n! ✏α1β1···αnβn Fα1β1 · · · Fαnβn

(rα + aα) qα

anom Jα anom Eα = T rαJα S, anom + µ (rαJα anom PA(F))

Tαβ = ε uα uβ + P Pαβ + 2 q(α uβ) + Παβ Jα = q uα + να

qα = qα

anom + qα diss

να = να

anom + να diss

Παβ = Παβ

anom + Παβ diss

✦ Hydrodynamic equations of motion in d = 2n dimensions ✦ Adiabaticity condition on anomalous transport: Loganayagam ’11

Off-shell anomalous effective action

✦ Solution to the adiabaticity equation for anomalous transport derived from

an anomalous effective action:

Sanom = C 6 Z (D ^ A ^ F ˆ D ^ ˆ A ^ ˆ F) C 6 Z

M5

(A ^ F ^ F ˆ A ^ ˆ F ^ ˆ F) + C 3 Z pg

• µ2 !α + µ Bα

Dα (4.

✦ CS terms ensure gauge invariance of the full system via inflow.

T αβ

anom = 4 C

3 µ3 !(αuβ) + C µ2 B(αuβ) Jα

anom = C µ2 !α + C µ Bα ,

!α = 1 2 ✏αβγδ uβ rγuδ

Eα = F αβ uβ , Bµ = 1 2 ✏µναβ uνFαβ , Fαβ ⌘ 2 r[αAβ]

✦ The solution to the adiabaticity equation is the anomalous transport data in

the entropy frame

On-shell anomalous effective action

✦ Translate the solution of adiabaticity equation to Landau frame using ideal

fluid eoms

T ↵

(Landau) = (" + P)u↵u + Pg↵ + . . . ,

J↵

(Landau) = ⇢ u↵ + C µ2

✓ 1 − 2 3 ⇢ µ " + P ◆ !↵ + Cµ ✓ 1 − 1 2 ⇢ µ " + P ◆ B↵ + . . . , J↵

s (Landau) = s u↵ − C

s " + P ✓2 3 µ3 !↵ + 1 2 µ2 B↵ ◆ + . . . ,

Son Surowka ’09 ✦ Thus for a particular on-shell embedding we recover the requisite

anomalous transport.

✦ The story generalizes in an obvious fashion to all even dimensions; by a

suitable rewriting we can also talk about non-abelian generalization.

A shadow of the anomaly

✦ The anomalous action demands contribution from the hydrodynamical

shadow gauge field to ensure chemical shift invariance.

ˆ Aα = Aα + µ uα , ˆ Fαβ ⌘ 2 r[α ˆ Aβ]

✦ Can argue that without the shadow terms the action fails to satisfy the

desired symmetries in d > 2.

✦ In d=2 the shadow disappears: no transverse plane to cast shadows. ✦ Unfortunately, the shadow enters the naive dynamical equations.

rαJα

cov = C

8 ✏αβγδ ⇣ Fαβ Fγδ ˆ Fαβ ˆ Fγδ ⌘ = C 2 ? ⇣ F ^ F ˆ F ^ ˆ F ⌘

rµT µν = F να (Jcov)α + C 8 µ ✏αβγδ ˆ Fαβ ˆ Fγδ uν

Uneclipsed equilibrium

✦ The shadow terms disappear in equilibrium since the transverse electric

field is zero

ˆ Eα ⌘ Eα µ aα P αβrβµ

✦ Evaluating the action on stationary backgrounds reproduces the equilibrium

partition function ds2 = −e2(~

x)(dt2 + ai(~

x)dxi)2 + gij(~ x)dxidxj A↵ = (A0(~ x), Ai(~ x)) ,

˜ Aα ≡ (A0 + µ(0), Ai − A0 ai)

Sanom

• eq

=

• eq
• eq

Z ✓ = C 3 Z d3x √g3 ✓ A0 2 T0 ✏ijk ˜ Ai ˜ Fjk + A2 4 T0 ✏ijk ˜ Ai fjk ◆ the time-circle reduction and denote the 3-dimensional

✦ Claim: The Schwinger-Keldysh (doubled) theory constructed out of the

action described above leads to the correct current + Ward identities.

Looking ahead: Open issues

✦ Are the constraints exhaustive? ✴ are there invariants of Sdiff which are not expressible in terms of

hydrodynamical variables? [Unlikely, but no proof yet].

✦ First principles understanding of why certain pieces of transport are fixed in

the effective action.

✴ what singles out the second order transport data for neutral NDF? ✴ why does Hall viscosity vanish? ✦ vis a vis anomalies: ✴ can the Schwinger-Keldysh construction be generalized to gravitational/

mixed anomalies?

✴ connections to holographic set-ups?

Looking ahead: physical questions

✦ How physical are non-dissipative fluids? ✴ linearized solutions are non-diffusive/unattenuated, i.e., no imaginary

part to dispersion relations.

✴ spectrum comprises of normal modes alone. ✦ Characteristic initial value problem ✴ evolution of Cauchy data: shocks, turbulence, etc.. ✦ Holographic avatars ✴ black holes dual to NDF? Perhaps in higher derivative gravity? ✴ Connections to horizon dynamics?