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Anomalous Hydrodynamics and Non-Equilibrium EFT

Paolo Glorioso

July 19, 2018

PG, H. Liu, S. Rajagopal [1710.03768]

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Hydrodynamics is the universal low-energy description of quantum systems in a local thermal state.

Thermodynamic quantities are well-defined in mesoscopic regions

∂µT µν = 0, Tµν = (ε+ p)uµuν + pgµν + · · ·

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Two fundamental limitations:

Does not capture hydrodynamic fluctuations. E.g. long-time tails, etc. Current methods (e.g. stochastic hydro) are not systematic.

Too phenomenological. Constitutive relations satisfy constraints such as the second law of thermodynamics. Lack of a general derivation of these constraints.

Recently, an effective field theory (EFT) for systems in local thermal equilibrium was formulated. The formalism provides a Lagrangian for hydrodynamics, and addresses these two issues.

It is then of primary importance to generalize this EFT to include anomalies.

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Goal of this talk: provide a general effective field theory which captures the low-energy physics of a system with anomalies in local thermal equilibrium.

∂µT µν = F νµJµ, ∂µJ

µ = cF ∧ F

Lagrangian with “pions,” symmetries, constraints from unitarity/causality

derivation of constraints on transport from symmetry principles

J i = ξBB i + ξωω

i + · · ·

systematic treatment of hydrodynamic fluctuations.

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Outline

1 Non-equilibrium EFT

2 EFT for hydrodynamics with chiral anomaly

3 Additional remarks

4 Conclusions & Outlook

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Non-equilibrium EFT Consider QFT in a state ρ0. Evolution of ρ0:

ρ(t) = U(t)ρ0U †(t) Tr(ρ0 · · · )

Closed time path or Schwinger-Keldysh contour.

Correlators of an operator O conjugated to background A are encoded in the Schwinger-Keldysh generating functional:

e iW [A1,A2] =

∫ ρ0

Dψ1Dψ2e iS[ψ1,A1]−iS[ψ2,A2] =

∫ Dχ1Dχ2e

i Ieff[χ1,A1;χ2,A2;ρ0]

Ieff[χ1, χ2; ρ0]: effective action for the slow modes χ1, χ2. Local, includes dissipation and fluctuation effects caused by fast modes.

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Example: Brownian motion

Heavy particle coupled to thermal bath, integrate out bath:

Leff = mẋr ẋa − νẋrxa + i

2 σx2a

xr = 1

2 (x1 + x2), xa = x1 − x2

This is the functional representation of the Langevin equation.

mẍr + νẋr = ξ, 〈ξ(t)ξ(0)〉 = σδ(t)

Unitarity: σ ≥ 0 Thermal equilibrium: σ = 2νβ

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EFT for hydrodynamics with chiral anomaly

We want low-energy theory for the correlation functions of Tµν and Jµ.

Background fields conjugated to Tµν and Jµ

=⇒ g1µν , g2µν , A1µ,A2µ

Conservation of Tµν1,2 and J µ 1,2 implied by invariance under diffeomorphisms

and gauge transformations. Degrees of freedom are then the parameters of these transformations:

=⇒ Xµ1,2(σ a) : diffeomorphisms, ϕ1,2(σ

a) : gauge transformations

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EFT for hydrodynamics with chiral anomaly Xµ1,2(σ

0, σi ): trajectory of fluid element σi in physical spacetime.

ϕ1,2(σ 0, σi ): U(1) phase of fluid element σi .

Generating functional:

e iW [g1,A1;g2,A2] =

∫ DXµ1 Dϕ1DX

µ 2 Dϕ2 e

i Ieff[X1,ϕ1,g1,A1;X2,ϕ2,g2,A2]

Ieff: effective action for anomalous hydrodynamics 9 / 16

EFT for hydrodynamics with chiral anomaly

Symmetries

Time shift: σ0 → σ0 + f (σi )

Relabeling of fluid elements:

σi → σ′i (σj)

Diagonal shift ϕ1 → ϕ1 + χ(σi )

ϕ2 → ϕ2 + χ(σi )

These symmetries express that degrees of freedom can be redefined at a given time slice σ0 =const.

They are the very definition of a fluid.

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EFT for hydrodynamics with chiral anomaly

Three steps to implement anomaly.

1. Anomalous Ward identity:

δA1,2µ = −∂µλ1,2, δϕ1,2 = λ1,2

δIeff = c

∫ (λ1F1 ∧ F1 − λ2F2 ∧ F2)

Decompose into gauge-invariant and anomalous parts:

Ieff = Iinv + Ianom, δIinv = 0

All the actions are local. Choose:

Ianom = c

∫ (ϕ1F1 ∧ F1 − ϕ2F2 ∧ F2)

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EFT for hydrodynamics with chiral anomaly

2. Diagonal shift symmetry:

δϕ1 = χ(σ i ), δϕ2 = χ(σ

i ) =⇒ δχIeff = 0

Satisfied even in the presence of anomalies. However:

Ianom = c

∫ (ϕ1F1 ∧ F1 − ϕ2F2 ∧ F2)

=⇒ δχIanom = c ∫ χ(F1 ∧ F1 − F2 ∧ F2) 6= 0

Anomaly cancellation condition for diagonal shifts:

δχIinv = −c ∫ χ (F1 ∧ F1 − F2 ∧ F2)

This partially determines parity-odd terms in Iinv in terms of the anomaly coefficient c.

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EFT for hydrodynamics with chiral anomaly 3. Dynamical KMS invariance:

together with time-reversal invariance (or a generalization of it) gives

Ieff[KMS(χ1),KMS(χ2)] = Ieff[χ1, χ2], KMS2 = Id

This completely determines parity odd terms in Ieff, in terms of c , up to three constants a1, a2, a3.

From Ieff one recovers J iodd = ξBB

i + ξωω i

CME and CVE coefficients ξB(T , µ), ξω(T , µ) are determined by c , and the constants a1, a2, a3.

CPT invariance ⇒ a1, a3 = 0. Only c , a2 are left. 13 / 16

Relation to global anomalies For stationary background configurations, and setting to zero dynamical fields, the effective action factorizes:

e iIeff[g1,A1;g2,A2] = e iW̄ [g1,A1]e−iW̄ [g2,A2] + · · ·

Continuing W̄ to Euclidean time, gives the thermal partition function

e iW̄ [g ,A] → e−β0F [g ,A] = Z

Placing the system on a background with nontrivial topology, and performing a suitable “large” diffeomorphism gives a global anomaly:

Z → e−ixa2Z

where x is a constant related to the topology. a2 is then related to a global anomaly. [Golkar,Sethi ’15]

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Hydrodynamic fluctuations

Traditional method of computing hydrodynamic fluctuations: stochastic hydrodynamics

∂µT µν = ξµ, ∂µJ

µ = ξ, ξµ, ξ: thermal noise

Successful predictions (long time tails, scale dependence of transport, ...) E.g. shear viscosity in 3 + 1d : η = η0 + c1|ω/T |1/2

However, approximations are not controlled. Lack of a systematic method.

Non-equilibrium EFT allow to evaluate hydrodynamic fluctuations using standard QFT methods. [Gao, PG, Liu ’18]

This provides a systematic framework to study the interplay between anomaly and hydrodynamic fluctuations.

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Conclusions Non-equilibrium EFT of anomalous fluids

Anomalous transport completely determined by symmetries

Systematic approach to compute fluctuations (generalizes stochastic hydro)

Future directions Quantum and thermal fluctuations with anomalies

Turbulence?

Dynamical electromagnetic field, MHD, ...

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