Effective Field Theory of
Dissipative Fluids
Hong Liu

Paolo Glorioso

Michael Crossley

arXiv: 1511.03646

Conserved quantities
Consider a long wavelength disturbanceof a system in thermal equilibrium

non-conserved quantities: relax locally,conserved quantities: cannot relax locally, only via transports

Conserved quantities

Gapless andlong-livedmodes(only ones ina medium)

There should exist a universal low energy effective theory.

Hydrodynamics
Thermal equilibrium:
Promote these quantities to dynamical variables: (local equilibrium)
slowly varying functionsof spacetime

Express expectation values of the stress tensor and conserved
current in terms of derivative expansion of these variables:
constitutive relations.
Equations of motion:

d+1 variables, d+1 equations

Despite the long and glorious
history of hydrodynamics
It does not capture fluctuations.

Fluctuations
There are always statistical fluctuations …..
Important in many contexts:
Long time tailtransports,
dynamical aspects of phase transitions,non-equilibrium states,
turbulence,
finite size systems ….
At low temperatures, quantum fluctuations can also be important.

Phenomenological level: stochastic hydro (Landau, Lifshitz): noises with local Gaussian distribution
Expect:
1. interactions among noises
2. interactions between dynamical variables and noises
3. fluctuations of dynamical variables themselves
particularly important for non-equilibrium situations.
Until now not known how to treat such nonlinear
effects systematically. Not even clear it is a good question.

Constraints
Current formulation of hydrodynamics is awkward.
Constitutive relations : not enough to just write down the mostgeneral derivative expansion consistent with symmetries.
Phenomenological constraints: solutions should satisfy:
1. Entropy condition
2. Onsager relations: linear response matrix must be symmetricawkward: use solutions to constrain equations of motion
Microscopic derivation?

Are these complete?

develop hydrodynamics as a bona fide low energy effective field
theory of a general many-body system at finite temperature
Action principle which incorporates
both dissipations and noises
1. gives a full interacting theory of noises.
2. Microscopic origin and completeness ofphenomenological constraints
3. New constraints (nonlinear Onsager relations)Should be distinguished from an action which just reproduces standard
eoms (which may not capture fluctuations correctly)

Effective theory approach may also make it easier to generalizehydrodynamics EOM to less familiar situations, say with
momentum dissipations, anomalies.....

Searching for an action principle for hydrodynamics
has been a long standing open problem, dating backat least to G. Herglotz in 1911 ….....
All results at non-dissipative level ….
Many activities since 70’s to understandhydrodynamic fluctuations ….....

Results
Approach: put a relativistic quantum many-body ssystem in a
curved spacetime1. Hydrodynamics with classical statistical fluctuations
is described by a supersymmetric quantum field theory

See also Haehl, Loganayagam, Rangamani

2. Hydrodynamics with quantum fluctuations also incorporated
is described by a “quantum-deformed” (supersymmetric)
quantum field theory.

Part II: formulation

Transition amplitudes v.s. expectation values
We are interested in an effective theory describing nonlinear
dynamics around a state.

Closed time path (CTP) or Schwinger-Keldysh contour
Should double all degrees of freedom
Should be contrasted with EFT describing transition amplitudes,

Hydro effective field theory
At long distances and large times:
All correlation functions of
the stress tensor andconserved currents inthermal equilibriumhydrodynamicmodes

EFT approach:
1. What are

do not work?

2. What are the symmetries of
3. Integration measure?

?

Dynamical variables: integrating in
Toy example: a single conserved current

1. Current conservation:

2. W must be nonlocal : Non-locality solely due to integratingout hydro modes
Integrate in hydro modes:(a):local(c):(b): Ensure 1 is satisfied

EOMs must be equivalent to current conservations

Proposal: (use the usual Stueckelberger trick )is a local action.

: hydro modes

Satisfy the following consistency requirements:
1.
2.

Eoms ofare equivalent to current conservations.

Dynamical variables (II)For stress tensor, we put the system in a curved spacetime

Conservation of stress tensor:

Integrate in hydro modes:

1.

Promote spacetime coordinates todynamical fields

2. X eoms are equivalent toconservation of stress tensor
an emergent spacetime with coordinates

Interpretation of:label individual fluid elements,

internal time: motion of a fluid element in physical spacetime

So we just recovered the Lagrange description of a fluid!
As a starting point, we could simply double the degrees of freedom
in the Lagrange description.

A bit history:
Using a single copy ofas dynamical variable for an
ideal fluid action dated back to G. Herglotz in 1911.
Covariant

was used by Taub in 1954.

Rediscovered in 2005 by Dubovsky, Gregoire, Nicolis and Rattazziin hep-th/0512260 and further developed by Dubovsky, Hui,
Nicolis and Son in arXiv:1107.0731 , ......
Nickel and Son showed the covariant version arises naturally
from holography (arXiv:1103.2137).
Doubled copies appeared in Haehl, Loganayagam, Rangamani
arXiv:1502.00636, and Crossley, Glorioso, HL, Wang
arXiv:1504.07611.

Standard hydro variables (which are now derived quantities)

A significant challenge: ensure the eoms from the actionof X and can be solely expressed in terms of these velocity
type of variables. (e.g. solids v.s. fluids)

Symmetries (I)Now need to specify the symmetries ofNote that it is defined in fluid spacetime
Interpretation of:label individual fluid elements,

Require the action to be invariant under:

define what is a fluid!

internal timeIt turns out these symmetries indeed do magic for you:at the level of equations of motion, they ensure all dependence
on dynamical variables can be expressed in
Recover standard formulation of hydrodynamics(modulo phenomenological constraints)

Full non-linear fluid fluctuating dynamics encoded
in non-trivial differential geometry:

This would be the full the story in a usual situation.

Symmetries (II)We are considering EFT for asystem defined with CTP:
The generating functional has the following properties:• Reflectivity condition:• KMS condition plus PT imply a Z2 symmetry on W:• Unitarity condition:

Full bosonic theory
Reflectivity condition can be easily imposed, leading to acomplex action.
Imaginary part of the action non-negative
Imposing KMS condition is very tricky.
proposal: local KMS condition, a Z2 symmetry on the action
All the constraints from entropy current
condition and linear Onsager relations
New constraints on equations of motion from nonlinear
Onsager relations.

Fermions and Supersymmetry
Unitarity condition:

See also Haehl et al
arXiv: 1510.02494
1511.07809

is a “topological” condition on the measure of path integrals
Introduce fermionic partners (“ghost” fieds) for dynamicalvariables and require the action to have a BRST type symmetry.
At a quadratic level in dynamical fields, one finds that local KMS
condition leads to an emergent fermionic symmetry.

But not clear how to write down a nonlinear action with such an algebra.
Requires a “quantum-deformed” SUSY

Classical limit:
become standard supersymmetry in time direction.
In this limit one can write down a supersymmetric completionof the full bosonic hydrodynamic action.
Note that in the classical limit, path integral remains,
capturing statistical fluctuations.

Example: nonlinear stochastic diffusion
Consider the theory for a single conserved current, where
the relevant physics is diffusion.
Dynamical variables:
Roughly,(or): standard diffusion mode,: the noise.

If ignoring interactions of noise

A variation of Kardar-Parisi-Zhang equation

Summary
An EFT for general dissipative fluids.
Recovers the standard hydrodynamics as equationsof motion, constitutive relations, constraints.
Encodes quantum and thermal fluctuations
systematically in a path integral expansion.
Full non-linear fluid fluctuating dynamics encoded
in non-trivial differential geometry.
Fermionic excitations and Emergent supersymmetry.

Future directions
Formalism:
Non-relativistic limit ,superfluids,
Anisotropic, inhomogeneous,“quantum-deformed” Supersymmetry…....
Applications:Longtime tails, running of viscosities,
Non-equilibrium steady states, dynamical flows of QGP
Dynamical aspects of classical andquantum phase transitions
Scaling behavior in hydro behavior via fixed points
of QFTs, such as KPZ scaling, turbulence ….
….........

Thank You