Schwinger-Keldysh supersymmetry
and a gauge theory of entropy
Felix Haehl(Durham University → UBC)YITP, 16 June 2016

based on:

FH, R. Loganayagam, M. Rangamani[1510.02494], [1511.07809], [work in progress]
see also [1312.0610], [1412.1090], [1502.00636]

Summary

This talk is about three statements:
1

A satisfactory field theoretic understanding of mixed state evolution(in particular EFT with dissipation) is missing and desirable.

2

Unitarity implies a universal supergeometry underlying any such fieldtheory, which keeps track of the basic entanglement structure(‘backbone’) of the mixed initial state.

3(Near-)thermal EFTs have an emergent U (1)T symmetry of gauged
thermal translations. This explains dissipation as a symmetry
breaking, local entropy current, the second law etc.

Felix Haehl, 1/10

Field theory and dissipation
Many interesting and/or realistic systems are in mixed states due to
tracing out part of Htot (‘environment’)
Example: low-energy EFT of thermal systems (fluids, black holes, ...)
UV/IR coupling (‘entanglement’)apparent non-unitarity: dissipation, information loss, local entropy
current, second law, ...
I horizon, complementarity, entanglement, ...
IICFTLhΨL |CFTR|ψR i

Felix Haehl, 2/10

Field theory and dissipation
Many interesting and/or realistic systems are in mixed states due totracing out part of Htot (‘environment’)
Example: low-energy EFT of thermal systems (fluids, black holes, ...)
UV/IR coupling (‘entanglement’)apparent non-unitarity: dissipation, information loss, local entropy
current, second law, ...
I horizon, complementarity, entanglement, ...
IITask 1: How to formulate QFT evolution of density matrices?
IWell known aspect: Schwinger-Keldysh formalism
FF

Doubling of fields and symmetries to evolve h · | and | · i
SSK = S[ΦR ] − S[ΦL ]Task 2: How to do RG on such systems?
II

Schwinger, Keldysh,
Feynman-Vernon, ’60s

Are the two copies going to interact? How?
Understand macroscopic irreversibility as a symmetry breaking?
Felix Haehl, 2/10

Schwinger-Keldysh and unitaritybt

SK generating functional:iεZSK [JR , JL ] = TrnU [JR ] ρinitial U † [JL ]o→ time ordered (R) →← anti-time ordered (L) ←i(ε − β)An interesting consequence of unitarity:source alignment ⇒ localization:IZSK [JR = JL ≡ J] = Tr ρinitial

Only sensitive to initial state correlations (entanglement structure)

Felix Haehl, 3/10

Schwinger-Keldysh and unitaritybt

SK generating functional:iεZSK [JR , JL ] = TrnU [JR ] ρinitial U † [JL ]o→ time ordered (R) →← anti-time ordered (L) ←i(ε − β)An interesting consequence of unitarity:source alignment ⇒ localization:IZSK [JR = JL ≡ J] = Tr ρinitial

Only sensitive to initial state correlations (entanglement structure)change basis:

OR JR − OL JL = Oav Jdiff + Odiff Jav|{z}| {z }→0
R+Lav ≡,2diff ≡ R − L→J⇒ The sector of difference operators Odiff ≡ OR − OLdecouples for any SK theory:
I(1)

(n)

hTSK Odiff · · · Odiff i = 0Correlations of Odiff protected by (topological) symmetry
Felix Haehl, 3/10

Schwinger-Keldysh and unitarity
Can we cook up formalism where this localization is manifest?
c.f. gauge theory:
I characterize pure gauge modes using nilpotent, Grassmann-odd BRSTcharges: [pure gauge] = QBRST (...) = QBRST (...)
I correlators of BRST-exact fields vanish
Efficient way to formulate SK localization: SK BRST cohomology
I every operator Ô represented by a quadruplet {O , O , O , O }RLGḠ
I BRST charges Q,Qdefinetopologicalsector:SKSKOR , OLQSKQSKOGQSK−QSKOGOR − OLIunitarity ⇒ correlators of SK BRST-exact operators vanish
Felix Haehl, 4/10

SK supergeometry
Convenient way to ascertain topological structure:superspace (xµ , θ, θ̄), where QSK = ∂θ̄ and QSK = ∂θQuadrupling of fields ⇔ lift fields to superfields:
QSKQSKL + θO
O(S) = OR +O+ θ̄ OG + θ̄θ (OR − OL )G2QSKQSKThis structure is robust under RG and universal for any unitary SKtheory• Note: for simple dissipative systems (e.g. Langevin theory of Brownian motion) thequadrupling is textbook material. BRST charges and superspace merely reformulation.

Felix Haehl, 5/10

Supergeometry of low-energy SK theories

Low-energy SK EFT: what are the symmetries, symmetry breakings,effective degrees of freedom Φ? (see also Hong Liu’s talk)
Many things as usual. New features:(1) Superspace, with quadrupling of naı̈ve IR fields Φ → Φ(S)(2) Write topological field theory of initial correlations:
ˆˆ

(top)
SSK = dd x Q , [Q , L(Φ)] = dd x dθ dθ̄ L(Φ(S) )(3) Then de-align sources JR 6= JL to deform away from topological limitFelix Haehl, 6/10

Low energy, near-equilibrium: emergent U (1)TWe are particularly interested in SK evolution amongst the class of
mixed states whose low-energy dynamics is locally thermalIII.e., UV modes are in thermal equilibrium
EFT of IR modes reflects this! (fluctuation-dissipation, second law, ...)

Proposal for implementing thermality in EFT
Invariance under emergent gauge symmetryof thermal translations: U (1)T

[1510.02494][1502.00636][1412.1090]? C.f. Euclidean theory: translation invariance in thermal circle? Microscopic origin of U (1)T : KMS condition? Two more BRST charges associated with U (1)T? U (1)T current = entropy current + ghost terms? Apparent non-unitarity (dissipation) ↔ ghost non-decoupling?!

Felix Haehl, 7/10

Toy model: Langevin particle
Consider Brownian motion of Langevin particle at x(t):−Eom ≡ md2 x ∂Udx+ν
=N+2dt∂x
dt
Martin-Siggia-Rose ’73

Martin-Siggia-Rose (MSR) construction:

De Dominicis-Peliti ’78

ˆδEomei SGaussian noise [N]δx ˆˆδEom= [dx] [df ][dψ][dψ] exp i dt f Eom + i ν f 2 + ψψ
δx[dx]

[dN] δ(Eom + N) detCan write this in superspace:
ˆˆ= [dx][df ][dψ][dψ] exp idt dθ dθ̄m2

dx(S)!

2dt− U (x(S) ) − i ν Dθ x(S) Dθ̄ x(S)WZ

where x(S) = x + θ ψ̄ + θ̄ ψ + θ̄θ fand Dθ refers to A = At dt + Aθ dθ + Aθ̄ dθ̄Felix Haehl, 8/10

Status and prospects
Step 1: In simple examples (Langevin theory) this works beautifully(SUSY structure well-known; U (1)T symmetry completes the picture nicely)
I

Viscosity proportional to CPT breaking order parameter hFθθ̄ i =6 0[1511.07809]

Felix Haehl, 9/10

Status and prospects
Step 1: In simple examples (Langevin theory) this works beautifully(SUSY structure well-known; U (1)T symmetry completes the picture nicely)
I

Viscosity proportional to CPT breaking order parameter hFθθ̄ i =6 0[1511.07809]

Step 2: We are trying to derive hydrodynamics with thisI

Advantage of hydro: we gave complete solution and classification oftransport ⇒ very sharp goal for what the SK EFT has to achieve[1412.1090], [1502.00636]

I

Have already written a SUSY EFT with thermal gauge symmetry,
which describes all of dissipative transport[1511.07809], see also Crossley-Glorioso-Liu [1511.03646]

I

W.i.p.: various other classes of transport also seem to work nicely

Felix Haehl, 9/10

Status and prospects
Step 1: In simple examples (Langevin theory) this works beautifully(SUSY structure well-known; U (1)T symmetry completes the picture nicely)
I

Viscosity proportional to CPT breaking order parameter hFθθ̄ i =6 0[1511.07809]

Step 2: We are trying to derive hydrodynamics with thisI

Advantage of hydro: we gave complete solution and classification oftransport ⇒ very sharp goal for what the SK EFT has to achieve[1412.1090], [1502.00636]

I

Have already written a SUSY EFT with thermal gauge symmetry,
which describes all of dissipative transport[1511.07809], see also Crossley-Glorioso-Liu [1511.03646]

I

W.i.p.: various other classes of transport also seem to work nicely

Step 3: If hydro works, move on to dual black holesI

SK doubling and ghosts: what will they teach us about dissipation,complementarity, unitarity etc. in gravity?
Felix Haehl, 9/10

Summary1

A satisfactory field theoretic understanding of mixed state evolution(in particular EFT with dissipation) is missing and desirable.

2

Unitarity implies a universal supergeometry underlying any suchfield theory, which keeps track of the basic entanglement structure(‘backbone’) of the mixed initial state.

3(Near-)thermal EFTs have an emergent U (1)T symmetry ofgauged thermal translations. This explains dissipation as asymmetry breaking, local entropy current, the second law etc.
Or, for the mathematically inclined:I

Near-thermal EFTs are deformations of TQFTs associated with theuniversal balanced equivariant cohomology of thermal translations.

I

Hydrodynamics is a deformation of a gauged topological σ-model.

Vafa-Witten ’94, Dijkgraaf-Moore ’96

Felix Haehl, 10/10

Further DetailsU (1)T thermal gauge invariance
Microscopic KMS condition:e−iδβKMS↓O(t) ≡ O(t − iβ) = O(t)Macroscopically, ensure KMS by introducing gauge (super-)field for’thermal translations’A = Aa dσ a + Aθ dθ + Aθ̄ dθ̄µµµDθ X(S)≡ ∂θ X(S)+ Aθ £β X(S)(and so on)

U (1)T transformations act as thermal translations, e.g.:
Φ(S) 7→ Φ(S) + Λ(S) £β Φ(S)Felix Haehl, 11/10

Effective action for dissipation in fluids
Fluids get their dynamics from σ-model maps:
X µ (σ): d-worldvolume → phys. spacetime
Effective action for dissipative sector (at any order in ∇µ ):pˆ−g(S)  ((ab)(cd))(S)(dissipation)(S)diη
DgDSeff∼d σ dθ dθ̄gθθ̄abcdworld
1 + β a Aavolume⇒ T ab ∼ i Fθθ̄ η ((ab)(cd)) £β gab + noise terms
Ghost bilinears responsible for dissipationhFθθ̄ i: order parameter for dissipation
I Can derive Jarzynski as SUSY Ward identity (⇒ Second Law)
I Variation w.r.t. A gives entropy currentaII

Felix Haehl, 12/10