Anomalous Transportfrom

Kubo Formulas

Karl LandsteinerInstituto de Física Teórica, UAM-CSIC

in collaboration withI. Amado, E. Megías, L. Melgar, F. Peña-Benitez,

[arXiv:1102.4577 (JHEP 1105.5006), arXiv:1103.5006 (Phys. Rev. Lett. 107 (2011) 021601), arXiv:1107.0368 (JHEP 1109 (2011) 121 ), arXiv:1111.2823 ]

UW-Seattle, 13 March 2012

OutlineMovie

CME and CVE (flash review)

Kubo formulas

Hydrodynamics

Chem. Potentials

Currents and Counterterms

Holography

Outlook

[Kharzeev, McLarren, Warringa, Nucl. Phys A803,227, arXiv:0706.1026][Fukushima, Kharzeev, Warringa, Phys. Rev. D78, 074033, arXiv:0706.1026][Kharzeev, Warringa, Phys. Rev. D80 (2009) 034028, arXiv:0907.5007]

RHIC’s hot quark soup

www.youtube.com/watch?v=kXy5EvYu3fw

The CME

B

J

+

-Electric current

Net chirality

Magnetic Field

P-odd charge separation

Text

[Kharzeev, McLarren, Warringa],[Fukushima, Kharzeev, Warringa][Newman, Son]

[parity violating currents: Vilenkin ’80, Giovannini, Shaposhnikov ’98, Alekseev, Chaianov, Fröhlich ’98]

The CVE

B

J5 Axial Current

Temperature

P-odd axial charge separation

Text

[Vilenkin ’80, CVE Kharzeev&Son ’10, Keren-Zur&Oz ’11]

Text

Angular Momentum

[Banerjee et al, Erdmenger et al.]

Kubo formulasChiral magnetic conductivity

Kubo formula, general symmetry group

[Kharzeev, Warringa]�J = σ �B

σAB = limpj→0

�

i,k

i

2pj�ijk�JA

i JBk �

����ω=0

Kubo formulas chiral fermions

chemical potentials and Cartan generators

1-loop graphq

q + kJAi (k)

JBj (!k)

ω̃fn = (2n+ 1)πT − iµf

S(q)f g =δf g

2

�

t=±∆t(iω̃

f , �q)P+γµq̂µt

∆t(iω̃f , q) =

1

iω̃f − tEq

Kubo formulas

Anomalycoeff

GAB =1

2

�

f,g

T gA fT

fB g

1

β

�

ω̃f

�d3q

(2π)3�ijntr

�Sf

f (q)γiSf

f (q + k)γj�

Kubo formulas

∼ dABC

TA

TB TC

AμA pure gauge

AνB AρC

Kubo formulasfinite density: charge transport => energy transport

energy flux sourced by magnetic fields

at ω=0 reverse order of operators

conductivity ?

[Neiman, Oz], [Loganayagam], [Kharzeev, Yee]

Tμν sourced by metric

Ag “gravitomagnetic field” -> chiral gravitomagnetic effect

chiral vortical effect: fluid velocities

Kubo formulas

Kubo formulasas before: general symmetry group

Integration constant -> gravitational anomaly!

q

q + k

JAi (k) T0j(!k)

Kubo formulas

∼ tr (TA)TA

gμν gλρ

AμA pure gauge

σABB =

1

4π2dABCµC

σ�V =

1

12π2dABCµAµBµC + bAµA

T 2

12

σAV = σ�,A

B =1

8π2dABCµBµC + bA

T 2

24

also energy flux

limpj→0

i

2pj

�

i,k

�ijk �T0iT0k�|ω=0 �= 0

4 types of conductivities:

Kubo formulas

σel = limω→0

−i

ω�JiJi�|k=0

dW

dt=

ω

2fI(−ω)ρIJ(ω)fJ(ω)

Compare with electric conductivity

Kubo formulas

σAB = limpj→0

�

i,k

i

2pj�ijk�JA

i JBk �

����ω=0

Rate of dissipation

Not in spectral density and at zero frequency

Dissipationless transport!

Tµν = (�+ P )uµuν + Pgµν + τµνdiss + τµνanom

νµ,Aanom = σABB Bµ

B + σAVω

µ

Qµanom = σA,�

B BµA + σ�

Vωµ

τµνanom = Qµuν +Qνuµ

BµA =

1

2�µνρλuνFA,ρλ

ωµ = �µνρλuν∂ρuλ

Hydrodynamicshydrodynamics:

Jµ,A = ρAuµ + νµ,Adiss + νµ,Aanom

dissipative terms: shear- and bulk viscosities, diffusivity and electric conductivity

[Son,Surowka] [Neiman, Oz], [Loganayagam]

Hydrodynamics

ξ coefficients are different from σ’s

Landau frame:

such that

uµ → uµ + δuµ

uµTµν = 0

anomalous terms only in current

νµanom = ξBBµ + ξV ωµ Qµ = 0

ψ(t− iβ) = ±ψ(t) ψ(t− iβ) = ±eβµqψ(t)

Chemical Potentials• Textbook: H → H − µQ

Z =�

n

�n|e−β(H−µQ)|n� =�

n

�n|e−βHeβµqn |n�

• Alternative: deform state space instead of Hamiltonian (state vs. coupling)

• Tacitly assumed: [H,Q] =0 !

[Landsmann, Waert] [T.S. Evans]

ψ(ti − iβ) = ±eβµqψ(ti)

titf

ti − iβ

Chemical Potentials• Keldysh-Schwinger:

• define initial (equilibrium) state through

• but do (real) time evolution with H’ where [H’,Q] <>0!

• non-equilibrium setup (quantum quench)

Formalism Hamiltonian Boundary conditions

(A) H − µQ Ψ(ti) = −Ψ(ti − iβ)(B) H Ψ(ti) = −e

βµΨ(ti − iβ)

Formalism Hamiltonian Boundary conditions

(A) H − µQ Ψ(ti) = −Ψ(ti − iβ)(B) H Ψ(ti) = −e

βµΨ(ti − iβ)

A0 = µ δAµ = ∂µχ χ = −µt

Chemical Potentials

• Anomalous charge: (B) seems better suited, b

• (A) not anymore equivalent to (B), ANOMALY!

• What’s the difference? Gauge trafo!

• Gauge invariant: Parallel transport

Formalism Hamiltonian Boundary conditions

(A) H − µQ Ψ(ti) = −Ψ(ti − iβ)(B) H Ψ(ti) = −e

βµΨ(ti − iβ)

δS =

�d4xχ �µνρλ

�C1FµνFρλ + C2R

αβµνR

βαλρ

�

SΘ =

�d4xΘ �µνρλ

�C1FµνFρλ + C2R

αβµνR

βαλρ

�

δΘ = −χ δ(S + SΘ) = 0

H − µ(Q+ 4

�d3x�C1�

0ijkAi∂jAk + C2K

0�

Kµ = �µνρλΓαβν

�∂ρΓ

βαλ +

2

3ΓσαλΓ

βσρ

��JΘ = 4C1µ �B

Chemical Potentials• Formalism (A’) (with anomaly):

• Spurious “axion”: (B) -> (A’) switches on θ !

Formalism Hamiltonian Boundary conditions

(A) H − µQ Ψ(ti) = −Ψ(ti − iβ)(B) H Ψ(ti) = −e

βµΨ(ti − iβ)

Chemical Potentials

A 0finite T part

(a) (b) (c)

• (A’) equivalent to (B) (b) and (c) cancel each other

Formalism Hamiltonian Boundary conditions

(A) H − µQ Ψ(ti) = −Ψ(ti − iβ)(B) H Ψ(ti) = −e

βµΨ(ti − iβ)

Chemical Potentials

A0 = μA0 = 0

θ = -μτ

A0 = 0A0 =-μ

• Holography: (B) and (A’)

Formalism Hamiltonian Boundary conditions

(A) H − µQ Ψ(ti) = −Ψ(ti − iβ)(B) H Ψ(ti) = −e

βµΨ(ti − iβ)

dABC

V AΨ1 1 1Ψ2 -1 1

∂µJµV �= 0

Sct = c

�d4x�µνρλVµAνF

Vρλ

∂µJµV = 0

∂µJµA =

1

48π2�µνρλ

�3FV

µνFVρλ + FA

µνFAρλ

�

dAAA = 2 dAV V = dV AV = dV V A = 2

�JV = 0

�JV =µA

2π2�BV

Currents and Countertems• A Tale of two Symmetries: V vs. A

• Symmetric

Indicates

• Cure: Bardeen counterterm

(A), (A’), (B)

(A’), (B)

(A)

�JA =µ

2π2�BV

[Bardeen ’69], [Rebhan, Schmitt, Stricker] [Gynther, K.L., Pena-Benitez, Rebhan]

Formalism Hamiltonian Boundary conditions

(A) H − µQ Ψ(ti) = −Ψ(ti − iβ)(B) H Ψ(ti) = −e

βµΨ(ti − iβ)

νµ = · · ·+ 3C1�µνρλAλFρλ

Currents and Counterterms• Which current?• Consistent current Jµ =

δS

δAµ

• Anomaly not in covariant form

• CS term in constitutive relation

• Covariant current Jµ → Jµ + Y µ

• Covariant form of anomaly

O(4) in derivatives!

[Bardeen, Zumino]

• Holography: Aµ = Aµ +A(2)

µ

r2+ · · ·

Holography

SEM =1

16πG

�d5x

√−g

�R+ 2Λ− 1

4FMNFMN

�

SCS =1

16πG

�d5x �MNPQRAM

�κ3FNPFQR + λRA

BNPRB

AQR

�

SGH =1

8πG

�

∂d4x

√−hK

SCSK = − 1

2πG

�

∂d4x

√−hλnM �MNPQRANKPLDQK

LR

S = SME + SCS + SGH + SCSK

Holography

mixed gauge gravitational Chern Simons term

[Newman], [Banerjee et al.], [Erdmenger et al.] [Yee] [Rebhan, Schmitt, Stricker] [Khalaydzyan, Kirsch], [Hoyos, Nishioka, OBannon]

GMN − ΛgMN =1

2FMLFN

L − 1

8F 2gMN + 2λ�LPQR(M∇B

�FPLRB

N)QR

�

∇NFNM = −�MNPQR�κFNPFQR + λRA

BNPRB

AQR

�

8πGT ij =

√−γ

�Ki

j −Kγij + 4λ�(iklm

�1

2F̂klR̂j)m +∇n(AkR̂

nj)lm)

��

∂

16πGJ i = −√−γ

�Ei + �ijkl

�4κ

3AjF̂kl −

λ

2πGDjKsjDkK

sl

��

∂

DiJi = − 1

16πG�ijkl

�κ3F̂ijF̂kl + λR̂m

sijR̂smkl

�

DiTij = −JmF̂mj +AjDiJ

i

−κ

48πG=

N

96π2

−λ

48πG=

N

768π2

Holographymixed gauge gravitational Chern Simons term

�JJ� = −ikz

�µ

4π2− β

12π2

�

�JT � = −ikz

�µ2

8π2+

T 2

24

�

�TT � = −ikz

�µ3

12π2+

µT 2

12

�

HolographyKubo formulas: fluctuations

background: charged AdS black hole

correlators are

same as weak coupling!non-renormalization

(no T^3 terms)

[Neiman, Oz], [Loganayagam]

DiscussionAnomalies -> parity violating transport

Kubo formulas

Surprise: mixed gauge gravitational anomaly contributes

Holography with gravitational CS term

(non)-renormalization?

Higher dimension

Superfluids

Hydrodynamics and derivative expansion? (fluid/gravity)

Observable effects? [Karen-Zur, Oz], [Kharzeev, Son], [Kharzeev, Yee]

[Son,Surowka], [Longanyagam], [Longanayagam, Surowka],

[Longanyagam], [Longanayagam, Surowka],[Kharzeev, Yee]

[Chapman, Neiman, Oz]

[Lin], [Neiman, Oz], [Batthacharya et al.]