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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.]