Chiral Transport
Kenji Fukushima
The University of Tokyo

August 22, 2016 @ RIKEN-Wako

1

SetupLet’s think about Chiral Matter(massless right-handed particles)
For vector gauge theories
left-handed contributions should be added(chiral anomaly is considered mostly, andslightly about mixed gravitational anomaly)

August 22, 2016 @ RIKEN-Wako

2

Quantum Anomaly
Axial U(1) rotation by qZ2eµ⌫⇢µS = dx ✓(x) @µ j +"Fµ⌫ F⇢
232⇡Z2eµ⌫⇢= dx @µ ✓(x) j µ"A⌫ @ ⇢ A28⇡

Conserved Current(Chern-Simons Current)

cf. Gauge-variant: massless pole canceled by dipole ghosts
August 22, 2016 @ RIKEN-Wako

3

CS current may not be physical?
ij =e2 i0jk"
A0 @ j Ak =24⇡

e2i
A0 B
24⇡

In electromagnetism a constant vector potential is irrelevant

Aµ ! Aµ + @ µ 'A0 can be non-trivial in Euclidean spacetime

i@0 + eA0 (x)µ !

@4 + ieA4 (x)µSee: A. Yamamoto, 1210.8250

August 22, 2016 @ RIKEN-Wako

4

CS current can be physical!
A0 $ iA4 ⇠µgauged away→anomaly

e2
j=
µB
4⇡ 2

Especially if µA for right-handed and -µA for left-handed:
Chiral Magnetic Effect

e2
jV =
µA B
22⇡

Especially if µV for right-handed and µV for left-handed:
Chiral Separation Effect

e2
jA =
µV B
22⇡

August 22, 2016 @ RIKEN-Wako

5

Diagrammatic Calculation

B
jV/A
Kubo formulaj=ABihjji
k!0 2kµ A/VA ⇠ lim

August 22, 2016 @ RIKEN-Wako

6

Not easy in reality…
Where can we find either left-handedor right-handed chiral and charged fermions?
How can we implement chemical potentials
separately for left- and right-handed particles?
HIC:hji = 0

hj · ji =6 0background???

Weyl semimetals: P and/or T broken
T : monopoles at ± k,

P : monopole at k, anti-monopole atAugust 22, 2016 @ RIKEN-Wakok
7

Electromagnetic Backgrounds
P- and T-odd external backgrounds

E · B 6= 0j?

Similar problem to “Classical” Hall Effectj Hall ⇠ E ⇥ B
August 22, 2016 @ RIKEN-Wako

8

Lorentz Boost for CHE0B

B 'B

j 0 = neBoost0E ' (v/c)Bnec
0jHall = j ' v · ne = B EClassical Hall Conductivity

Put E from the beginning to realize parallel E and B
August 22, 2016 @ RIKEN-Wako

9

Lorentz Boost for CME
BCME

j Ohm = EE
Ohm’s Law
Hall Current
August 22, 2016 @ RIKEN-Wako

10

Schwinger Mechanism
yyK′

K ′′

⊗ηyK⃗
B
⃗j
⃗
E

Fukushima-Kharzeev-Warringa (2010)

⃗ ′′
θ
B
⃗′
jyB′′′⃗⃗jEz
θ z⃗j ′ z⃗ ′′ESchwinger process in K’✓ 0 ◆✓◆2 0 02etransformation
Ez B zBm′ ⇡z
0FIG. 2: Lorentz
fromaframe
Kin which the@t j0 =
coth
⇡ exp
200|
4⇡
|eE
electric field (E), magnetic
field E(B),andthecurrent
densityzz(j) are
parallelto each other,
Current
generation
rate to a frame K in which B and✓◆j have a component
to✓E. ◆3 perpendicular22e B y Ez B zBzm ⇡@t jy =
coth
⇡ exp
222
⇡ Ez + B z
Ez|eEz |

August 22, 2016 @ RIKEN-Wako′µEz′ sinh η ey + Bz′ cosh η e.
Sincejpoints in the zz′µ11

Further Simplification

August 22, 2016 @ RIKEN-Wako

12

Lorentz Boost for CME
j CME = (E · B)B / B 2

B

Only this is externalE
j Ohm = E

j = ( Ohm +CME )E

2/B
CME

Son-Spivak (2012)

August 22, 2016 @ RIKEN-Wako

13

Magnetoresistance (MR)
Lorentz force = “Classical” MR
Perpendicular E and B are Lorentz force free

Negative “magnetoresistance”

2: Magnetoresistance in field parallel to current
ZrTe5 . (a) Temperature dependence FIG.
of resistivity
in ZrTe5 in magAugust 22, 2016 @ RIKEN-Wako

14

temperatures. For clarity, the resistivity curves were s~
cleavage plane (B k b). The inset shows the electron di↵raction fromHow to compute (negative) MR?
Boltzmann Equation for Massless Fermions@f@f@f+ ẋ
+ ṗ
= Icoll [f ]@t@x@pChiral Kinetic Theory
Son-Yamamoto (2012)
Stephanov-Yin (2012)

Established in…
Massless Limit (no chirality mixing)
Adiabatic Limit (no antiparticle mixing / no vacuum fluct)
August 22, 2016 @ RIKEN-Wako

15

Chiral Kinetic Theory@f@f@f+ ẋ+ ṗ= Icoll [f ]@t@x@p
1ẋ =p̂ + E ⇥ b + (p̂ · b)B
1+B·b
1ṗ =E + p̂ ⇥ B + (E · B)b
1+B·bp̂
Approximate solution
b=±(sign=helicity)of Weyl equations
22|p|August 22, 2016 @ RIKEN-Wako

16

Axial AnomalyZ(1 + B · b)f (p, x)pZ j(x) =p̂ + E ⇥ b + (p̂ · b)B f (p, x)⇢(x) =pCME0-th (1-st) moment equation:@⇢(x)
1+ r · j(x) = 2 E · Bf (0, x)@t4⇡
Adiabatic approx.
August 22, 2016 @ RIKEN-Wako

17

@hpi
+ r · hpẋi = ⇢(x)E + j(x) ⇥ B ,@t Moment Equations

1-st (2-nd) moment equation:
mentum continuity equation, whichisZ
3d p ⇥1 + B · b) f (p, x) , hpẋi =p p̂ + E ⇥ b +3@hpi(2⇡)+ r · hpẋi = ⇢(x)E + j(x) ⇥ B ,@tt, the energy equation can be derived as
Momentum conservation

Energy conservation@hpi+ r · hpẋi Z= Ed3· pj(x)⇥,(1 + B · b) f (p,@tx) , hpẋi =p p̂ + E ⇥ b +3(2⇡)
No anomalous term for higher moment equations (at p=0)

Maxwell equations
are used asnt, the energy equation
canbederivedZ
August 22, 2016 @ RIKEN-Wako 3⇥
d p18

Closed-set of Equations@f@f@f+ ẋ+ ṗ= Icoll [f ]@t@x@pr·E =⇢Z

(1 + B · b)f (p, x)pZ j(x) =p̂ + E ⇥ b + (p̂ · b)B f (p, x)⇢(x) =

pr·B =0@Br⇥E =@t
For a given initial condition,@Er⇥B =j+the time evolution is deterministic@tIn principle, numerical simulation is possible…
Research frontier at the present…

Icoll [f ] = 0 gives a “Chiral Vlasov-Maxwell Equations”
August 22, 2016 @ RIKEN-Wako

19

Simple Example
Neglect electric fields and time-derivatives
Z and look for a static configuration

j=p⇥p̂ + (p̂ · b)B f =r ⇥ (r ⇥ B) =j⇤

r2 B =

ABAr ⇥ B =2ABj

Static configurations must be “inhomogeneous”
characterized by a wave-number sA

Chiral Plasma Instability

Akamatsu-Yamamoto (2013)

August 22, 2016 @ RIKEN-Wako

20

Toward Chiral MHD
Vlasov-Maxwell equations are too heavy to solve numerically
Moment equations after the momentum integrations (6+1→3+1)
Closure condition for truncation
Using the equation of state instead of the energy conservation

Magnetohydrodynamics (MHD) Alfven“chiral Alfven wave”
N. Yamamoto (2015)

Chiral MHD is still unknown…
Difficulty is that hpi, hp̂i, j, u are all very di↵erent

cf. ideal MHD (infinite conductivity)
Magnetic lines always move with fluid : no B-flux reconnection
Reconnection favored by anomalous current?
cf.cf. CME from magnetic reconnection Hirono-Kharzeev-Yin (2015,2016)
August 22, 2016 @ RIKEN-Wako

21µν frame,µανβ F equationsµν2- fluidµuν+∂µ handµνsidesofPthese
intoaccounterightrestE=, Bβ uα=) takeuην F(10)Tαβ . ∂ · u, (9)τ=−ηP(∂
u−P+σE,ν=−σTP∂
2 ϵ ζ− 3αβ
ν
one
finds
that
satisfieable.
Equation(16) yields
eep
terms
of sidesorderO(p)in!equations
theconstitutiveequations
Tthese"11.15.-q,
47.75.+f,
11.25.Tq,
12.38.Mhembers:
e right
handoftakeintoaccountact
that
external
field
performs
workonthesystem,µ O(p2 ) µin the equations
satisfiedµ j µ (or termsµν of orderor T µν andwhere
ξ, ξB , D,+σE,(10)
ν
=−σT
P
∂
e
fact
that
external
field
performs
workon
thesystem,
νdthe
∂
µν
µν
µν
µanomaly.
Note
thatrighthandsides
are(Tofwhere
P Note= g that+ uAµuandtheentropy
production
Tthe
f motion).
is, not
dynamical.
quiring∂s≥ (0µD=Td(µ̄),ξ=d(µ̄)),BB sides B
BPdtheanomaly.
Notethat
therighthandareof
2 positive. However, in the presence of∂toµ̄ theis
manifestly
r-rateO(p)orO(p)
inourpowercounting,
which
areusing
the followin
ativistichydrodynamics
isimporproportionalvorticity
[19]
∂µ D
2In
thepresenceof
anexternal
background
field
theµν
µν
µν
derO(p)orO(p)
inourpower
counting,
which
aresons
whereP=g+uu,
andtheentropy
production
anomalies
the equations
last
termwhere
in Eq.
(8)canhave either
sign, of hydrodynamiceq(in nuclear
physics,
astrophysics,
intheorderhydrodynamicequations.
ydrodynamic
obtaintheformd(µ̄)
isanotherfunctionµ̄.
FromEqs.
B
thin
the
order
hydrodynamicequations.
Anomalous
Hydrodynamics
Son-Surowka(2009)-andrate
ismanifestly
positive.
However,inthepresenceofcanoverwhelm
the
other
terms.
Therefore,thehyµµµν
µexample, hydrodynamicmodelsareand(18)
weget∂
D
j=nu−
σT(g+uµu2
hestress-energy
tensorandthecurrentareµThe
stress-energy
tensorandthecurrent
anomalies
thelastterm
inmodified.
Eq. (8)arecan have either sign,
drodynamicequations
haveoftothe
bedescribingthe∂ω
=−µν evolution
νλµ fire- µµ∂
T=Fj,
∂j=
CEB.
(5)
µλ
µµϵ2n+
1 make1 µνλρ
canoverwhelm
thed otherterms.
Therefore,thehyTheandmostgeneralonecanisto′′ addµy-ioncollisions.
Themodification
relativistichy—(µ̄) ϵ− Canom=ρ ,0,(uν ∂µ̄λ u
B (µ̄) = d (µ̄), ω dB=µ2
following
toP theU(1)andentropy
currents,µν termsµµν νhaveµνµν
2
drodynamicequations
tobemodified.
ϵ+µνµνµν∂µ B = −2ωns
been
proposed
manyyears
ethehave
T=(ϵ+)uu+
Pg+
τ,
(6)
T=(ϵ+P)uu+
Pg+
τ,
(6)
where
wedescribe
havedefinedtheelectricandmagnetic
fieldsin is to add
nDThe
mostgeneralmodification
onecanmake
whichcanbeintegrated.
We
findations
thedynamicsof
anµµµ!
"nµµµ1µµνµµναβ
j=nu+ν,(7)µnν Fisαβthechargedensity, σµ= whereµuµ restµν
he theory
fluidframe,
E=F,touµνtime,+B
ϵ Bentropy.currents,
j=nu+
ν(7)
2+σE
ξω+ξ,(11)ν=−σTP∂thefollowingtermstheU(1)andϵ th+ic
atlarge
distance
and
B
ν
one
finds
thatgξ
isthenewkinetic
coeﬃcient.
1 account
1T
The right hand sides of theseequations
take=into
2 Vortical Effect3(CVE)
Chirald(µ̄)
Cµ̄,d(µ̄)=Cµ̄.
(namicvariables
are
the
local
velocsatisfiedBµ.he fact!
"µµνthat µexternalµµµµ ν +Evenin (12)a parity-invarianttheor
2 system,
3 To proceed
fieldperforms
the2 sτ =andµDB Bworksuµ ν−theDω+order, µon
erearetermsofO(p)which
incorpoµµµνu=−1),
localtemperatureµν anomaly.
Tareend+ σE+sides
ξω+ξµincorpoBallowed, a (11)ν ν=µ−σTP that
∂ν ofBTopthe
Notetheright
hand
areofcurrent
ξωisbysymmeteτ
andterms
orderO(p)whichanewpairofte,inparticular,dissipativeeﬀects.
Following
LandauSothenewkinetic
coeﬃcientsareT indexpotential(s)µO(p(x),2 ) wherethe∂µ D −)drder
O(p)
orin
ourpowercounting,
whichareµνµcurrent.
Thisterm
containsanewµB areinLifshitz,
particular,dissipative
Landau$only#eﬀects.
it=uµ νµ. =Re0. # sdT + ndµ,
whereξ,charges.
ξsBµwe,=D,canandDfunctions
Tandµ alwaysµrequireµuµ τ ofFollowingµ$3
2
servedThehydrodynamic
suν + DωB, eﬀectµis as important(12)1 nµ as thB
withintheµνorderhydrodynamicequations.
µν
2 +2Dnµµ −anditsµ+eeeLifshitz,
find
τandν
from
therequirement
oftheexisτ =ν C =µ0.
, ξuBµand=− % sdT.D (n=variables;
Crequireµ initial− uµ condition,Tξarbitrary
quiring∂µ swe≥can0 of
foralways
an&time evolution
these∂µ3ϵ+P Nevertheless, this2 ϵ term+∂TP hasBsµ µνsion.
The
stress-energy
tensorandthecurrentareµνµnceofτtheanentropy
with
using
following
identities
follow from theidealfindand ν current
from∂sµthe
requirement
ofderivathe exisoftheconservation
laws
Twhich= non-negative
0,%=µµνµshydrodynamic
fact,
if
one
follows
thestandard
whereξ,ξ,D,andD
arefunctionsofTandµ.
Re2nD
B
B∂Pextendµ uν ∂µtoe,of∂µan
sbyentropy≥the0. constitutive
Transforming
T non-negative+ µ∂µcharges.—Itj using
equations,
Extension
multipleiseasytoµ̄ ∂equations
eentedcurrentswith
derivaµthenewtermseems
tobe disalloµν ∂µ s ≥ theµcase
νAugustµνµν
quiringforanarbitrary
initialcondition,andϵ+UP22,
2016@charges.
RIKEN-Wako
22theaµµ0a +of
many
Here
weconsider
onlyT=(ϵ+P)uu+Pgτ,(6)µµνµndjintermsofu,T,andµ.
∂∂µ s using≥ 0.
Transforminguν ∂µwhich
T an +µ∂µfrom
j currentusing
2
entropywithmanifes
nD
thefollowing
identities
follow
theidealBµµµµ chargesµthat commute with each other. We denote

Best among Knowns

Chiral Vortical EffectVilenkin (1980)
Solving the Dirac eq. in a rotating frame at T > 0j=!

Z0f (p) =p✓22◆µT+!
24⇡
12

Skeptical about T=0 limit because of the boundary effect
Rotation alone induces “nothing” (causality+finite size)
Ebihara-KF-Mameda (2016)

Field-theoretical computation interpreted as anomaly?
August 22, 2016 @ RIKEN-Wako

23τ =sides−ηP ofPthese(∂αequationsuβ +∂β uα ) take− ζ−into∂ · u, (9)right handaccount
3η P!µ"satisfiee
act thatexternal
fieldperforms
workonthesystem,µµ+
σE,Kubo Formula(10)νCalculation= −σT P µν ∂νdfromTthe right hand sides are oftheanomaly.
Notethat2
rs O(p)orO(pcounting,whichareµν ) inµνour power
where P= g + uµ uν , andthe entropy
production
inorderhydrodynamic
equations.
Landsteiner-Megias-Pena-Benitez(2011)- therate
is manifestly
positive.
However,in the presence of

anomalies
last term(8) canare
haveKubo
eithersign,
he stress-energy
tensorand intheEq.
Anomalousthe
Hydrodynamics→current
Deriving
theFormulas

and µcan overwhelm
xz the other terms. Therefore, the hy—u = (1, v , 0, v ) gµ⌫ = ⌘µ⌫ + hµ⌫ Aµ = (0, ax , 0, az )drodynamicequations have to be modified.
eµνµ νµνµνT=(ϵ+P)uu+Pg+
τ, make is(6)The most general
to add⇠ v modification one cannµthe jfollowingg= nuµterms+ ν µ ,to the U(1) and entropy currents,(7)⇠ @v + @h ⇠ @a.
!µ"e µν ν µ =µ −σT P µν ∂ν+ σE µ + ξω µ + ξB B µ ,(11)To pand ν are terms ofTorder O(p) which incorpo)e τµ µµµµµin
particular,dissipativeeﬀects.
Following
Landau (12) a news=su−
ν+Dω+DB
B i ,Xe
T= limµν hj i T 0j i µ"
sdT + nLifshitz,
we canCVE
alwaysrequireuτ=uν=0.
zijnsµµ!=0kn !02knµνµijs
ξ, ξνB , D,
andtheDBrequirement
are functions ofof theT andµ. Refindτwhereandfromexis%µ∂quiring∂µ s current≥ 0 forsµanwith
arbitrary
initial condition,ande of anentropy
non-negative
derivaAugust 22, 2016 @ RIKEN-Wako
24µµνµthe Transforming
following identities
followthe ideal∂≥ 0.
uν ∂µwhich
T +µ∂µfromj usinge∂µ s using

=Rjkih ,Rkjih =Rkjhi ,Rkjih + Rikjh + Rjikh

Mixed Gravitational Anomaly
II.

CHIRAL ANOMALIES

Landsteiner-Megias-Pena-Benitez (2011)

Conductivity calculated for general SU(N) groupdimensions
reads,
From group theoretical structure:µrµ jA = CF ✏µ⌫⇢

we can see,Fµ⌫ F⇢ + CR ✏µ⌫⇢ Rµ⌫ ↵ R⇢ ↵2µ termB ⇠ µ!

2T term

From where T 2 ???

µµ⌫⇢@µManyjA= 4C@(✏A@A),Fµ⌫⇢papers and discussions but…

Modified at finite T ?
@ RIKEN-Wakoµ August 22, 2016µ⌫⇢jA = 4CF ✏A⌫ F ⇢.

25

values of these coefficients are a necessary and sufficient conexp
dition for the presence of anomalies [10]. Therefore the nonthe
vanishing values of the transport coefficients (1) and (2) haveB
to be attributed to the presence of chiral and gravitationalcha
Landsteiner-Megias-Pena-Benitez (2011)
anomalies.
spe
Since the gravitational anomaly is fourth order in derivamig
tives it is a bit surprising to find it contributing to first ordersugtransport coefficients. One possible intuitive explanation one
rise
could think of is that the gravitational field in the presence ofnumµmatter gives rise to a fluid velocity u e.g. through frame dragcon
ging effects and that this might effectively reduce the number
numof derivatives that enter in the hydrodynamic expansion.
The holographic calculation in AdS/CFT [6] did not showW
2 . This not surprising sinceaJensen-Loganayagam-Yarom
contribution proportional to T(2012,2013)
con
onlyaholographicgauge
Chern-Simons
termwasincluded.
Anomaly inflow:pita
Holographic
modelingof
thegravitational
anomalycalls
how2n-dim gauge anomaly = (2n+2)-dim chiral anomaly
Gluever also for inclusion of a mixed gauge-gravitational ChernAugust 22, 2016 @ RIKEN-Wako
26 cionSimons term of the form
A ∧ R ∧ R [9].
000Mixed Gravitational AnomalyJust for curiosity…
What is the Chern-Simons current with the Kerr metric?
(r,= cos ✓, , t)Taking the extremal limit (T=0) for simplicity:
32 ! 3 ( 1 + 2r!)[ 4 + 48r3 ! 3 ( 1 + r!)
jr =( 2 + 4r2 ! 2 )5j =32! 4 [ 648r4 ! 44 2 r!( 3 + 8r!)]'32! 354(3 + 56r2 ! 2 ) + 72 2 (r2 ! 2 + 2r4 ! 4 )] 32( 3 +'6( 2 + 4r2 ! 2 )5

Is this physical or unphysical???
August 22, 2016 @ RIKEN-Wako272)! 4

SummaryResearch Frontier□ Experimental Data and Interpretation□ Chiral Magnetohydrodynamics□ Chiral Vortical Effect : Better for HIC
Confusing and misleading (causality)

Future Directions□ Astrophysical Applications□ Particle Production with Mixed Anomaly(Chiral Anomaly → Schwinger Mechanism)□ Chern-Simons Gravity Theory
August 22, 2016 @ RIKEN-Wako28