Transport from gravitational Chern-Simons: Theme and Variation

Karl LandsteinerInstituto de Física Teórica UAM-CSIC

Gauge/Gravity, Würzburg, 29-07-2018

Outline

• Anomalous Transport

• 3 Examples

• Translation breaking

• Quenching the CME

• Odd viscosity

• Outlook

[Maldacena] [Witten] [Gubser, Klebanov, Polyakov]

Holography: ds2 = r2(�f(r)dt2 + d⌥x2) +dr2

f(r)r2

AdS = spherical (hyperbolic) cow of sQGP

⌘

s=

1

4⇡[Policastro, Son, Starinets][Kovtun, Son, Starinets]

Anomalous Transport

SCS =

Z(

3A ^ F ^ F + �A ^ tr(R ^R)

�SCS =

Z

@✓⇣3F ^ F + � tr(R ^R)

⌘

Chiral anomaly“Gravitational” Chiral anomaly

Anomalous Transport

~J = 8µ ~B +�4µ2 + 32⇡2�T 2

�2~!

~J✏ =�4µ2 + 32⇡2�T 2

�~B +

✓4

3µ3 + 32⇡2�µT 2

◆2~!

•Chiral Magnetic Effect•Chiral Vortical Effect•Chemical Potential “k“ chiral anomaly•Temperature “l“, grav. Anomaly

[Banerjee, Bhattacharya, Bhattacharyya, Dutta, Loganayagam, Surowka][Erdmenger, Haack, Kaminski, Yarom], [Megias, Melgar, K.L., Pena-Benitez]

Gravitational anomaly• Objection: 4-th order in derivatives, too high

• BUT: AdS has 5-th dimension

DµJµ =

1

768⇡2✏µ⌫⇢�R↵

�µ⌫R�

↵⇢�

Zd5xA ^ tr (R ^R) =

Zd5xA ^ tr

⇣R(4) ^R(4) +A ^D(K ^DK)

⌘

• Is there even in flat space• Vanishes at the boundary• Picks up a contribution from the horizon• Suggest “low energy anomaly” • even in flat space, 2 space time derivatives

@µJµ = D(K ^DK)

Kij ⇡@

@rgij

• BUT: K is covariant tensor w.r.t. to 4-dim spacetime

Quantum Currents from 5D• At the black hole horizon

ds2 = r2f(r)(�dt2 + 2 �Ad�xdt) + r2d�x2 +dr2

r2f(r)

• Following Hawking

• Current

•NO! not anomaly but quantum current (Luttinger) T 2

12⌃Eg. ⌃Bg = ⌃� ⌃Jq

•The Chiral Vortical Effect (CVE) !!

Quantum Currents from 5D

⇧Jq =b

12T 2⇧�

�⇥⇥T

T⇥⇥� ⇥v

[Megias, Melgar, K.L.,Pena-Benitez][Azayanagi, Loganayagam, Ng, Rodriguez],[Chapman, Neiman, Oz],[Grozdanov, Poovuttikul]hydro+geometry: [Jensen,Loganayagam,Yarom], global anomaly: [Golkar, Sethi]

•Current Jµ =p�gFµr � �

2⇡G✏µ⌫⇢�K�

⌫D⇢K��

[Copetti, Fernandez-Pendas, K.L.]

• Luttinger ~Eg = �~rT

T~Bg = r⇥ ~v

D(K ^DK) / f 0(rh)2 @ ~A

@t(~r⇥ ~A)

D(K ^DK) / 64⇡2T 2 ~Eg · ~Bg

Chiral Vortical Effect~J = 64⇡2�T 2~!

[POSTER -> C. Copetti]

Comments• Black hole horizon• Without AdS • Holographic RG flow• Without Holography: couple to black hole• BH does RG flow for you• Boundary condition: no outgoing current at horizon

RegularityFinte T=1/b

[Jensen, Loganayagam, Yarom] (Cone)[Stone, Kim]

Also [Wilczek, Robinson]

• Global anomaly: [Golkar, Sethi], [Glorioso, Liu]

Translation breaking• In Holography “linear axion” background (massless scalar)

S =

Zd5x

p�g

�|@�|2

�, � ⇡ kx+ · · ·

• Background breaks translations eoms are homogeneous• Graviton has mass

1. Intuition: Momentum density, broken symmetry2. Intuition: Energy current is dissipationless

T 0i = T i0

• Charge (Momentum) = Current (Energy-current)

[Copetti, Fernandez-Pendas, K.L., Megias]

Translation breaking• Can extrinsic curvature term be seen in UV?

f = 1� k2

r2� (1� k2

4)r4H

r4ds2 = �fr2dt2 +

dr2

r2f+ r2d~x2

Unusual power!

• Extrinsic curvature as additional variable

�Son�shell =

Z

@

p�g(tµ⌫�gµ⌫ + uµ⌫�Kµ⌫)

• Energy momentum tensor (Ward identity)

⇥µ⌫ = tµ⌫ + uµ�K⌫�

• New term is due to gravitational Chern-Simons term

Symmetry breaking - II• CME and CVE without new term

• Impossible in unitary theory• CVE = 2 CME for energy current by Kubo formulas

• Including the new term

• All is well!• Energy current being dissipation wins!

[Copetti, Fernandez-Pendas, K.L., Megias]

J i =�4µ2 + 32⇡2�T 2

�2⌦i

T 0i =�4µ2 + 32⇡2�T 2 � 4�k2

�Bi

T 0i =�4µ2 + 32⇡2�T 2

�Bi

CME• Aug. 2008: "The Chiral Magnetic Effect” [Fukushima, Kharzeev, Warringa]

~J =µ5

2⇡2~B

B

JE

n- 0 1

• Topology of QCD fields

• CME induces charge separation

• First observable of anomalous transport

Quenching the CME

•CME and CVE depend on equilibrium quantities •Natural question: anomaly induced transport far from equilibrium physics

• Possible importance for Heavy Ion Collisions (magnetic field has already decayed in hydrodynamic regime)

• Holography allows both: study fast time evolution, quenches and anomalous transport

• Study CME via gravitational Chern-Simons term

T, µ

[E. Lopez, G. Milans del Bosch, K.L.]

• “Minimal” setup: inject energy

• Equilibrium: energy — temperature T0 T• CME in energy-momentum tensor• First near equilibrium = hydro

What to look for

Tµ⌫ = (✏+ p)uµu⌫ + p⌘µ⌫ + �̂B(uµB⌫ + u⌫Bµ) ,

Jµ = ⇢uµ + �BBµ ,

JXµ = ⇢Xuµ + �B,XBµ

�̂B = 0

�B = 24↵µ� ⇢

✏+ p

�12↵µ2 + 32�⇡2T 2

�

�B,X = � ⇢X✏+ p

�12↵µ2 + 32�⇡2T 2

�

• Landau frame

• Energy current = Momentum density

What to look for

T0i = Ti0

• Momentum density = conserved charge

32�⇡2T 20~B = (✏+ p)~v

• Monitor response in tracer U(1) current

~JX = 32⇢X✏+ p

(T 20 � T 2)⇡2�~B

• Removing constants: benchmark near equilibrium curve

jX =T 2/T 2

0 � 1

T 4/T 40

1.5 2.0 2.5 3.0 3.5 4.0t

0.05

0.10

0.15

0.20

0.25jX

Holographic quench• Very slow quenches

2tT0=15.3

2tT0=12.7

2tT0=10.2

2tT0=7.64

2tT0=5.09

-0.6 -0.4 -0.2 têt

0.15

0.2

0.25jX

1.5 2.0 2.5 3.0 3.5 4.0t

0.05

0.10

0.15

0.20

0.25jX

Holographic quench• Fast quenches

jX ë jXeq

0.8 1 1.20.95

1

1.05

tT

f0HtL

jX

0.0 0.2 0.4 0.6 0.8 1.0 1.2tT

0.02

0.04

0.06

0.08

0.10

0.12

0.14

2⌧T < 1

Holography for Weyl Semi-Metals[K.L., Liu, Sun]

WSM

pz

�

pz

�

pz

�

Critical Trivial

• Holographic model for quantum phase transition

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

M

b

ΣAHE

8Αb

⇥xy = 8�A5z(0)

[POSTER -> J. Fernandez-Pendas]

Odd viscosity• Hall viscosity in 2D Quantum Hall states• Time reversal breaking necessary• 2D : invariant e tensor• 3D: need some anisotropy

[Avron, Seiler, Zograf]

[Landau, Lifshytz Vol. 10]

Vij =1

2(@ivj + @jvi)

• In total: 3 shear, 2 “bulk” and 2 odd viscosities

⌧xy = ⌘?Vxy � ⌘H? (Vxx � Vyy)

⌧xz = ⌘kVxz + ⌘Hk Vyz

⌧yz = ⌘kVyz � ⌘Hk Vxz

Odd viscosity

⌘3 = ⌘?

~b

x

y

z

⌘4 = ⌘k

~b

pressure

stress

Odd viscosity• Odd viscosities by adding gravitational anomaly term• Probe IR region of geometry: Low T

T/b=0.05T/b=0.04T/b=0.03

��� ��� ��� ��� ��� ��� ����

�

��

��

��

��

��

η�∥

��

transverse

T/b=0.05T/b=0.04T/b=0.03

��� ��� ��� ��� ��� ��� ����

��

��

��

��

���

��

η�⊥�ζ��

parallel

• Prediction from Holography !• Again: gravitational Anomaly at first order !

Summary

• Holography is efficient discovery tool for transport • Fate of anomalous transport under symmetry breaking

• New questions:

• What is the extrinsic curvature in field theory?

• Anomalous transport far from equilibrium (QGP)

• Odd viscosity and grav. anomaly?