Spin transportvia

gauge/gravity duality

Taro Kimura (RIKEN)

collaboration withK. Hashimoto (Osaka/RIKEN) and N. Iizuka (Kyoto)

based on arXiv:1304.3126 [hep-th]

Why spin?

electronics + spin = spintronics

Spin transport

• Spin-depend conduction

~Jz = ~J" � ~J#

cf. electric current : ~J = ~J" + ~J#

WARNING!!

This spin current is not conserved.

1. approximately conserved situation2. improve the definition of the current

@tSz + ~r · ~Jz = Tz

1. Introduction

2. Spin current definition

3. Analysis via gauge/gravity duality

4. Summary

Contents

Spin current definition

•Electric current

Jµ =�S

�Aµ@µJ

µ = 0

U(1) gauge symmetry

•Stress tensor

Tµ⌫ =�S

�gµ⌫@µT

µ⌫ = 0

Translation symmetry

�µ , i@µ�⇤�� i�⇤@µ�

spin connection•Spin current

Jµ

ab=

�S

�! abµ

@µJµ

ab= 0

Local Lorentz symmetry(rotation symmetry)

cf. [Wen-Zee ’92] [Froehlich-Studer ’93]

i�µ

✓@µ � i

2! abµ ⌃ab

◆�m

�

⌃ab =i

4[�a, �b]Local Lorentz generator :

Jµ

ab=

1

2 �µ⌃ab

Jµa ⌘ ✏0abc J

µ bc

=1

2 �µ (�a ⌦ )

r = @ � �Covariant derivative :

• Remark

“spin current” = total angular momentum current

totalRelativistic :

spin orbitNon-relativistic :

• Spin current definition

• spin connection associated with local Lorentz symmetry

• fermionic contribution → spin-depend current

• a proper non-relativistic limit required

Summary

1. Introduction

2. Spin current definition

3. Analysis via gauge/gravity duality

4. Summary

Contents

Strongly correlated spin transport

Strongly correlated spin transport

• A method to approach such a difficult system

Strongly correlated spin transport

Gauge/gravity duality

also AdS/CFT, holography,..

Strongly correlated QFT Classical gravity

• Generating function

ZQFT =

ZD� e�S[�]+

Rdx J�

ZSUGRA = e�Sgravity[�0]

GKP-Witten relation

not quantum anymore, but classical

just solve Einstein equation

• Source term for QFT

Lsource

= AµJµ, gµ⌫T

µ⌫ , ! abµ Jµ

ab

• “Dictionary” between QFT and gravity

QFTBH horizon

gravity

Example :AdS4 Schwarzschild geometry

QFT

gravity

• Background energy flow

• Strongly correlated QFT (Conformal Field Theory)• d = 2+1

d = 2+1

d = 3+1

• Finite temperature

BH horizon

• “Boosted” AdS4 Schwarzschild black brane solution

off-diagonal

• “Boosted” AdS4 Schwarzschild black brane solution

• Temperature :

• “boost” :

ds

2 =� U(r)dt2 +1

U(r)dr

2 + r

2dy

2

+�r

2 � a

2U(r)

�dx

2 � 2aU(r)dtdx

T =3r04⇡

t �! t+ at

U(r) =r3 � r30

rwith

Background energy flow

Perturbation

• Gravity action

• Fluctuations :

Sgravity =

Zd

4x

p�g (R[g]� 2⇤)

+

Zd

3x

p��⇥

�gxy

= ✏ e�i!tr2h(r)

�gty

= ✏ e�i!tr2f(r)

• Einstein equations (e.o.m.)

f 0(r) =

✓a+

1

a

r3

r30 � r3

◆�1

h0(r)

h(r) +1

!2

r3 � r30r3

d

dr

r3 � r30

(1� a2)r3 + a2r30r4

d

drh(r)

�

ry-comp. :

ty -comp. :

= 0

!

r0=

3

4⇡

!

Tdimensionless ratio :

• Einstein equations (e.o.m.)

f 0(r) =

✓a+

1

a

r3

r30 � r3

◆�1

h0(r)

!

2

r

20

h(x) = x

2(x3 � 1)d

dx

1� x

3

x

2(1� a

2 + a

2x

3)

d

dx

h(x)

�

ry-comp. :

ty -comp. :

x =r0

r

with

• Near the horizon

• Near the boundary

• Two independent solutions

Boundary conditionsx ⇠ 1

x ⇠ 0

h(x) ⇠ exp

✓� i

3

!

r0log(1� x)

◆in-going condition

h(x) ⇠ h0 +O(x2)

h(x) ⇠ h3 x3 +O(x5)

Non-normalizable mode

Normalizable mode

• Thermal spin Hall conductivity

J z

x

= �sH ry

T

sH = � J z

x

ry

T=

! xy (N)x

i!Tg(NN)ty

/ h3

f0

sign flip

non-zero in DC limit

5 10 15 20

-2

2

4

6

8

10

!

T

a = 0.03, 0.5, 0.9

• Another spin transport coefficient

J z

x

= �↵s rx

µz

↵s = � J z

x

rx

µz

=! xy (N)x

i! ! xy (NN)x

/ h3

h0

non-zero in DC limit

!

T5 10 15 20

-0.8

-0.6

-0.4

-0.2

• Spin transport via gauge/gravity duality

• Example : pure gravity on “boosted” AdS4 Schwarzschild

•ω-dependence of thermal spin Hall conductivity and more

Summary

Summary

• Definition of the spin current as a conserved one

• spin connection associated with local Lorentz symmetry

• Spin transport via gauge/gravity duality

• AC conductivities estimated

Summary

• TO DO

• Non-relativistic limit

• Analysis with U(1) EM field

Summary