A Holographic Model of the Kondo Effect Andy OBannon University - - PowerPoint PPT Presentation

Andy O’Bannon

University of Oxford October 29, 2013

A Holographic Model

• f the

Kondo Effect

71 High Street, Oxford

OUT OF BUSINESS

Credits

Based on 1310.3271 Jackson Wu Johanna Erdmenger

National Center for Theoretical Sciences, Taiwan Max Planck Institute for Physics, Munich

Carlos Hoyos

Tel Aviv University

Outline:

• The Kondo Effect
• The CFT Approach
• A Top-Down Holographic Model
• A Bottom-Up Holographic Model
• Summary and Outlook

July 10, 1908

Heike Kamerlingh Onnes liquifies helium Leiden, the Netherlands

(1 atm)

T ≈ 4.2 K

Shortly Thereafter

Leiden, the Netherlands Begins studying low-temperature properties of metals

T ≈ 1 to 10 K

April 8, 1911

Heike Kamerlingh Onnes discovers superconductivity

R

“for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium”

1913

Onnes receives the Nobel Prize in Physics

Smith and Fickett, J. Res. NIST, 100, 119 (1995)

Ag

ΘD ≈ 200 K

Ag

Resistivity measures electron scattering cross section

ΘD ≈ 200 K

Debye Temperature

Quantized vibrational modes of a solid = Phonons Minimum wavelength: 2 x (lattice spacing) Maximal Frequency lowest temperature at which maximal-energy phonon excited

ΘD

T ΘD

electron-phonon scattering

ΘD ≈ 200 K

Ag

ρ(T) ∝ T

ρ(T) = ρ0 + a T 2 + b T 5

electron-phonon scattering

T ΘD

ΘD ≈ 200 K

Ag

ρ(T) = ρ0 + a T 2 + b T 5

T ΘD

electron-electron scattering

ΘD ≈ 200 K

Ag

ρ(T) = ρ0 + a T 2 + b T 5

T ΘD

electron-impurity scattering

ΘD ≈ 200 K

Ag

T ΘD

increasing concentration of impurities

ΘD ≈ 200 K

Ag

The Kondo Effect

r e s i s t a n c e temperature ~10 K

a b

476 A.M. Tsvelick and P. B. Wiegmann

• Fig. 1.13

E

• Q_

9 8- 7-

6'

5- 4- 5- 2-

1

*o (L._gq, Ce) B 6 % .*° Oat %Ce ~****** ~- 0.61 at%Ce . . . . 1.20 at % Ce

• ••
• vvv

1.80 at %Ce °

k

..-

2.90 at % Ca :

~ " . +..

..,.

• ,q.,,,,.
• Jj

~,,p

• .....................

..~Z.~

0.05 0.1 0.2 0.5 1 2 5 I0 20 50 100200

T~ K

Electrical resistivity of LaB s and four (La, Ce)B 6 samples as a function of temperature (after Samwer and Winzer 1976).

• Fig. 1.14

E

• q_

0 + ° ° ° Jl°,,o°,.,% ( L__q, Ce] B 6

0.5÷ ........ "--.[5..

0.8 T .... "%";, 1.2 at % Ce

1 T ....... ~ .... ~, "x."-:'. .5 T .................................

~..'.'.!.~::,.%

2 T ............. ""-" "~ I:~ ."

......... ::::':!!t~ . . . . ..

4 T ...................... " ..... ~...-"~'" "" .......

6T

' '

• '

i ' ' ' ' 'o 'o

0.02 0.05 O.I 2 05 1 2 5 I0 2 5 I00

T/K

Electrical resistivity of an (La, Ce)B 6 sample with 1"2 at.~ Ce versus temperature for various magnetic fields (after Samwer and Winzer 1976).

Samwer and Winzer, Z. Phys B, 25, 269, 1976

MAGNETIC Impurities

Curie:

Exact results in the theory of magnetic alloys

• Fig. 1.1

469 80

"T 60

• E

2

40

20

• 2

/

__, t ~'

(La Ce) B 6 ~xod x°

• o

* 0.072 °×,8

• 0.1.3

xoX

• exo

×o~ x° t21

~oo ~ .~o ~'° 420~ °xx° ~°x°<c~°~ ~'"

• [9

x°~xx /"/"* // _ 18 E

0.2

0.4

T/K

2 4

8 T/K

Inverse impurity susceptibility per mole Ce, 1/) of (La, Ce)B 6 alloys with 0'072 and 0'13 at.% Ce as a function of temperature T. The insert shows the behaviour at low temperature

• n an expanded scale. The dashed line is an extrapolation of the Curie-Weiss law fitted

to the data (after Felsch 1978).

2

• Fig. 1.2

0.8 f (La' Ce)a6

""" .................. 2!

........ 2 ......... 22

iiio ;;

LS D 1.8

• -- Free iron

* 2.9 ...... Ion in Crystal Field

• 5.0

0.6 0.4 0.2 ~t

IO0 200 300

T/K

Z T versus temperature T for (La, Ce)B 6 alloys. The dashed line is the behaviour of free Ce 3 +

• ions. The dotted line is the behaviour of Ce 3

+ ions in the cubic crystal field of LaB 6 (after Felsch 1978).

Felsch, Z. Phys B, 29, 211, 1978

Pauli: χ ∝ T 0

χ ∝ T −1

Fermi liquid Free magnetic moment

The Kondo Hamiltonian

Conduction electrons

ckσ c†

kσ

σ =↑, ↓

Dispersion relation , Spin SU(2)

HK =

• k,σ

(k) c†

kσckσ + gK

S ·

• kσkσ

c†

kσ

1 2 σσckσ

ε(k) = k2 2m − εF

ckσ → eiαckσ

Charge U(1)

gK > 0

Anti-Ferromagnetic

gK

Kondo coupling

gK < 0

Ferromagnetic Spin of magnetic impurity

• S
• Pauli matrices

The Kondo Hamiltonian

HK =

• k,σ

(k) c†

kσckσ + gK

S ·

• kσkσ

c†

kσ

1 2 σσckσ

concentration of impurities DECREASES

ρ(T)

decreases

T

⇒

UV cutoff

=

c, ˜ c

∝

as

ρ(T) = ρ0 + a T 2 + b T 5 + c g2

K − ˜

c g3

K ln (T/εF )

εF

gK < 0

Ferromagnetic

DECREASES

ρ(T)

decreases

T

⇒

as

gK < 0

Ferromagnetic

ρ(T) = ρ0 + a T 2 + b T 5 + c g2

K − ˜

c g3

K ln (T/εF )

concentration of impurities UV cutoff

=

c, ˜ c

∝

εF

ρ(T) = ρ0 + a T 2 + b T 5 + c g2

K − ˜

c g3

K ln (T/εF )

concentration of impurities INCREASES

ρ(T)

decreases

T

⇒

UV cutoff

=

c, ˜ c

∝

as

εF

Anti-Ferromagnetic

gK > 0

ρ(T) = ρ0 + a T 2 + b T 5 + c g2

K − ˜

c g3

K ln (T/εF )

“Kondo temperature”

O(g3

K)

O(g2

K)

term is same order as term when

TK ≈ εF e− c

˜ c 1 gK

ρ(T) = ρ0 + a T 2 + b T 5 + c g2

K − ˜

c g3

K ln (T/εF )

Breakdown of Perturbation Theory

Cross section for electron scattering off a MAGNETIC impurity INCREASES as energy DECREASES

ρ(T) = ρ0 + a T 2 + b T 5 + c g2

K − ˜

c g3

K ln (T/εF )

βgK ∝ −g2

K + O(g3 K)

Asymptotic freedom!

TK ∼ ΛQCD

The Kondo Problem

What is the ground state? We know the answer! The coupling diverges at low energy!

Solutions of the Kondo Problem Numerical RG (Wilson 1975) Fermi liquid description (Nozières 1975) Bethe Ansatz/Integrability (Andrei, Wiegmann, Tsvelick, Destri, ... 1980s) Conformal Field Theory (CFT) (Affleck and Ludwig 1990s) Large-N expansion

(Anderson, Read, Newns, Doniach, Coleman, ...1970-80s)

Quantum Monte Carlo (Hirsch, Fye, Gubernatis, Scalapino,... 1980s)

UV IR

The electrons SCREEN the impurity’s spin

Fermi liquid + decoupled spin

“Kondo resonance” A MANY-BODY effect Produces a MANY-BODY RESONANCE

UV IR

A SINGLE electron binds with the impurity Anti-symmetric singlet of SU(2)

1

• 2 (|i e |i e)

Fermi liquid + decoupled spin

“Kondo singlet” Intuitive SINGLE-BODY Description

Fermi liquid + decoupled spin

UV IR

Fermi liquid

+ electrons EXCLUDED from impurity location

+ NO spin

Fermi liquid + NON-MAGNETIC impurity Fermi liquid + decoupled spin

UV IR

476 A.M. Tsvelick and P. B. Wiegmann

• Fig. 1.13

E

• Q_

9 8- 7-

6'

5- 4- 5- 2-

1

*o (L._gq, Ce) B 6 % .*° Oat %Ce ~****** ~- 0.61 at%Ce . . . . 1.20 at % Ce

• ••
• vvv

1.80 at %Ce °

k

..-

2.90 at % Ca :

~ " . +..

..,.

• ,q.,,,,.
• Jj

~,,p

• .....................

..~Z.~

0.05 0.1 0.2 0.5 1 2 5 I0 20 50 100200

T~ K

Electrical resistivity of LaB s and four (La, Ce)B 6 samples as a function of temperature (after Samwer and Winzer 1976).

• Fig. 1.14

E

• q_

0 + ° ° ° Jl°,,o°,.,% ( L__q, Ce] B 6

0.5÷ ........ "--.[5..

0.8 T .... "%";, 1.2 at % Ce

1 T ....... ~ .... ~, "x."-:'. .5 T .................................

~..'.'.!.~::,.%

2 T ............. ""-" "~ I:~ ."

......... ::::':!!t~ . . . . ..

4 T ...................... " ..... ~...-"~'" "" .......

6T

' '

• '

i ' ' ' ' 'o 'o

0.02 0.05 O.I 2 05 1 2 5 I0 2 5 I00

T/K

Electrical resistivity of an (La, Ce)B 6 sample with 1"2 at.~ Ce versus temperature for various magnetic fields (after Samwer and Winzer 1976).

Samwer and Winzer, Z. Phys B, 25, 269, 1976

Kondo Effect in Many Systems

Quantum dots

8

Cu, Ag, Au, Mg, Zn, ... doped with Cr, Fe, Mo, Mn, Re, Os, ... 200nm

Alloys

Goldhaber-Gordon, et al., Nature 391 (1998), 156-159. Cronenwett, et al., Science 281 (1998), no. 5376, 540-544.

Enhance the spin group

Generalizations

SU(2) → SU(N)

Observation of the SU(4) Kondo state in a double quantum dot

• A. J. Keller1, S. Amasha1,†, I. Weymann2, C. P. Moca3,4, I. G. Rau1,‡, J. A. Katine5,

Hadas Shtrikman6, G. Zar´ and3, and D. Goldhaber-Gordon1,*

n, Poland

ulet” Group, Institute of Physics, Budapest University

• f Technology and Economics, H-1521 Budapest, Hungary

arXiv:1306.6326v1 [cond-mat.mes-hall] 26 Jun 2013

Enhance the spin group

Generalizations

SU(2) → SU(N)

arXiv:1310.6563v1 [cond-mat.str-el] 24 Oct 2013

SU(12) Kondo Effect in Carbon Nanotube Quantum Dot

Igor Kuzmenko1 and Yshai Avishai1,2

1 Department of Physics, Ben-Gurion University of the Negev Beer-Sheva, Israel 2 Department of Physics, Hong Kong University of Science and Technology, Kowloon, Hong Kong

(Dated: October 25, 2013)

Multiple “channels” or “flavors” Enhance the spin group Representation of impurity spin

Generalizations

SU(2) → SU(N)

c → cα

α = 1, . . . , k

simp = 1/2 → Rimp

U(1) × SU(k)

IR fixed point: “Non-Fermi liquids” NOT always a fermi liquid

Generalizations

Kondo model specified by Apply the techniques mentioned above...

N, Rimp, k

Open Problems

Entanglement Entropy Quantum Quenches Multiple Impurities

Kondo:

Form singlets with each other Competition between these can produce a QUANTUM PHASE TRANSITION

Form singlets with electrons

• Si ·

Sj

Open Problems UBe13

UPt3

CeCu6

YbAl3

CePd2Si2

YbRh2Si2

Multiple Impurities

Heavy fermion compounds

NpPd5Al2 CeCoIn5

Open Problems UBe13

UPt3

CeCu6

YbAl3

CePd2Si2

YbRh2Si2

Multiple Impurities

Heavy fermion compounds

NpPd5Al2 CeCoIn5

Open Problems

Example

• J. Custers et al., Nature 424, 524 (2003)

Kondo lattice

YbRh2Si2

1

0.0 0.1 0.2 0.3

LFL AF NFL YbRh2Si2 H || c

2

T (K) H (T)

ρ ∼ T 2

ρ ∼ T

Multiple Impurities

Heavy fermion compounds

Solutions of the Kondo Problem Numerical RG (Wilson 1975) Fermi liquid description (Nozières 1975) Bethe Ansatz/Integrability (Andrei, Wiegmann, Tsvelick, Destri, ... 1980s) Conformal Field Theory (CFT) (Affleck and Ludwig 1990s) Large-N expansion

(Anderson, Read, Newns, Doniach, Coleman, ...1970-80s)

Quantum Monte Carlo (Hirsch, Fye, Gubernatis, Scalapino,... 1980s)

The Kondo Lattice

The Kondo Lattice... “... remains one of the biggest unsolved problems in condensed matter physics.”

Alexei Tsvelik QFT in Condensed Matter Physics (Cambridge Univ. Press, 2003)

“... remains one of the biggest unsolved problems in condensed matter physics.”

Alexei Tsvelik QFT in Condensed Matter Physics (Cambridge Univ. Press, 2003)

Let’s try AdS/CFT!

The Kondo Lattice...

GOAL

Find a holographic description

• f the

Kondo Effect

Solutions of the Kondo Problem Numerical RG (Wilson 1975) Fermi liquid description (Nozières 1975) Bethe Ansatz/Integrability (Andrei, Wiegmann, Tsvelick, Destri, ... 1980s) Conformal Field Theory (CFT) (Affleck and Ludwig 1990s) Large-N expansion

(Anderson, Read, Newns, Doniach, Coleman, ...1970-80s)

Quantum Monte Carlo (Hirsch, Fye, Gubernatis, Scalapino,... 1980s)

Outline:

• The Kondo Effect
• The CFT Approach
• A Top-Down Holographic Model
• A Bottom-Up Holographic Model
• Summary and Outlook

Kondo interaction preserves spherical symmetry

gK3( x) S · c†( x) 1 2 c( x)

Reduction to one dimension

restrict to momenta near restrict to s-wave

kF

CFT Approach to the Kondo Effect

Affleck and Ludwig 1990s

c( x) ≈ 1 r

• e−ikF rL (r) − e+ikF rR (r)

L R

r

L L

r

ψL(−r) ≡ ψR(+r)

r = 0 r = 0

RELATIVISTIC chiral fermions “speed of light”

vF

CFT Approach to the Kondo Effect

˜ gK ≡ k2

F

2π2vF × gK

CFT!

=

HK = vF 2π +∞

−∞

dr

• †

LirL + (r) ˜

gK S · †

L

L

k ≥ 1

Spin SU(N)

U(1)

SU(k) SU(N)

J = ψ†

LψL

JA = ψ†

L tA ψL

• J = †

L

L

Kac-Moody Current Algebra

z ≡ τ + ir

JA(z) =

• n∈Z

z−n−1JA

n

[JA

n , JB m] = if ABCJC n+m + N n

2 δABδn,−m

SU(k)N

N counts net number of chiral fermions

CFT Approach to the Kondo Effect

Full symmetry:

(1 + 1)d

conformal symmetry

SU(N)k × SU(k)N × U(1)kN

HK = vF 2π +∞

−∞

dr

• †

LirL + (r) ˜

gK S · †

L

L

CFT Approach to the Kondo Effect

J = ψ†

LψL

JA = ψ†

L tA ψL

Kondo coupling:

S · J

U(1)

SU(k) SU(N)

HK = vF 2π +∞

−∞

dr

• †

LirL + (r) ˜

gK S · †

L

L

• J = †

L

L

UV IR

SU(N)k × SU(k)N × U(1)Nk SU(N)k × SU(k)N × U(1)Nk

Eigenstates are representations

• f the Kac-Moody algebra

UV IR

SU(N)k × SU(k)N × U(1)Nk SU(N)k × SU(k)N × U(1)Nk

|c, s, f

|c, s, f

Fusion Rules

s ⊕ simp = s

UV IR

SU(N)k × SU(k)N × U(1)Nk SU(N)k × SU(k)N × U(1)Nk

Fusion Rules

s ⊕ simp = s

Example:

|s − simp| ≤ s ≤ min{s + simp, k − (s + simp)}

(for k − (s + simp) > 0)

SU(2)k

UV IR

decoupled spin at r = 0

L L

r

ψL(0−) = ψL(0+)

ψL(0−) = −ψL(0+) π/2 phase shift

UV IR

L L

r

ψ (r) = A cos kr + B sin kr

ψ (r) = A| sin kr| + B sin kr decoupled spin at r = 0 π/2 phase shift

CFT Approach to the Kondo Effect Take-Away Messages

Central role of the Kac-Moody Algebra PHASE SHIFT Kondo coupling:

S · J

Outline:

• The Kondo Effect
• The CFT Approach
• A Top-Down Holographic Model
• A Bottom-Up Holographic Model
• Summary and Outlook

GOAL

Find a holographic description

• f the

Kondo Effect

What classical action do we write

• n the gravity side of the correspondence?

How do we describe holographically...

1 2 3 The chiral fermions? The impurity? The Kondo coupling?

Top-down:

AdS solution to a string or supergravity theory

Bottom-up:

AdS solution of some ad hoc Lagrangian Holography

Open strings

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

3-3

5-5

7-7 3-7

7-3 3-5

7-5

5-7

and and and and and

5-3

Top-Down Model

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

3-3

5-5

7-7 3-7

7-3 3-5

7-5

5-7

and and and and and CFT with holographic dual

5-3

Top-Down Model

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

3-3

5-5

7-7 3-7

7-3 3-5

7-5

5-7

and and and and and Decouple

5-3

Top-Down Model

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

3-3

5-5

7-7 3-7

7-3

5-3

3-5

7-5

5-7

and and and and and (1+1)-dimensional chiral fermions

Top-Down Model

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

3-3

5-5

7-7 3-7

7-3 3-5

7-5

5-7

and and and and and the impurity

5-3

Top-Down Model

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

3-3

5-5

7-7 3-7

7-3 3-5

7-5

5-7

and and and and and Kondo interaction

5-3

Top-Down Model

Previous work

Mück 1012.1973 Kachru, Karch, Yaida 0909.2639, 1009.3268 Faraggi and Pando-Zayas 1101.5145 Jensen, Kachru, Karch, Polchinski, Silverstein 1105.1772 Karaiskos, Sfetsos, Tsatis 1106.1200 Harrison, Kachru, Torroba 1110.5325 Benincasa and Ramallo 1112.4669, 1204.6290 Faraggi, Mück, Pando-Zayas 1112.5028 Itsios, Sfetsos, Zoakos 1209.6617

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

3-3

5-5

7-7 3-7

7-3 3-5

7-5

5-7

and and and and and Absent in previous constructions

5-3

Top-Down Model

The D3-branes

N = 4

SU(Nc) YM

SUSY

1 2 3 4 5 6 7 8 9 Nc D3 X X X X

λ ≡ g2

Y MNc

3-3 strings

βλ = 0

(3 + 1)- dimensional

CFT!

The D3-branes

N = 4

SU(Nc) YM

SUSY

1 2 3 4 5 6 7 8 9 Nc D3 X X X X

λ ≡ g2

Y MNc

3-3 strings

(3 + 1)- dimensional

g2

Y M → 0

Nc → ∞

λ fixed

The D3-branes

N = 4

SU(Nc) YM

SUSY

1 2 3 4 5 6 7 8 9 Nc D3 X X X X

λ ≡ g2

Y MNc

3-3 strings

(3 + 1)- dimensional

g2

Y M → 0

Nc → ∞

λ → ∞

1 2 3 4 5 6 7 8 9 Nc D3 X X X X

N = 4 SYM

Type IIB Supergravity

Nc → ∞ =

The D3-branes

λ → ∞

AdS5 × S5

g2

Y MNc ∝ L4 AdS/α2

g2

Y M ∝ gs

LAdS ≡ 1

1 2 3 4 5 6 7 8 9 Nc D3 X X X X

The D3-branes

• S5 F5 ∝ Nc

F5 = dC4

N = 4 SYM

Type IIB Supergravity

Nc → ∞ =

λ → ∞

AdS5 × S5

Anti-de Sitter Space

r = ∞

r = 0

boundary

x

ds2 = dr2 r2 + r2 −dt2 + dx2 + dy2 + dz2

Poincaré horizon

Anti-de Sitter Space

x

ds2 = dr2 r2 + r2 −dt2 + dx2 + dy2 + dz2

UV IR

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

3-3

5-5

7-7 3-7

7-3 3-5

7-5

5-7

and and and and and Decouple

5-3

Top-Down Model

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

5-5

7-7

SYM

U(N5)

(5 + 1)-dim.

SYM

(7 + 1)-dim. U(N7)

g2

Dp ∝ gs α p−3

2

g2

Y M ∝ gs

g2

Y MNc ∝ 1/α2

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

g2

D5N5 ∝ gYM

N5 √Nc

g2

D7N7 ∝ N7

Nc

5-5

7-7

SYM

U(N5)

(5 + 1)-dim.

SYM

(7 + 1)-dim. U(N7)

g2

Dp ∝ gs α p−3

2

Probe Limit

Nc → ∞

g2

Y M → 0

g2

D5N5 ∝ gY M

N5 √Nc → 0

g2

D7N7 ∝ N7

Nc → 0

N7 , N5 fixed

N7/Nc → 0 and N5/Nc → 0

Probe Limit

becomes a global symmetry

U(N7) × U(N5)

Total symmetry:

(plus R-symmetry)

SU(Nc)

• × U(N7) × U(N5)
• gauged

global SYM theories on D7- and D5-branes decouple

3-3

5-5

7-7 3-7

7-3

5-3

3-5

7-5

5-7

and and and and and (1+1)-dimensional chiral fermions

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

Top-Down Model

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X

8 Neumann-Dirichlet (ND) intersection

The D7-branes

Harvey and Royston 0709.1482, 0804.2854 Buchbinder, Gomis, Passerini 0710.5170

Neumann Dirichlet

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X

The D7-branes

Harvey and Royston 0709.1482, 0804.2854 Buchbinder, Gomis, Passerini 0710.5170

1/4 SUSY

(1+1)-dimensional chiral fermions

N7

ψL

SUSY

8 Neumann-Dirichlet (ND) intersection

N = (0, 8)

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X

The D7-branes

(1+1)-dimensional chiral fermions

N7

L

ψL

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X

The D7-branes

S3-7 =

• dx+dx−ψ†

L (i∂− − A−) ψL

(1+1)-dimensional chiral fermions

N7

ψL

SU(Nc) × U(N7) × U(N5)

Nc

N 7

singlet

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X

The D7-branes

SU(Nc)N7 × SU(N7)Nc × U(1)NcN7

Kac-Moody algebra (1+1)-dimensional chiral fermions

N7

ψL

S3-7 =

• dx+dx−ψ†

L (i∂− − A−) ψL

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X

The D7-branes

Do not come from reduction from (3+1) dimensions Genuinely relativistic

Differences from Kondo

(1+1)-dimensional chiral fermions

N7

ψL

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X

The D7-branes

SU(Nc) is gauged!

• J = †

L

L

Differences from Kondo

(1+1)-dimensional chiral fermions

N7

ψL

Gauge Anomaly!

The D7-branes

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X

SU(Nc) is gauged!

Harvey and Royston 0709.1482, 0804.2854 Buchbinder, Gomis, Passerini 0710.5170

Gauge Anomaly!

The D7-branes

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X

SU(Nc) is gauged!

Probe Limit!

SU(Nc) SU(Nc) gY M gY M

In the probe limit, the gauge anomaly is suppressed...

N7

∝ g2

Y MN7

Nc

gD7 gD7

∝ g2

D7Nc

... but the global anomalies are not.

g2

D7 ∝ 1/Nc

U(N7) U(N7)

In the probe limit, the gauge anomaly is suppressed... ... but the global anomalies are not.

SU(Nc)N7 → SU(Nc)

SU(N7)Nc × U(1)NcN7 → SU(N7)Nc × U(1)NcN7

AdS3 × S5

Probe D7-branes

N = 4 SYM

Nc → ∞ =

λ → ∞

Probe ψL

=

Type IIB Supergravity

AdS5 × S5

AdS3 × S5

Probe D7-branes

N = 4 SYM

Nc → ∞ =

λ → ∞

Probe ψL

=

Type IIB Supergravity

AdS5 × S5

ds2 = dr2 r2 + r2 −dt2 + dx2 + dy2 + dz2 + ds2

S5

J = A

Current Gauge field U(N7)

U(N7)

AdS3 × S5

Probe D7-branes

N = 4 SYM

Nc → ∞ =

λ → ∞

Probe ψL

=

Type IIB Supergravity

AdS5 × S5

Kac-Moody Algebra Chern-Simons Gauge Field

Gukov, Martinec, Moore, Strominger hep-th/0403225 Kraus and Larsen hep-th/0607138

=

rank and level

• f

algebra

=

rank and level

• f

gauge field

J

=

A

Current Gauge field

J

=

A

Current Gauge field

Gauge field on D7-brane Decouples on field theory side... ...but not on the gravity side!

U(N7)Nc

AdS3 × S5

Probe D7-branes along

= −1 2TD7(2πα)2

• P[F5] ∧ tr
• A ∧ dA + 2

3A ∧ A ∧ A

• + . . .

SD7 = +1 2TD7(2πα)2

• P[C4] ∧ tr F ∧ F + . . .

= −Nc 4π

• AdS3

tr

• A ∧ dA + 2

3A ∧ A ∧ A

• + . . .

U(N7)Nc Chern-Simons gauge field

Answer #1

Chern-Simons Gauge Field in AdS3 The chiral fermions:

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

3-3

5-5

7-7 3-7

7-3 3-5

7-5

5-7

and and and and and the impurity

5-3

Top-Down Model

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N5 D5 X X X X X X

The D5-branes

Gomis and Passerini hep-th/0604007

(0+1)-dimensional fermions

N5

χ

1/4 SUSY

8 ND intersection

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N5 D5 X X X X X X

The D5-branes

Gomis and Passerini hep-th/0604007

(0+1)-dimensional fermions

N5

χ

8 ND intersection

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N5 D5 X X X X X X

The D5-branes

S3-5 =

• dt χ†(i∂t − At − Φ9)χ

SU(Nc) × U(N7) × U(N5)

Nc

singlet

N 5

(0+1)-dimensional fermions

N5

χ

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N5 D5 X X X X X X

The D5-branes

SU(Nc) is “spin”

“Abrikosov pseudo-fermions”

Abrikosov, Physics 2, p.5 (1965)

“slave fermions”

• S = †

Integrate out

N5 = 1

Gomis and Passerini hep-th/0604007

χ

. . .

R =

}

Det (D) = TrRPexp

• i
• dt (At + Φ9)
• Q = χ†χ

charge

U(N5) = U(1)

Probe D5-branes

N = 4 SYM

Nc → ∞ =

λ → ∞

Probe

=

Type IIB Supergravity

AdS5 × S5

AdS2 × S4

χ

ds2 = dr2 r2 + r2 −dt2 + dx2 + dy2 + dz2 + ds2

S5

Probe D5-branes

N = 4 SYM

Nc → ∞ =

λ → ∞

Probe

=

Type IIB Supergravity

AdS5 × S5

AdS2 × S4

χ

Probe D5-branes

N = 4 SYM

Nc → ∞ =

λ → ∞

Probe

=

Type IIB Supergravity

AdS5 × S5

AdS2 × S4

χ

Q

=

Electric flux

J

=

Current Gauge field a

U(N5)

U(N5)

Probe D5-brane along AdS2 × S4

Camino, Paredes, Ramallo hep-th/0104082

electric field

AdS2

√−gf tr

• ∂AdS2 = Q = χ†χ

Q

Dissolve strings into the D5-brane

frt = ∂rat − ∂tar

Answer #2

Yang-Mills Gauge Field in AdS2

electric flux

=

Rimp

The impurity:

3-3

5-5

7-7 3-7

7-3 3-5

7-5

5-7

and and and and and Kondo interaction

5-3

1 2 3 4 5 6 7 8 9 Nc D3 X X X X N7 D7 X X X X X X X X N5 D5 X X X X X X

Top-Down Model

1 2 3 4 5 6 7 8 9 N5 D5 X X X X X X N7 D7 X X X X X X X X

The Kondo Interaction

2 ND intersection Complex scalar!

SU(Nc) × U(N7) × U(N5)

singlet

O ≡ ψ†

Lχ

N5

N 7

TACHYON

The Kondo Interaction

1 2 3 4 5 6 7 8 9 N5 D5 X X X X X X N7 D7 X X X X X X X X

m2

tachyon = − 1

4α

D5 becomes magnetic flux in the D7 SUSY completely broken

The Kondo Interaction

SU(Nc) is “spin”

• S ·

J = |†

L|2 + O(1/Nc)

“double trace”

• J = †

L

L

• S = †
• S ·

J = † · †

L

L

• ij ·

kl = iljk − 1 Nc ijkl

AdS3 × S5

Probe D7-branes

N = 4 SYM

Nc → ∞ =

λ → ∞

Probe ψL

=

Type IIB Supergravity

AdS5 × S5

Probe D5-branes Probe

=

AdS2 × S4

χ

=

Bi-fundamental scalar

AdS2 × S4 O ≡ ψ†

Lχ

Answer #3

Bi-fundamental scalar in The Kondo interaction:

AdS2

r = ∞

r = 0

x

tr f 2

• AdS2
• AdS2

|DΦ|2+V (Φ†Φ) Nc

• AdS3

A ∧ F

DΦ = ∂Φ + iAΦ − iaΦ

Top-Down Model

r = ∞

r = 0

x

tr f 2

• AdS2
• AdS2

|DΦ|2+V (Φ†Φ) Nc

• AdS3

A ∧ F

Top-Down Model

N, k, Rimp

r = ∞

r = 0

x

tr f 2

• AdS2
• AdS2

|DΦ|2+V (Φ†Φ) Nc

• AdS3

A ∧ F

Top-Down Model

U(k)N

N, k, Rimp

r = ∞

r = 0

x

tr f 2

• AdS2
• AdS2

|DΦ|2+V (Φ†Φ) Nc

• AdS3

A ∧ F

What is V (Φ†Φ) ?

Top-Down Model

We don’t know.

r = ∞

r = 0

x

tr f 2

• AdS2
• AdS2

|DΦ|2+V (Φ†Φ) Nc

• AdS3

A ∧ F

Top-Down Model

What is V (Φ†Φ) ?

Calculation in

Gava, Narain, Samadi hep-th/9704006 Aganagic, Gopakumar, Minwalla, Strominger hep-th/0009142

Difficult to calculate in AdS5 × S5

R9,1

Top-Down Model

What is V (Φ†Φ) ?

Calculation in

Gava, Narain, Samadi hep-th/9704006 Aganagic, Gopakumar, Minwalla, Strominger hep-th/0009142

Switch to bottom-up model!

R9,1

Top-Down Model

Outline:

• The Kondo Effect
• The CFT Approach
• A Top-Down Holographic Model
• A Bottom-Up Holographic Model
• Summary and Outlook

r = ∞

r = 0

x

tr f 2

• AdS2
• AdS2

|DΦ|2+V (Φ†Φ)

Bottom-Up Model

Nc

• AdS3

A ∧ F

DΦ = ∂Φ + iAΦ − iaΦ

r = ∞

r = 0

x

tr f 2

• AdS2
• AdS2

|DΦ|2+V (Φ†Φ) Nc

• AdS3

A ∧ F

We pick V (Φ†Φ)

Bottom-Up Model

r = ∞

r = 0

x

tr f 2

• AdS2
• AdS2

|DΦ|2+V (Φ†Φ) Nc

• AdS3

A ∧ F

V (Φ†Φ) = m2Φ†Φ

Bottom-Up Model

S = SCS + SAdS2

SCS = − N 4π

• tr
• A ∧ dA + 2

3A ∧ A ∧ A

• SAdS2 = −
• d3x δ(x)√−g

1 4trf 2 + |DΦ|2 + V (Φ†Φ)

• DΦ = ∂Φ + iAΦ − iaΦ

V (Φ†Φ) = m2Φ†Φ

Bottom-Up Model

S = SCS + SAdS2

SCS = − N 4π

• tr
• A ∧ dA + 2

3A ∧ A ∧ A

• SAdS2 = −
• d3x δ(x)√−g

1 4trf 2 + |DΦ|2 + V (Φ†Φ)

• Bottom-Up Model

Kondo model specified by

N, Rimp, k

SCS = − N 4π

• tr
• A ∧ dA + 2

3A ∧ A ∧ A

• SAdS2 = −
• d3x δ(x)√−g

1 4trf 2 + |DΦ|2 + V (Φ†Φ)

• Bottom-Up Model

Kondo model specified by

N, Rimp, k

U(k)N

F = dA f = da

Single channel

. . .

Rimp =

U(1) gauge fields

Probe limit Chern-Simons

AdS2

U(1)Nc

Equations of Motion

µ, ν = r, t, x

m, n = r, t

Φ = eiψφ

εmµνFµν = −4π N δ(x)Jm ∂n √−g gnqgmpfqp

• = −Jm

∂mJm = 0

∂m √−g gmn∂nφ

• = √−g gmn(Am − am + ∂mψ)(An − an + ∂nψ)φ + 1

2 √−g ∂V ∂φ

Jm ≡ 2√−g gmn (An − an + ∂nψ) φ2

Ansatz:

Equations of Motion

at(r) φ(r)

Ax(r)

Static solution After gauge fixing, only non-zero fields:

frt = a

t(r)

Frx = A

x(r)

Jt(r) = −2√−g gttatφ2

Equations of Motion

∂r √−g grr ∂rφ

• − √−g gtt a2

t φ − √−g m2 φ = 0

εtrxFrx = −4π N δ(x)Jt(r)

∂r √−g grrgttfrt

• = −Jt(r)

Jt(r) = −2√−g gttatφ2

Boundary Conditions

c = ˜ gK ˜ c

Witten hep-th/0112258

φ (r) = c r−1/2 log r + ˜ c r−1/2 + . . .

We choose Breitenlohner-Freedman bound

m2 =

√−gf rt

• ∂AdS2 = Q

Our double-trace (Kondo) coupling:

φ (r) = 0

T > Tc

ψ†

Lχ = 0

ψ†

Lχ = 0

A holographic superconductor in Hawking temperature

T

=

AdS2

T < Tc

AdS-Schwarzschild black hole

gf tr

• ∂AdS2 = 0

gf tr

• ∂AdS2 = 0

φ(r) = 0

Hawking temperature

T

=

AdS-Schwarzschild black hole Superconductivity???

φ (r) = 0

T > Tc

ψ†

Lχ = 0

ψ†

Lχ = 0

T < Tc

gf tr

• ∂AdS2 = 0

gf tr

• ∂AdS2 = 0

φ(r) = 0

Hawking temperature

T

=

AdS-Schwarzschild black hole The large-N Kondo effect!

φ (r) = 0

T > Tc

ψ†

Lχ = 0

ψ†

Lχ = 0

T < Tc

gf tr

• ∂AdS2 = 0

gf tr

• ∂AdS2 = 0

φ(r) = 0

Solutions of the Kondo Problem Numerical RG (Wilson 1975) Fermi liquid description (Nozières 1975) Bethe Ansatz/Integrability (Andrei, Wiegmann, Tsvelick, Destri, ... 1980s) Conformal Field Theory (CFT) (Affleck and Ludwig 1990s) Large-N expansion

(Anderson, Read, Newns, Doniach, Coleman, ...1970-80s)

Quantum Monte Carlo (Hirsch, Fye, Gubernatis, Scalapino,... 1980s)

Large-N Approach to the Kondo Effect

Spin SU(N)

Rimp = anti-symm.

k = 1

N → ∞

fixed with NgK

SU(N) × U(1) × U(1)

• singlet

bi-fundamental

• S = †

O(τ) ≡ c†(0, τ)χ(τ)

Large-N Approach to the Kondo Effect

O = 0

O = 0

Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, 216403 (2003)

Spin SU(N)

Rimp = anti-symm.

k = 1

N → ∞

fixed with NgK

T > Tc

Tc TK

T < Tc

O = 0

Large-N Approach to the Kondo Effect

Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, 216403 (2003)

Spin SU(N)

Rimp = anti-symm.

k = 1

N → ∞

fixed with NgK

T > Tc

Represents the binding of an electron to the impurity

T < Tc

O = 0

Large-N Approach to the Kondo Effect

“(0+1)-DIMENSIONAL SUPERCONDUCTIVITY”

Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, 216403 (2003)

Spin SU(N)

Rimp = anti-symm.

k = 1

N → ∞

fixed with NgK

T > Tc

U(1) × U(1) → U(1)

T < Tc

O = 0

O = 0

Large-N Approach to the Kondo Effect

Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, 216403 (2003)

Spin SU(N)

Rimp = anti-symm.

k = 1

N → ∞

fixed with NgK

T > Tc

The phase transition is an ARTIFACT of the large-N limit!

The actual Kondo effect is a crossover

T < Tc

O = 0

O = 0

Hawking temperature

T

=

AdS-Schwarzschild black hole The large-N Kondo effect!

φ (r) = 0

T > Tc

ψ†

Lχ = 0

ψ†

Lχ = 0

T < Tc

gf tr

• ∂AdS2 = 0

gf tr

• ∂AdS2 = 0

φ(r) = 0

∂r √−g grr ∂rφ

• − √−g gtt a2

t φ − √−g m2 φ = 0

εtrxFrx = −4π N δ(x)Jt(r)

∂r √−g grrgttfrt

• = −Jt(r)

Jt(r) = −2√−g gttatφ2

The Phase Shift

εtrxFrx = −4π N δ(x)Jt(r)

∂r √−g grrgttfrt

• = −Jt(r)

Jt(r) = −2√−g gttatφ2

The Phase Shift

T > Tc

φ(r) = 0

Jt(r) = 0

√−gf rt = Q √−gf rt = Q

T > Tc

φ(r) = 0

Jt(r) = 0

The Phase Shift

UV IR

εtrxFrx = −4π N δ(x)Jt(r)

∂r √−g grrgttfrt

• = −Jt(r)

Jt(r) = −2√−g gttatφ2

The Phase Shift

T > Tc T < Tc

φ(r) = 0

Jt(r) = 0

Screening of the Impurity

√−gf rt

• horizon

√−gf rt

• ∂AdS2 = Q = 1/2

εtrxFrx = −4π N δ(x)Jt(r)

∂r √−g grrgttfrt

• = −Jt(r)

Jt(r) = −2√−g gttatφ2

The Phase Shift

T > Tc T < Tc

φ(r) = 0

Jt(r) = 0

εtrxFrx = −4π N δ(x)Jt(r)

Jt(r) = −2√−g gttatφ2

The Phase Shift

magnetic flux electric charge density

T > Tc T < Tc

φ(r) = 0

Jt(r) = 0

Integrate up to some r Compactify , integrate over

x x

The Phase Shift εtrxFrx = 2 ∂rAx(r) = −4π N δ(x)Jt(r)

Ax|r − Ax|∂AdS = −2π N δ(x)

• drJt(r)
• dx Ax|r −
• dx Ax|∂AdS = −2π

N

• drJt(r)

ei

R dxAx

T > Tc T < Tc

φ(r) = 0

Jt(r) = 0

The Phase Shift

Kraus and Larsen hep-th/0607138

• dxAx = 0
• dxAx = 0

UV IR

T > Tc T < Tc

φ(r) = 0

Jt(r) = 0

The Phase Shift

√−gf rt = Q √−gf rt < Q

ei

R dxAx

UV IR

The Resistivity

(What we know so far)

ρ

T

Tc

? ? ? ? ?

∝ T ∆irr.

The Resistivity

(What we know so far)

ρ

T

Tc

? ? ? ? ?

∝ T ∆irr.

No logarithm! Absent at large N Minimum?

Outline:

• The Kondo Effect
• The CFT Approach
• A Top-Down Holographic Model
• A Bottom-Up Holographic Model
• Summary and Outlook

Summary

What is the holographic dual of the Kondo effect? Holographic superconductor in coupled as a defect with a special boundary condition on the scalar

AdS2

AdS3

to a Chern-Simons gauge field in

Outlook

• Multi-channel?
• Other impurity representations?
• Spin as global symmetry?
• Entanglement entropy?
• Quantum Quenches?
• Multiple impurities? Kondo lattice?
• Suggestions welcome!

Thank You.