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A Holographic Model of the Kondo Effect Andy OBannon Max Planck Institute for Physics Munich, Germany August 2, 2013 Credits Work in progress with: Johanna Erdmenger Max Planck Institute for Physics, Munich Carlos Hoyos Tel Aviv

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SLIDE 1

A Holographic Model

  • f the

Kondo Effect

Andy O’Bannon

Max Planck Institute for Physics Munich, Germany August 2, 2013

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SLIDE 2

Credits

Work in progress with:

Jackson Wu Johanna Erdmenger

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

Carlos Hoyos

Tel Aviv University

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SLIDE 3

Outline:

  • The Kondo Effect
  • The CFT Approach
  • Top-Down Holographic Model
  • Bottom-Up Holographic Model
  • Summary and Outlook
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SLIDE 4

The Kondo Effect

The screening

  • f a magnetic moment

by conduction electrons at low temperatures

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SLIDE 5
  • µ ∝ g

S

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SLIDE 6

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)

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SLIDE 7

gK

Kondo coupling Spin of magnetic moment

  • S
  • Pauli matrices

The Kondo Hamiltonian

HK =

  • k,σ

(k) c†

kσckσ + gK

S ·

  • kσkσ

c†

kσ

1 2 σσckσ

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SLIDE 8

Running of the Coupling

Asymptotic freedom!

βgK ∝ −g2

K + O(g3 K)

TK ∼ ΛQCD

UV

The Kondo Temperature

gK → 0

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SLIDE 9

The Kondo Problem

IR

Running of the Coupling

At low energy, the coupling diverges! What is the ground state?

βgK ∝ −g2

K + O(g3 K)

gK → ∞

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SLIDE 10

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)

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SLIDE 11

UV IR

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

1

  • 2 (|i e |i e)

Fermi liquid + decoupled spin

“Kondo singlet”

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SLIDE 12

Fermi liquid + decoupled spin

UV IR

Fermi liquid

+ electrons EXCLUDED from impurity location

+ NO magnetic moment

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SLIDE 13

Heavy fermion compounds

Quantum dots

8

...with Cr, Fe, Mo, Mn, Re, Os, ... impurities

alloys of Cu, Ag, Au, Mg, Zn, ...

UBe13

UPt3

CeCu6

YbAl3

CePd2Si2

YbRh2Si2

200nm

The Kondo Effect in Many Systems

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SLIDE 14

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)

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SLIDE 15

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

Generalizations

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

N, k, Rimp

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SLIDE 16

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

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SLIDE 17

Open Problems

Heavy fermion compounds

  • 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

Entanglement Entropy Quantum Quenches Multiple Impurities

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SLIDE 18

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)

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SLIDE 19

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)

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SLIDE 20

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)

Let’s try AdS/CFT!

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SLIDE 21

GOAL

Find a holographic description

  • f the

Kondo Effect

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SLIDE 22

GOAL

Find a holographic description

  • f the

Kondo Effect

Single Impurity ONLY

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SLIDE 23

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)

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SLIDE 24

Outline:

  • The Kondo Effect
  • The CFT Approach
  • Top-Down Holographic Model
  • Bottom-Up Holographic Model
  • Summary and Outlook
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SLIDE 25

Kondo interaction preserves spherical symmetry

Reduction to one spatial 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)
  • gK3(

x) S · c†( x)

  • 2 c(

x)

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SLIDE 26

L R

r

L L

r

ψL(−r) ≡ ψR(+r)

r = 0 r = 0

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SLIDE 27

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

TL

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SLIDE 28

k ≥ 1

Spin SU(N)

U(1)

SU(k) SU(N)

J = ψ†

LψL

  • J = †

L

T L

JA = ψ†

L tA ψL

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SLIDE 29

Kac-Moody 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

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SLIDE 30

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

TL

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SLIDE 31

CFT Approach to the Kondo Effect

J = ψ†

LψL

  • J = †

L

T L

JA = ψ†

L tA ψL

Kondo coupling:

S · J

U(1)

SU(k) SU(N)

HK = vF 2π +∞

−∞

dr

  • †

LirL + (r)˜

gK S · †

L

TL

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SLIDE 32

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

Determine how representations re-arrange between UV and IR

RUV

primaries ⊗ Rimp = RIR primaries

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SLIDE 33

CFT Approach to the Kondo Effect Take-Away Messages

Central role of the Kac-Moody Algebra Kondo coupling:

S · J

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SLIDE 34

Outline:

  • The Kondo Effect
  • The CFT Approach
  • Top-Down Holographic Model
  • Bottom-Up Holographic Model
  • Summary and Outlook
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SLIDE 35

GOAL

Find a holographic description

  • f the

Kondo Effect

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SLIDE 36

What classical action do we write

  • n the gravity side of the correspondence?
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SLIDE 37

How do we describe holographically...

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

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SLIDE 38

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

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SLIDE 39

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

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SLIDE 40

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

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SLIDE 41

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

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SLIDE 42

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

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SLIDE 43

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

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SLIDE 44

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

3-3 strings

  • S5 F5 ∝ Nc

F5 = dC4

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SLIDE 45

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

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SLIDE 46

Probe Limit

becomes a global symmetry

U(N7) × U(N5)

Total symmetry:

(plus R-symmetry)

SU(Nc)

  • × U(N7) × U(N5)
  • gauged

global

N7/Nc → 0 and N5/Nc → 0

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SLIDE 47

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

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SLIDE 48

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 ψL

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

SU(Nc) × U(N7) × U(N5)

Nc

N 7

singlet

Skenderis, Taylor hep-th/0204054

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SLIDE 49

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

Kac-Moody algebra

Harvey and Royston 0709.1482, 0804.2854 Buchbinder, Gomis, Passerini 0710.5170 Skenderis, Taylor hep-th/0204054

SU(Nc)N7 × SU(N7)Nc × U(1)NcN7

(1+1)-dimensional chiral fermions ψL

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SLIDE 50

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 ψL

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SLIDE 51

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

T L

Differences from Kondo

(1+1)-dimensional chiral fermions ψL

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SLIDE 52

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!

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SLIDE 53

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

SU(Nc)N7 → SU(Nc)

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

... but the global anomalies are not.

Probe Limit

N7/Nc → 0

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SLIDE 54

AdS3 × S5

Probe D7-branes

N = 4 SYM

Nc → ∞ =

λ → ∞

Probe ψL

=

Type IIB Supergravity

AdS5 × S5

J = A

Current Gauge field U(N7)

U(N7)

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SLIDE 55

Kac-Moody Algebra Chern-Simons Gauge Field

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

=

rank and level

  • f

Kac-Moody algebra = rank and level

  • f

gauge field

J = A

Current Gauge field U(N7)

U(N7)

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SLIDE 56

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

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SLIDE 57

Answer #1

Chern-Simons Gauge Field in AdS3 The chiral fermions:

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SLIDE 58

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

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SLIDE 59

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 Camino, Paredes, Ramallo hep-th/0104082

(0+1)-dimensional fermions χ

SU(Nc) × U(N7) × U(N5)

Nc

singlet

N 5

Skenderis, Taylor hep-th/0204054

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SLIDE 60

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”

  • S = †

T

“Abrikosov pseudo-fermions”

Abrikosov, Physics 2, p.5 (1965)

“slave fermions”

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SLIDE 61

Integrate out

N5 = 1

Gomis and Passerini hep-th/0604007

χ

. . .

R =

}

charge

U(N5) = U(1)

Q = χ†χ

Det (D) = TrRP exp

  • i
  • dt At
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SLIDE 62

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)

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SLIDE 63

Probe D5-brane along AdS2 × S4

Camino, Paredes, Ramallo hep-th/0104082

electric field

AdS2

frt

√−gf tr

  • ∂AdS2 = Q = χ†χ

Q

Dissolve strings into the D5-brane

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SLIDE 64

Answer #2

Yang-Mills Gauge Field in AdS2 The impurity:

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SLIDE 65

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

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SLIDE 66

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

Complex scalar!

SU(Nc) × U(N7) × U(N5)

singlet

O ≡ ψ†

Lχ

N5

N 7

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SLIDE 67

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

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SLIDE 68

The Kondo Interaction

SU(Nc) is “spin”

  • J = †

L

T L

  • S = †

T

  • Tij ·

Tkl = iljk − 1 Nc ijkl

  • S ·

J = |†

L|2 + O(1/Nc)

  • S ·

J = † T · †

L

TL

“double trace”

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SLIDE 69

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

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SLIDE 70

Answer #3

Bi-fundamental scalar in The Kondo interaction:

AdS2

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SLIDE 71

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

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SLIDE 72

r = ∞

r = 0

x

tr f 2

  • AdS2
  • AdS2

|DΦ|2+V (Φ†Φ) Nc

  • AdS3

A ∧ F

Top-Down Model

N, k, Rimp

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SLIDE 73

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

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SLIDE 74

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

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SLIDE 75

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

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SLIDE 76

r = ∞

r = 0

x

tr f 2

  • AdS2
  • AdS2

|DΦ|2+V (Φ†Φ) Nc

  • AdS3

A ∧ F

Switch to bottom-up model!

Top-Down Model

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SLIDE 77

Outline:

  • The Kondo Effect
  • The CFT Approach
  • Top-Down Holographic Model
  • Bottom-Up Holographic Model
  • Summary and Outlook
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SLIDE 78

r = ∞

r = 0

x

tr f 2

  • AdS2
  • AdS2

|DΦ|2+V (Φ†Φ) Nc

  • AdS3

A ∧ F

Bottom-Up Model

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SLIDE 79

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

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SLIDE 80

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

slide-81
SLIDE 81

r = ∞

r = 0

x

tr f 2

  • AdS2
  • AdS2

|DΦ|2+V (Φ†Φ) Nc

  • AdS3

A ∧ F

Bottom-Up Model

We choose Breitenlohner-Freedman bound

m2 =

slide-82
SLIDE 82

φ (r) = 0

T > Tc

ψ†

Lχ = 0

ψ†

Lχ = 0

A holographic superconductor in AdS2

T < Tc

gf tr

  • ∂AdS2 = 0

gf tr

  • ∂AdS2 = 0

φ(r) = 0

Phase Transition

slide-83
SLIDE 83

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

Phase Transition

slide-84
SLIDE 84

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

Phase Transition

slide-85
SLIDE 85

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)

slide-86
SLIDE 86

φ (r) = 0

T > Tc

ψ†

Lχ = 0

ψ†

Lχ = 0

T < Tc

gf tr

  • ∂AdS2 = 0

gf tr

  • ∂AdS2 = 0

φ(r) = 0

Phase Transition

Tc ∝ TK

slide-87
SLIDE 87

φ (r) = 0

T > Tc

ψ†

Lχ = 0

ψ†

Lχ = 0

T < Tc

gf tr

  • ∂AdS2 = 0

gf tr

  • ∂AdS2 = 0

φ(r) = 0

Phase Transition Represents the binding of an electron to the impurity

slide-88
SLIDE 88

φ (r) = 0

T > Tc

ψ†

Lχ = 0

ψ†

Lχ = 0

T < Tc

gf tr

  • ∂AdS2 = 0

gf tr

  • ∂AdS2 = 0

φ(r) = 0

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

The actual Kondo effect is a crossover

slide-89
SLIDE 89
  • Entropy?
  • Heat capacity?
  • Susceptibility?
  • Resistivity?
slide-90
SLIDE 90

Outline:

  • The Kondo Effect
  • The CFT Approach
  • Top-Down Holographic Model
  • Bottom-Up Holographic Model
  • Summary and Outlook
slide-91
SLIDE 91

Summary

What is the holographic dual of the Kondo effect? Holographic superconductor in coupled as a defect

AdS2

AdS3

to a Chern-Simons gauge field in

slide-92
SLIDE 92

Outlook

  • Multi-channel?
  • Other impurity representations?
  • Entanglement entropy?
  • Quantum Quenches?
  • Kondo Lattice?