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

A Holographic Model of the Kondo Effect Andy O’Bannon 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 University Jackson Wu National Center for Theoretical Sciences, Taiwan

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

The Kondo Effect The screening of a magnetic moment by conduction electrons at low temperatures

µ ∝ g � � S

The Kondo Hamiltonian 1 � ( k ) c † c † � k σ c k σ + g K � � H K = 2 � � σσ � c k � σ � S · k σ k, σ k σ k � σ � c † Conduction electrons , c k σ k σ Spin SU (2) σ = ↑ , ↓ c k σ → e i α c k σ Charge U (1) ε ( k ) = k 2 Dispersion relation 2 m − ε F

The Kondo Hamiltonian 1 � ( k ) c † c † � k σ c k σ + g K � � H K = 2 � � σσ � c k � σ � S · k σ k, σ k σ k � σ � � Spin of magnetic moment S � � Pauli matrices g K Kondo coupling

Running of the Coupling β g K ∝ − g 2 K + O ( g 3 K ) Asymptotic freedom! UV g K → 0 The Kondo Temperature T K ∼ Λ QCD

Running of the Coupling β g K ∝ − g 2 K + O ( g 3 K ) At low energy, the coupling diverges! IR g K → ∞ The Kondo Problem What is the ground state?

Solutions of the Kondo Problem Numerical RG (Wilson 1975) Fermi liquid description (Nozières 1975) Bethe Ansatz/Integrability (Andrei, Wiegmann, Tsvelick, Destri, ... 1980s) Large-N expansion (Anderson, Read, Newns, Doniach, Coleman, ...1970-80s) Quantum Monte Carlo (Hirsch, Fye, Gubernatis, Scalapino,... 1980s) Conformal Field Theory (CFT) (Affleck and Ludwig 1990s)

Fermi liquid UV + decoupled spin An electron binds with the impurity Anti-symmetric singlet of SU (2) 1 � 2 ( | � i � e � � | � i � e � ) IR “Kondo singlet”

Fermi liquid UV + decoupled spin Fermi liquid + NO magnetic moment + electrons EXCLUDED IR from impurity location

The Kondo Effect in Many Systems alloys of Cu, Ag, Au, Mg, Zn, ... ...with Cr, Fe, Mo, Mn, Re, Os, ... impurities Heavy fermion compounds CeCu 6 CePd 2 Si 2 YbAl 3 YbRh 2 Si 2 UBe 13 UPt 3 8 Quantum dots 200nm

Generalizations Enhance the spin group SU (2) → SU ( N ) Representation of impurity spin s imp = 1 / 2 → R imp Multiple “channels” or “flavors” c → c α α = 1 , . . . , k U (1) × SU ( k )

Generalizations Kondo model specified by N, k, R imp Apply the techniques mentioned above... IR fixed point: NOT always a fermi liquid “Non-Fermi liquids”

Open Problems Entanglement Entropy Quantum Quenches Multiple Impurities Kondo: Form singlets with electrons S i · � � Form singlets with each other S j Competition between these can produce a QUANTUM PHASE TRANSITION

Open Problems Entanglement Entropy Quantum Quenches Multiple Impurities 0.3 Heavy fermion compounds YbRh 2 Si 2 H || c 0.2 YbRh 2 Si 2 ρ ∼ T T (K) NFL Kondo lattice 0.1 ρ ∼ T 2 LFL AF J. Custers et al., Nature 424, 524 (2003) 0.0 2 0 1 H (T)

Solutions of the Kondo Problem Numerical RG (Wilson 1975) Fermi liquid description (Nozières 1975) Bethe Ansatz/Integrability (Andrei, Wiegmann, Tsvelick, Destri, ... 1980s) Large-N expansion (Anderson, Read, Newns, Doniach, Coleman, ...1970-80s) Quantum Monte Carlo (Hirsch, Fye, Gubernatis, Scalapino,... 1980s) Conformal Field Theory (CFT) (Affleck and Ludwig 1990s)

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)

The Kondo Lattice “... remains one of the biggest unsolved problems Let’s try AdS/CFT! in condensed matter physics.” Alexei Tsvelik QFT in Condensed Matter Physics ( Cambridge Univ. Press, 2003)

GOAL Find a holographic description of the Kondo Effect

GOAL Find a holographic description Single Impurity ONLY of 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) Large-N expansion (Anderson, Read, Newns, Doniach, Coleman, ...1970-80s) Quantum Monte Carlo (Hirsch, Fye, Gubernatis, Scalapino,... 1980s) Conformal Field Theory (CFT) (Affleck and Ludwig 1990s)

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

CFT Approach to the Kondo Effect Affleck and Ludwig 1990s Reduction to one spatial dimension Kondo interaction preserves spherical symmetry x ) � � x ) � S · c † ( � g K � 3 ( � 2 c ( � x ) restrict to s-wave restrict to momenta near k F x ) ≈ 1 e − ik F r � L ( r ) − e + ik F r � R ( r ) � � c ( � r

L r R r = 0 ψ L ( − r ) ≡ ψ R (+ r ) L L r r = 0

CFT Approach to the Kondo Effect � + ∞ H K = v F � � � † g K � S · � † L � L i � r � L + � ( r )˜ dr T � L 2 π −∞ k 2 F g K ≡ × g K ˜ 2 π 2 v F RELATIVISTIC chiral fermions “speed of light” v F = CFT!

k ≥ 1 Spin SU ( N ) J = ψ † L ψ L U (1) J = � † � L � SU ( N ) T � L L t A ψ L J A = ψ † SU ( k )

z ≡ τ + ir � J A ( z ) = z − n − 1 J A n n ∈ Z n + m + N n [ J A n , J B m ] = if ABC J C 2 δ AB δ n, − m Kac-Moody Algebra SU ( k ) N N counts net number of chiral fermions

CFT Approach to the Kondo Effect � + ∞ H K = v F � � � † g K � S · � † L � L i � r � L + � ( r )˜ dr T � L 2 π −∞ Full symmetry: (1 + 1) d conformal symmetry SU ( N ) k × SU ( k ) N × U (1) kN

CFT Approach to the Kondo Effect � + ∞ H K = v F � � � † g K � S · � † L � L i � r � L + � ( r )˜ dr T � L 2 π −∞ J = ψ † L ψ L U (1) J = � † � L � T � L SU ( N ) L t A ψ L J A = ψ † SU ( k ) Kondo coupling: � S · � J

UV SU ( N ) k × SU ( k ) N × U (1) Nk Eigenstates are representations of the Kac-Moody algebra Determine how representations re-arrange between UV and IR R UV primaries ⊗ R imp = R IR primaries IR SU ( N ) k × SU ( k ) N × U (1) Nk

CFT Approach to the Kondo Effect Take-Away Messages Central role of the Kac-Moody Algebra Kondo coupling: � S · � J

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

GOAL Find a holographic description of the Kondo Effect

What classical action do we write on the gravity side of the correspondence?

How do we describe holographically... 1 The chiral fermions? 2 The impurity? 3 The Kondo coupling?

Top-Down Model 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 7 D7 X X X X X X X X N 5 D5 X X X X X X and and 3-3 5-5 7-7 and 7-3 3-7 Open strings 3-5 5-3 and 7-5 and 5-7

Top-Down Model 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 7 D7 X X X X X X X X N 5 D5 X X X X X X and and 3-3 5-5 7-7 and 7-3 3-7 3-5 5-3 and CFT with holographic dual 7-5 and 5-7

Top-Down Model 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 7 D7 X X X X X X X X N 5 D5 X X X X X X and and 3-3 5-5 7-7 and 7-3 3-7 3-5 5-3 and Decouple 7-5 and 5-7

Top-Down Model 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 7 D7 X X X X X X X X N 5 D5 X X X X X X and and 3-3 5-5 7-7 and 7-3 3-7 3-5 5-3 and (1+1) -dimensional chiral fermions 7-5 and 5-7

Top-Down Model 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 7 D7 X X X X X X X X N 5 D5 X X X X X X and and 3-3 5-5 7-7 and 7-3 3-7 the impurity 3-5 5-3 and 7-5 and 5-7

Top-Down Model 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 7 D7 X X X X X X X X N 5 D5 X X X X X X and and 3-3 5-5 7-7 Kondo interaction and 7-3 3-7 3-5 5-3 and 7-5 and 5-7

The D3-branes 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X 3-3 strings N = 4 SYM Type IIB Supergravity N c → ∞ = AdS 5 × S 5 λ → ∞ � F 5 = dC 4 S 5 F 5 ∝ N c

Top-Down Model 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 7 D7 X X X X X X X X N 5 D5 X X X X X X and and 3-3 5-5 7-7 and 7-3 3-7 3-5 5-3 and Decouple 7-5 and 5-7

Probe Limit N 7 /N c → 0 and N 5 /N c → 0 becomes a global symmetry U ( N 7 ) × U ( N 5 ) Total symmetry: SU ( N c ) � × U ( N 7 ) × U ( N 5 ) � �� � �� � gauged global (plus R-symmetry)

Top-Down Model 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 7 D7 X X X X X X X X N 5 D5 X X X X X X and and 3-3 5-5 7-7 and 7-3 3-7 3-5 5-3 and (1+1) -dimensional chiral fermions 7-5 and 5-7

The D7-branes 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 7 D7 X X X X X X X X Skenderis, Taylor hep-th/0204054 Harvey and Royston 0709.1482, 0804.2854 Buchbinder, Gomis, Passerini 0710.5170 (1+1) -dimensional chiral fermions ψ L SU ( N c ) × U ( N 7 ) × U ( N 5 ) N 7 N c singlet