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

Generalizations Enhance the spin group SU (2) → SU ( N ) arXiv:1306.6326v1 [cond-mat.mes-hall] 26 Jun 2013 Observation of the SU(4) Kondo state in a double quantum dot A. J. Keller 1 , S. Amasha 1, † , I. Weymann 2 , C. P. Moca 3,4 , I. G. Rau 1, ‡ , J. A. Katine 5 , Hadas Shtrikman 6 , G. Zar´ and 3 , and D. Goldhaber-Gordon 1,* 1 Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA 2 Faculty of Physics, Adam Mickiewicz University, Pozna´ n, Poland 3 BME-MTA Exotic Quantum Phases “Lend¨ ulet” Group, Institute of Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary 4 Department of Physics, University of Oradea, 410087, Romania 5 HGST, San Jose, CA 95135, USA 6 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 96100, Israel † Present address: MIT Lincoln Laboratory, Lexington, MA 02420, USA ‡ Present address: IBM Research – Almaden, San Jose, CA 95120, USA * Corresponding author; goldhaber-gordon@stanford.edu

Generalizations Enhance the spin group SU (2) → SU ( N ) arXiv:1310.6563v1 [cond-mat.str-el] 24 Oct 2013 SU(12) Kondo E ff ect in Carbon Nanotube Quantum Dot Igor Kuzmenko 1 and Yshai Avishai 1 , 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)

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, R imp , k 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 Multiple Impurities Heavy fermion compounds NpPd 5 Al 2 CeCoIn 5 CeCu 6 CePd 2 Si 2 YbAl 3 YbRh 2 Si 2 UBe 13 UPt 3

Open Problems Multiple Impurities Heavy fermion compounds NpPd 5 Al 2 CeCoIn 5 CeCu 6 CePd 2 Si 2 YbAl 3 YbRh 2 Si 2 UBe 13 UPt 3

Open Problems Multiple Impurities Heavy fermion compounds 0.3 Example 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

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

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 • A Top-Down Holographic Model • A Bottom-Up Holographic Model • Summary and Outlook

CFT Approach to the Kondo Effect Affleck and Ludwig 1990s Reduction to one dimension Kondo interaction preserves spherical symmetry x ) 1 x ) � S · c † ( � g K � 3 ( � � c ( � x ) 2 � 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 i � r � L + � ( r ) ˜ L � � � L dr 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 = � † � SU ( N ) L � � � 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 Current 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 i � r � L + � ( r ) ˜ L � � � L dr 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 i � r � L + � ( r ) ˜ L � � � L dr 2 π −∞ J = ψ † L ψ L U (1) J = � † � SU ( N ) L � � � L 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 IR SU ( N ) k × SU ( k ) N × U (1) Nk

UV SU ( N ) k × SU ( k ) N × U (1) Nk | c, s, f � s ⊕ s imp = s � Fusion Rules | c, s � , f � IR SU ( N ) k × SU ( k ) N × U (1) Nk

UV SU ( N ) k × SU ( k ) N × U (1) Nk Fusion Rules Example: SU (2) k s ⊕ s imp = s � | s − s imp | ≤ s � ≤ min { s + s imp , k − ( s + s imp ) } (for k − ( s + s imp ) > 0) IR SU ( N ) k × SU ( k ) N × U (1) Nk

UV ψ L (0 − ) = ψ L (0 + ) decoupled spin at r = 0 L L r π / 2 phase shift IR ψ L (0 − ) = − ψ L (0 + )

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

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

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 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?

Holography Top-down: AdS solution to a string or supergravity theory Bottom-up: AdS solution of some ad hoc Lagrangian

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

Previous work Kachru, Karch, Yaida 0909.2639, 1009.3268 Mück 1012.1973 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

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 Absent in previous 3-5 5-3 and constructions 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 (3 + 1)- dimensional SU ( N c ) YM SUSY N = 4 λ ≡ g 2 Y M N c β λ = 0 CFT!

The D3-branes 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X 3-3 strings (3 + 1)- dimensional SU ( N c ) YM SUSY N = 4 λ ≡ g 2 Y M N c g 2 Y M → 0 N c → ∞ λ fixed

The D3-branes 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X 3-3 strings (3 + 1)- dimensional SU ( N c ) YM SUSY N = 4 λ ≡ g 2 Y M N c g 2 Y M → 0 N c → ∞ λ → ∞

The D3-branes 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N = 4 SYM Type IIB Supergravity N c → ∞ = AdS 5 × S 5 λ → ∞ g 2 Y M ∝ g s g 2 Y M N c ∝ L 4 AdS / α � 2 L AdS ≡ 1

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

Anti-de Sitter Space ds 2 = dr 2 − dt 2 + dx 2 + dy 2 + dz 2 � r 2 + r 2 � x r = ∞ boundary r = 0 Poincaré horizon

Anti-de Sitter Space ds 2 = dr 2 − dt 2 + dx 2 + dy 2 + dz 2 � r 2 + r 2 � x UV IR

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

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 7-7 5-5 SYM SYM (7 + 1)-dim. U ( N 7 ) U ( N 5 ) (5 + 1)-dim. Dp ∝ g s α � p − 3 g 2 2 g 2 Y M ∝ g s g 2 Y M N c ∝ 1 / α � 2

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 7-7 5-5 SYM SYM (7 + 1)-dim. U ( N 7 ) U ( N 5 ) (5 + 1)-dim. Dp ∝ g s α � p − 3 g 2 2 N 5 D 7 N 7 ∝ N 7 g 2 g 2 D 5 N 5 ∝ g YM √ N c N c

Probe Limit g 2 N c → ∞ Y M → 0 N 7 , N 5 fixed N 7 /N c → 0 and N 5 /N c → 0 D 7 N 7 ∝ N 7 g 2 → 0 N c N 5 g 2 D 5 N 5 ∝ g Y M → 0 √ N c

Probe Limit SYM theories on D7- and D5-branes decouple 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 Harvey and Royston 0709.1482, 0804.2854 Buchbinder, Gomis, Passerini 0710.5170 8 Neumann-Dirichlet (ND) intersection Neumann Dirichlet

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 Harvey and Royston 0709.1482, 0804.2854 Buchbinder, Gomis, Passerini 0710.5170 8 Neumann-Dirichlet (ND) intersection 1 / 4 SUSY ψ L (1+1) -dimensional chiral fermions N 7 SUSY N = (0 , 8)

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 ψ L (1+1) -dimensional chiral fermions N 7 L

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 ψ L (1+1) -dimensional chiral fermions N 7 SU ( N c ) × U ( N 7 ) × U ( N 5 ) N 7 N c singlet � dx + dx − ψ † S 3-7 = L ( i ∂ − − A − ) ψ L

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 ψ L (1+1) -dimensional chiral fermions N 7 Kac-Moody algebra SU ( N c ) N 7 × SU ( N 7 ) N c × U (1) NcN 7 � dx + dx − ψ † S 3-7 = L ( i ∂ − − A − ) ψ L

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 ψ L (1+1) -dimensional chiral fermions N 7 Differences from Kondo Do not come from reduction from (3+1) dimensions Genuinely relativistic

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 ψ L (1+1) -dimensional chiral fermions N 7 Differences from Kondo SU ( N c ) is gauged! J = � † � L � � � L