A Holographic Model of the Kondo Effect (Part 1)

A Holographic Model of the Kondo Effect (Part 1) Andy O Bannon Quantum Field Theory, String Theory, and Condensed Matter Physics Kolymbari, Greece September 1, 2014

Credits Based on 1310.3271 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 gs

The Kondo Hamiltonian 1 H K = k, (k) c k c k + g K S k k c k 2 c k c k, c k Conduction electrons =, Spin SU(2) c k e i c k Charge U(1) (k) = k2 2m F Dispersion relation

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

Running of the Coupling g K g 2 K + O(g3 K ) Asymptotic freedom! UV g K 0 The Kondo Temperature T K QCD

Running of the Coupling g K g 2 K + O(g3 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)

UV Fermi liquid + decoupled spin (a) T >> T K (b) T < T K size 1/T K IR Kondo screening cloud

UV Fermi liquid + decoupled spin (a) T >> T K (b) T < T K size 1/T K IR On average, the NET spin of the Kondo screening cloud equals that of a single electron

UV Fermi liquid + decoupled spin The Kondo screening cloud spin binds with the impurity spin Anti-symmetric singlet of SU(2) 1 2 ( i e i e ) IR Kondo singlet

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

Kondo Effect in Many Systems Alloys Cu, Ag, Au, Mg, Zn,... doped with Cr, Fe, Mo, Mn, Re, Os,... Quantum dots 8 200nm Goldhaber-Gordon, et al., Nature 391 (1998), 156-159. Cronenwett, et al., Science 281 (1998), no. 5376, 540-544.

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 =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 Affleck, Laflorencie, Sørensen 0906.1809 Quantum Quenches Latta et al. 1102.3982 Multiple Impurities Kondo: Form singlets with electrons S i S j Form singlets with each other Competition between these can produce a QUANTUM PHASE TRANSITION

Open Problems Entanglement Entropy Affleck, Laflorencie, Sørensen 0906.1809 Quantum Quenches Latta et al. 1102.3982 Multiple Impurities 2.5 a CeCu 6-x Au x 2.0 Heavy fermion compounds T N 1.5 YbRh 2 Si 2 H c 0.3 0.2 T (K) Kondo 1.0 lattice 0.5 AF AF NFL LFL 0.1 T (K) 0.0 0.0 0.5 x 1.0 0 1 H (T) 2 0.0

Open Problems Entanglement Entropy Affleck, Laflorencie, Sørensen 0906.1809 Quantum Quenches Let s try AdS/CFT! Latta et al. 1102.3982 Multiple Impurities 2.5 a CeCu 6-x Au x 2.0 Heavy fermion compounds T N 1.5 YbRh 2 Si 2 H c 0.3 0.2 T (K) Kondo 1.0 lattice 0.5 AF AF NFL LFL 0.1 T (K) 0.0 0.0 0.5 x 1.0 0 1 H (T) 2 0.0

GOAL Find a holographic description of the Kondo Effect

GOAL Find a holographic description Single Impurity of the ONLY Kondo Effect

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 g K 3 (x )S c (x ) 2 c(x ) restrict to s-wave restrict to momenta near k F c(x ) 1 r e ik F r L (r) e +ik F r R (r)

L r r =0 R L( r) R(+r) L L r r =0

CFT Approach to the Kondo Effect H K = v F 2 + dr L i r L + (r) g K S L L g K k 2 F 2 2 v F g K RELATIVISTIC chiral fermions v F = speed of light chiral CFT!

Spin SU(N) k 1 J = L L U(1) J = L L SU(N) J A = L ta L SU(k)

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

CFT Approach to the Kondo Effect H K = v F 2 + dr L i r L + (r) g K S L L Full symmetry: (1 + 1)d chiral conformal symmetry SU(N) k SU(k) N U(1) kn

CFT Approach to the Kondo Effect H K = v F 2 + dr L i r L + (r) g K S L L J = L L U(1) J = L L SU(N) J A = L ta L SU(k) Kondo coupling: S J marginal

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 highest weight R imp = R IR highest weight 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

STRATEGY Follow the CFT approach Reproduce the Symmetries Reproduce the Kondo coupling S J

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 3-3 and 5-5 and 7-7 3-7 and 7-3 3-5 and 5-3 Open strings 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 3-3 and 5-5 and 7-7 3-7 and 7-3 3-5 and 5-3 7-5 and 5-7 CFT with holographic dual

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 3-3 and 5-5 and 7-7 3-7 and 7-3 3-5 and 5-3 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 3-3 and 5-5 and 7-7 3-7 and 7-3 3-5 and 5-3 (1+1)-dimensional 7-5 and 5-7 chiral fermions

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 3-3 and 5-5 and 7-7 3-7 and 7-3 the impurity 3-5 and 5-3 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 3-3 and 5-5 and 7-7 Kondo interaction 3-7 and 7-3 3-5 and 5-3 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 N c = Type IIB Supergravity AdS 5 S 5 S 5 F 5 N c F 5 = dc 4

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 N c = Type IIB Supergravity AdS 5 S 5 Our Kondo model will have TWO coupling constants t Hooft and Kondo

Anti-de Sitter Space ds 2 = dr2 r 2 + r2 dt 2 + dx 2 + dy 2 + dz 2 x r = boundary r =0 Poincaré horizon

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 3-3 and 5-5 and 7-7 3-7 and 7-3 3-5 and 5-3 Decouple 7-5 and 5-7

Probe Limit N 7 /N c 0 and N 5 /N c 0 U(N 7 ) U(N 5 ) becomes a global symmetry 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 3-3 and 5-5 and 7-7 3-7 and 7-3 3-5 and 5-3 (1+1)-dimensional 7-5 and 5-7 chiral fermions

0 1 2 3 4 5 6 7 8 9 N c D3 X X X X The D7-branes 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 L SU(N c ) U(N 7 ) U(N 5 ) N c N 7 singlet

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 Differences from Kondo (1+1)-dimensional chiral fermions L 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 (1+1)-dimensional chiral fermions L Differences from Kondo SU(N c ) is gauged! J = L 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 SU(N c ) is gauged! Gauge Anomaly! Harvey and Royston 0709.1482, 0804.2854 Buchbinder, Gomis, Passerini 0710.5170

Probe Limit N 7 /N c 0 In the probe limit, the gauge anomaly is suppressed... SU(N c ) N7 SU(N c )... but the global anomalies are not. SU(N 7 ) Nc U(1) Nc N 7 SU(N 7 ) Nc U(1) Nc N 7

N =4 SYM N c = Type IIB Supergravity AdS 5 S 5 Probe L = Probe D7-branes AdS 3 S 5

N =4 SYM N c = Type IIB Supergravity AdS 5 S 5 Probe L = Probe D7-branes AdS 3 S 5 ds 2 = dr2 r 2 + r2 dt 2 + dx 2 + dy 2 + dz 2 + ds 2 S 5

N =4 SYM N c = Type IIB Supergravity AdS 5 S 5 Probe L = Probe D7-branes AdS 3 S 5 J = Gauge field A U(N 7 ) Current U(N 7 )

Current = J Gauge field A Kac-Moody Algebra = Chern-Simons Gauge Field rank and level = rank and level of of algebra gauge field Gukov, Martinec, Moore, Strominger hep-th/0403225 Kraus and Larsen hep-th/0607138

Probe D7-branes along AdS 3 S 5 S D7 =+ 1 2 T D7(2 ) 2 P [C 4 ] tr F F +... = 1 2 T D7(2 ) 2 P [F 5 ] tr A da + 2 3 A A A +... = N c tr A da + 2 4 AdS 3 A A A +... 3 U(N 7 ) Nc Chern-Simons gauge field

Answer #1 The chiral fermions: Chern-Simons Gauge Field in AdS 3

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 3-3 and 5-5 and 7-7 3-7 and 7-3 the impurity 3-5 and 5-3 7-5 and 5-7

The D5-branes 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 5 D5 X X X X X X Skenderis, Taylor hep-th/0204054 Camino, Paredes, Ramallo hep-th/0104082 Gomis and Passerini hep-th/0604007 (0+1)-dimensional fermions SU(N c ) U(N 7 ) U(N 5 ) N c singlet N 5

The D5-branes 0 1 2 3 4 5 6 7 8 9 N c D3 X X X X N 5 D5 X X X X X X SU(N c ) is spin S = slave fermions Abrikosov pseudo-fermions Abrikosov, Physics 2, p.5 (1965)

Gomis and Passerini hep-th/0604007 N 5 =1 Integrate out Det (D) =Tr R P exp i dt A t } R =... Q = U(N 5 )=U(1) charge

N =4 SYM N c = Type IIB Supergravity AdS 5 S 5 Probe = Probe D5-branes AdS 2 S 4

N =4 SYM N c = Type IIB Supergravity AdS 5 S 5 Probe = Probe D5-branes AdS 2 S 4 ds 2 = dr2 r 2 + r2 dt 2 + dx 2 + dy 2 + dz 2 + ds 2 S 5

N =4 SYM N c = Type IIB Supergravity AdS 5 S 5 Probe = Probe D5-branes AdS 2 S 4 J = U(N 5 ) Current U(N 5 ) Gauge field a Q = Electric flux

Probe D5-brane along AdS 2 S 4 Camino, Paredes, Ramallo hep-th/0104082 Dissolve Q strings into the D5-brane AdS 2 electric field f rt gf tr AdS 2 = Q =

Answer #2 The impurity: Yang-Mills Gauge Field in AdS 2

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 3-3 and 5-5 and 7-7 Kondo interaction 3-7 and 7-3 3-5 and 5-3 7-5 and 5-7

The Kondo Interaction 0 1 2 3 4 5 6 7 8 9 N 5 D5 X X X X X X N 7 D7 X X X X X X X X Complex scalar! SU(N c ) U(N 7 ) U(N 5 ) singlet N 7 N 5 O L

The Kondo Interaction SU(N c ) is spin J = L L S = S J = L L ij kl = il jk 1 N c ij kl S J = L 2 + O(1/N c ) double trace

N =4 SYM N c = Type IIB Supergravity AdS 5 S 5 Probe L = Probe D7-branes AdS 3 S 5 Probe D5-branes Probe = AdS2 S 4 O L = Bi-fundamental scalar AdS 2 S 4

Answer #3 The Kondo interaction: Bi-fundamental scalar in AdS 2

Top-Down Model x r = N c A AdS 3 F tr f 2 AdS 2 D 2 +V ( ) AdS 2 r =0 D = +ia ia

Top-Down Model x r = N c A AdS 3 F tr f 2 AdS 2 D 2 +V ( ) AdS 2 r =0 What is V ( )?

Top-Down Model x r = N c A AdS 3 F tr f 2 AdS 2 D 2 +V ( ) AdS 2 r =0 We don t know.

Top-Down Model x r = N c A AdS 3 F tr f 2 AdS 2 D 2 +V ( ) AdS 2 Switch to bottom-up model! r =0

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

Bottom-Up Model x r = N c A AdS 3 F tr f 2 AdS 2 D 2 +V ( ) AdS 2 r =0

Bottom-Up Model x r = N c A AdS 3 F tr f 2 AdS 2 D 2 +V ( ) AdS 2 r =0 We pick V ( )

Bottom-Up Model x r = N c A AdS 3 F tr f 2 AdS 2 D 2 +V ( ) AdS 2 r =0 V ( ) = m 2

Boundary Conditions We choose m 2 = Breitenlohner-Freedman bound (r) = cr 1/2 + cr 1/2 log r +... Our double-trace (Kondo) coupling: c = g K c Witten hep-th/0112258 Berkooz, Sever, Shomer hep-th/0112264 gf rt AdS 2 = Q

Phase Transition T>T c gf tr AdS 2 =0 (r) =0 L =0 T<T c gf tr AdS 2 =0 (r) =0 =0 L A holographic superconductor in AdS 2

Phase Transition T>T c gf tr AdS 2 =0 (r) =0 L =0 T<T c gf tr AdS 2 =0 (r) =0 =0 L Superconductivity???

Phase Transition T>T c gf tr AdS 2 =0 (r) =0 L =0 T<T c gf tr AdS 2 =0 (r) =0 =0 L The large-n Kondo effect!

Phase Transition T>T c gf tr AdS 2 =0 (r) =0 L =0 T<T c gf tr AdS 2 =0 (r) =0 =0 L T c T K

Phase Transition T>T c gf tr AdS 2 =0 (r) =0 L =0 T<T c gf tr AdS 2 =0 (r) =0 =0 L Represents the formation of the Kondo singlet

Phase Transition T>T c gf tr AdS 2 =0 (r) =0 L =0 T<T c gf tr AdS 2 =0 (r) =0 =0 L The phase transition is an ARTIFACT of the large-n limit! The actual Kondo effect is a crossover

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

Summary What is the holographic dual of the Kondo effect? Holographic superconductor in AdS 2 coupled as a defect to a Chern-Simons gauge field in AdS 3

Outlook! Entropy? Heat Capacity? Resistivity? Multi-channel? Other impurity representations? Entanglement entropy? Quantum Quenches? Multiple Impurities?

Thank You.