A Holographic Model of the Kondo Effect (Part 1)
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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
2 Credits Based on Johanna Erdmenger Max Planck Institute for Physics, Munich Carlos Hoyos Tel Aviv University Jackson Wu National Center for Theoretical Sciences, Taiwan
3 Outline: The Kondo Effect The CFT Approach Top-Down Holographic Model Bottom-Up Holographic Model Summary and Outlook
4 The Kondo Effect The screening of a magnetic moment by conduction electrons at low temperatures
5 µ gs
6 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
7 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
8 Running of the Coupling g K g 2 K + O(g3 K ) Asymptotic freedom! UV g K 0 The Kondo Temperature T K QCD
9 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?
10 Solutions of the Kondo Problem Numerical RG (Wilson 1975) Fermi liquid description (Nozières 1975) Bethe Ansatz/Integrability (Andrei, Wiegmann, Tsvelick, Destri, s) Large-N expansion (Anderson, Read, Newns, Doniach, Coleman, s) Quantum Monte Carlo (Hirsch, Fye, Gubernatis, Scalapino, s) Conformal Field Theory (CFT) (Affleck and Ludwig 1990s)
11 UV Fermi liquid + decoupled spin (a) T >> T K (b) T < T K size 1/T K IR Kondo screening cloud
12 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
13 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
14 UV Fermi liquid + decoupled spin Fermi liquid + NO magnetic moment IR + electrons EXCLUDED from impurity location
15 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), Cronenwett, et al., Science 281 (1998), no. 5376,
16 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)
17 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
18 Open Problems Entanglement Entropy Affleck, Laflorencie, Sørensen Quantum Quenches Latta et al 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
19 Open Problems Entanglement Entropy Affleck, Laflorencie, Sørensen Quantum Quenches Latta et al 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 T (K) Kondo 1.0 lattice 0.5 AF AF NFL LFL 0.1 T (K) x H (T) 2 0.0
20 Open Problems Entanglement Entropy Affleck, Laflorencie, Sørensen Quantum Quenches Let s try AdS/CFT! Latta et al 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 T (K) Kondo 1.0 lattice 0.5 AF AF NFL LFL 0.1 T (K) x H (T) 2 0.0
21 GOAL Find a holographic description of the Kondo Effect
22 GOAL Find a holographic description Single Impurity of the ONLY Kondo Effect
23 Solutions of the Kondo Problem Numerical RG (Wilson 1975) Fermi liquid description (Nozières 1975) Bethe Ansatz/Integrability (Andrei, Wiegmann, Tsvelick, Destri, s) Large-N expansion (Anderson, Read, Newns, Doniach, Coleman, s) Quantum Monte Carlo (Hirsch, Fye, Gubernatis, Scalapino, s) Conformal Field Theory (CFT) (Affleck and Ludwig 1990s)
24 Outline: The Kondo Effect The CFT Approach Top-Down Holographic Model Bottom-Up Holographic Model Summary and Outlook
25 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)
26 L r r =0 R L( r) R(+r) L L r r =0
27 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!
28 Spin SU(N) k 1 J = L L U(1) J = L L SU(N) J A = L ta L SU(k)
29 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
30 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
31 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
32 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
33 CFT Approach to the Kondo Effect Take-Away Messages Central role of the Kac-Moody Algebra Kondo coupling: S J
34 Outline: The Kondo Effect The CFT Approach Top-Down Holographic Model Bottom-Up Holographic Model Summary and Outlook
35 GOAL Find a holographic description of the Kondo Effect
36 STRATEGY Follow the CFT approach Reproduce the Symmetries Reproduce the Kondo coupling S J
37 What classical action do we write on the gravity side of the correspondence?
38 How do we describe holographically... 1 The chiral fermions? 2 The impurity? 3 The Kondo coupling?
39 Top-Down Model 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 and and 5-3 Open strings 7-5 and 5-7
40 Top-Down Model 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 and and and 5-7 CFT with holographic dual
41 Top-Down Model 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 and and 5-3 Decouple 7-5 and 5-7
42 Top-Down Model 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 and and 5-3 (1+1)-dimensional 7-5 and 5-7 chiral fermions
43 Top-Down Model 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 and 7-3 the impurity 3-5 and and 5-7
44 Top-Down Model 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 and and 5-7
45 The D3-branes 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
46 The D3-branes 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
47 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
48 Top-Down Model 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 and and 5-3 Decouple 7-5 and 5-7
49 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)
50 Top-Down Model 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 and and 5-3 (1+1)-dimensional 7-5 and 5-7 chiral fermions
51 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/ Harvey and Royston , Buchbinder, Gomis, Passerini (1+1)-dimensional chiral fermions L SU(N c ) U(N 7 ) U(N 5 ) N c N 7 singlet
52 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/ Harvey and Royston , Buchbinder, Gomis, Passerini (1+1)-dimensional chiral fermions L Kac-Moody algebra SU(N c ) N7 SU(N 7 ) Nc U(1) Nc N 7
53 The D7-branes 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
54 The D7-branes 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
55 The D7-branes 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 , Buchbinder, Gomis, Passerini
56 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
57 N =4 SYM N c = Type IIB Supergravity AdS 5 S 5 Probe L = Probe D7-branes AdS 3 S 5
58 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
59 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 )
60 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/ Kraus and Larsen hep-th/
61 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 A A A +... = N c tr A da AdS 3 A A A U(N 7 ) Nc Chern-Simons gauge field
62 Answer #1 The chiral fermions: Chern-Simons Gauge Field in AdS 3
63 Top-Down Model 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 and 7-3 the impurity 3-5 and and 5-7
64 The D5-branes N c D3 X X X X N 5 D5 X X X X X X Skenderis, Taylor hep-th/ Camino, Paredes, Ramallo hep-th/ Gomis and Passerini hep-th/ (0+1)-dimensional fermions SU(N c ) U(N 7 ) U(N 5 ) N c singlet N 5
65 The D5-branes 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)
66 Gomis and Passerini hep-th/ N 5 =1 Integrate out Det (D) =Tr R P exp i dt A t } R =... Q = U(N 5 )=U(1) charge
67 N =4 SYM N c = Type IIB Supergravity AdS 5 S 5 Probe = Probe D5-branes AdS 2 S 4
68 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
69 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
70 Probe D5-brane along AdS 2 S 4 Camino, Paredes, Ramallo hep-th/ Dissolve Q strings into the D5-brane AdS 2 electric field f rt gf tr AdS 2 = Q =
71 Answer #2 The impurity: Yang-Mills Gauge Field in AdS 2
72 Top-Down Model 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 and and 5-7
73 The Kondo Interaction 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
74 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
75 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
76 Answer #3 The Kondo interaction: Bi-fundamental scalar in AdS 2
77 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
78 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 ( )?
79 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.
80 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
81 Outline: The Kondo Effect The CFT Approach Top-Down Holographic Model Bottom-Up Holographic Model Summary and Outlook
82 Bottom-Up Model x r = N c A AdS 3 F tr f 2 AdS 2 D 2 +V ( ) AdS 2 r =0
83 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 ( )
84 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
85 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/ Berkooz, Sever, Shomer hep-th/ gf rt AdS 2 = Q
86 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
87 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???
88 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!
89 Solutions of the Kondo Problem Numerical RG (Wilson 1975) Fermi liquid description (Nozières 1975) Bethe Ansatz/Integrability (Andrei, Wiegmann, Tsvelick, Destri, s) Large-N expansion (Anderson, Read, Newns, Doniach, Coleman, s) Quantum Monte Carlo (Hirsch, Fye, Gubernatis, Scalapino, s) Conformal Field Theory (CFT) (Affleck and Ludwig 1990s)
90 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
91 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
92 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
93 Outline: The Kondo Effect The CFT Approach Top-Down Holographic Model Bottom-Up Holographic Model Summary and Outlook
94 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
95 Outlook! Entropy? Heat Capacity? Resistivity? Multi-channel? Other impurity representations? Entanglement entropy? Quantum Quenches? Multiple Impurities?
96 Thank You.
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