A Holographic Model of the Kondo Effect

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1 A Holographic Model of the Kondo Effect Andy O Bannon University of Oxford October 29, 2013

2 71 High Street, Oxford

5 OUT OF BUSINESS

6 Credits Based on Johanna Erdmenger Max Planck Institute for Physics, Munich Carlos Hoyos Tel Aviv University Jackson Wu National Center for Theoretical Sciences, Taiwan

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

8 July 10, 1908 Leiden, the Netherlands Heike Kamerlingh Onnes liquifies helium T 4.2 K (1 atm)

9 Shortly Thereafter Leiden, the Netherlands Begins studying low-temperature properties of metals T 1 to 10 K

10 April 8, 1911 Heike Kamerlingh Onnes discovers superconductivity R

11 1913 Onnes receives the Nobel Prize in Physics for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium

12 Ag D 200 K Smith and Fickett, J. Res. NIST, 100, 119 (1995)

13 Ag D 200 K Resistivity measures electron scattering cross section

14 Debye Temperature Quantized vibrational modes of a solid = Phonons Minimum wavelength: 2 x (lattice spacing) Maximal Frequency D lowest temperature at which maximal-energy phonon excited

15 Ag D 200 K T D electron-phonon scattering (T ) T

16 Ag D 200 K T D electron-phonon scattering (T )= 0 + at 2 + bt 5

17 Ag D 200 K T D electron-electron scattering (T )= 0 + at 2 + bt 5

18 Ag D 200 K T D electron-impurity scattering (T )= 0 + at 2 + bt 5

19 Ag D 200 K T D increasing concentration of impurities

20 resistance The Kondo Effect a ~ 10 K temperature

21 9 E o Q_ ' *o (L._gq, Ce) B 6 %.* Oat %Ce ~****** ~ at%ce at % Ce vvv 1.80 at %Ce k at % Ca : ~ " ,. -,q.,,,,. Jj ~,,p ~Z.~ I 1! I I I I I I 1 I I T~ K Samwer and Winzer, Z. Phys B, 25, 269, 1976

22 MAGNETIC Impurities Fermi liquid Free magnetic moment Pauli: T 0 Curie: T 1 / / / / 80 "T 60 (.~ t / o E ) / // 20 /,/ #,r "/ I -2 0 / (La, Ce) B 6 t ~' ~xod x at % Ce * 0.072,8 o0.1.3 xox o exo o~ x xo o o o xoo t21 xo ~1 ~oo ~.~o ~' 420~ xx ~ x <c~ ~ ~'" -[9 x ~xx /"/"* // _ 18 E I ~ I T/K 0.4 I I I I 2 4 G 8 T/K Felsch, Z. Phys B, 29, 211, 1978 x

24 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

25 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 impurity Pauli matrices g K Kondo coupling g K < 0 Ferromagnetic g K > 0 Anti-Ferromagnetic

26 (T )= 0 + at 2 + bt 5 + cg 2 K cg 3 K ln (T/ F ) c, c concentration of impurities F = UV cutoff g K < 0 as T decreases Ferromagnetic (T ) DECREASES

27 (T )= 0 + at 2 + bt 5 + cg 2 K cg 3 K ln (T/ F ) c, c concentration of impurities F = UV cutoff g K < 0 as T decreases Ferromagnetic (T ) DECREASES

28 (T )= 0 + at 2 + bt 5 + cg 2 K cg 3 K ln (T/ F ) c, c concentration of impurities F = UV cutoff g K > 0 as T decreases Anti-Ferromagnetic (T ) INCREASES

29 (T )= 0 + at 2 + bt 5 + cg 2 K cg 3 K ln (T/ F )

30 (T )= 0 + at 2 + bt 5 + cg 2 K cg 3 K ln (T/ F ) Breakdown of Perturbation Theory O(gK) 3 term is same order as O(gK) 2 term when T K F e c c 1 g K Kondo temperature

31 (T )= 0 + at 2 + bt 5 + cg 2 K cg 3 K ln (T/ F ) Cross section for electron scattering off a MAGNETIC impurity INCREASES as energy DECREASES g K g 2 K + O(g3 K ) Asymptotic freedom! T K QCD

32 The Kondo Problem The coupling diverges at low energy! What is the ground state? We know the answer!

33 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)

34 UV Fermi liquid + decoupled spin The electrons SCREEN the impurity s spin A MANY-BODY effect Produces a MANY-BODY RESONANCE IR Kondo resonance

35 UV Fermi liquid + decoupled spin Intuitive SINGLE-BODY Description A SINGLE electron binds with the impurity Anti-symmetric singlet of SU(2) 1 2 ( i e i e ) IR Kondo singlet

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

37 UV Fermi liquid + decoupled spin Fermi liquid + IR NON-MAGNETIC impurity

38 9 E o Q_ ' *o (L._gq, Ce) B 6 %.* Oat %Ce ~****** ~ at%ce at % Ce vvv 1.80 at %Ce k at % Ca : ~ " ,. -,q.,,,,. Jj ~,,p ~Z.~ I 1! I I I I I I 1 I I T~ K Samwer and Winzer, Z. Phys B, 25, 269, 1976

39 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,

40 Generalizations Enhance the spin group SU(2) SU(N) arxiv: v1 [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. Zaránd 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, Poznań, Poland 3 BME-MTA Exotic Quantum Phases Lendület Group, Institute of Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary 4 Department of Physics, University of Oradea, , 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

41 Generalizations Enhance the spin group SU(2) SU(N) arxiv: v1 [cond-mat.str-el] 24 Oct 2013 SU(12) Kondo Effect 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)

42 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)

43 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

44 Open Problems Entanglement Entropy Quantum Quenches 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

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

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

47 Open Problems Multiple Impurities Heavy fermion compounds CeCu2.5 a 6-x Au x CeCu 6-x Au x 2.0 Example T N T N YbRh 2 Si 2 H c YbRh 2 Si 2 H c YbRh 2 Si 2 T 0.2 T (K) Kondo lattice AF AF J. Custers 0.0 et al., Nature 424, 524 (2003) x 1.0 AF AF NFL NFL 0 1 LFL T 2 H (T) T (K) LFL c d

48 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)

49 The Kondo Lattice

50 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)

51 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)

52 GOAL Find a holographic description of the Kondo Effect

53 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)

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

55 CFT Approach to the Kondo Effect Affleck and Ludwig 1990s Reduction to one dimension Kondo interaction preserves spherical symmetry g K 3 (x ) S c (x ) 1 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)

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

57 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 CFT!

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

59 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 Current Algebra N counts net number of chiral fermions

60 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 conformal symmetry SU(N) k SU(k) N U(1) kn

61 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

62 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

63 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

64 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

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

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

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

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

69 GOAL Find a holographic description of the Kondo Effect

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

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

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

73 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

74 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

75 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

76 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

77 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

78 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

79 Previous work Kachru, Karch, Yaida , Mück Faraggi and Pando-Zayas Jensen, Kachru, Karch, Polchinski, Silverstein Karaiskos, Sfetsos, Tsatis Harrison, Kachru, Torroba Benincasa and Ramallo , Faraggi, Mück, Pando-Zayas Itsios, Sfetsos, Zoakos

80 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 Absent in previous constructions 7-5 and 5-7

81 The D3-branes N c D3 X X X X 3-3 strings (3 + 1)- dimensional N =4 SUSY SU(N c ) YM g 2 YMN c =0 CFT!

82 The D3-branes N c D3 X X X X 3-3 strings (3 + 1)- dimensional N =4 SUSY SU(N c ) YM g 2 YMN c N c g 2 YM 0 fixed

83 The D3-branes N c D3 X X X X 3-3 strings (3 + 1)- dimensional N =4 SUSY SU(N c ) YM g 2 YMN c N c g 2 YM 0

84 The D3-branes N c D3 X X X X N =4 SYM N c = Type IIB Supergravity AdS 5 S 5 g 2 YM g s g 2 YMN c L 4 AdS/ 2 L AdS 1

85 The D3-branes N c D3 X X X X N =4 SYM Type IIB Supergravity N c = AdS5 S 5 S 5 F 5 N c F 5 = dc 4

86 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

87 Anti-de Sitter Space ds 2 = dr2 r 2 + r2 dt 2 + dx 2 + dy 2 + dz 2 x UV IR

88 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

89 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 + 1)-dim. U(N 7 ) SYM (5 + 1)-dim. U(N 5 ) SYM gdp 2 p 3 g s 2 g 2 YM g s g 2 YMN c 1/ 2

90 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 + 1)-dim. U(N 7 ) SYM (5 + 1)-dim. U(N 5 ) SYM gdp 2 p 3 g s 2 g 2 D7N 7 N 7 N c g 2 D5N 5 g YM N 5 N c

91 Probe Limit N c g 2 YM 0 N 7, N 5 fixed N 7 /N c 0 and N 5 /N c 0 g 2 D7N 7 N 7 N c 0 g 2 D5N 5 g YM N 5 N c 0

92 Probe Limit SYM theories on D7- and D5-branes decouple 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)

93 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

94 The D7-branes N c D3 X X X X N 7 D7 X X X X X X X X Harvey and Royston , Buchbinder, Gomis, Passerini Neumann-Dirichlet (ND) intersection Neumann Dirichlet

95 The D7-branes N c D3 X X X X N 7 D7 X X X X X X X X Harvey and Royston , Buchbinder, Gomis, Passerini Neumann-Dirichlet (ND) intersection 1/4 SUSY N 7 (1+1)-dimensional chiral fermions L N =(0, 8) SUSY

96 The D7-branes N c D3 X X X X N 7 D7 X X X X X X X X N 7 (1+1)-dimensional chiral fermions L L

97 The D7-branes N c D3 X X X X N 7 D7 X X X X X X X X N 7 (1+1)-dimensional chiral fermions L SU(N c ) U(N 7 ) U(N 5 ) N c N 7 singlet S 3-7 = dx + dx L (i A ) L

98 The D7-branes N c D3 X X X X N 7 D7 X X X X X X X X N 7 (1+1)-dimensional chiral fermions L Kac-Moody algebra SU(N c ) N7 SU(N 7 ) Nc U(1) NcN7 S 3-7 = dx + dx L (i A ) L

99 The D7-branes N c D3 X X X X N 7 D7 X X X X X X X X N 7 (1+1)-dimensional chiral fermions L Differences from Kondo Do not come from reduction from (3+1) dimensions Genuinely relativistic

100 The D7-branes N c D3 X X X X N 7 D7 X X X X X X X X N 7 (1+1)-dimensional chiral fermions L Differences from Kondo SU(N c ) is gauged! J = L L

101 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

102 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! Probe Limit!

103 In the probe limit, the gauge anomaly is suppressed... N 7 SU(N c ) SU(N c ) g YM g YM g 2 YMN 7... but the global anomalies are not. N c U(N 7 ) U(N 7 ) g D7 g D7 g 2 D7N c g 2 D7 1/N c

104 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

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

106 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

107 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 )

108 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/

109 Current = J Gauge field A U(N 7 ) Nc Gauge field on D7-brane Decouples on field theory side......but not on the gravity side!

110 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

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

112 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

113 The D5-branes N c D3 X X X X N 5 D5 X X X X X X Gomis and Passerini hep-th/ ND intersection 1/4 SUSY N 5 (0+1)-dimensional fermions

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 Gomis

114 The D5-branes N c D3 X X X X N 5 D5 X X X X X X Gomis and Passerini hep-th/ ND intersection N 5 (0+1)-dimensional fermions

115 The D5-branes N c D3 X X X X N 5 D5 X X X X X X N 5 (0+1)-dimensional fermions SU(N c ) U(N 7 ) U(N 5 ) N c singlet N 5 S 3-5 = dt (i t A t 9 )

116 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)

117 Gomis and Passerini hep-th/ N 5 =1 Integrate out Det (D) =Tr R P exp i dt (A t + 9 ) } R =. Q = U(N 5 )=U(1) charge

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

119 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

120 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

121 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 = r a t t a r gf tr AdS 2 = Q =

122 Answer #2 The impurity: Yang-Mills Gauge Field in AdS 2 R imp = electric flux

123 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

124 The Kondo Interaction N 5 D5 X X X X X X N 7 D7 X X X X X X X X 2 ND intersection Complex scalar! SU(N c ) U(N 7 ) U(N 5 ) singlet N 7 N 5 O L

125 The Kondo Interaction N 5 D5 X X X X X X N 7 D7 X X X X X X X X SUSY completely broken TACHYON m 2 tachyon = 4 1 D5 becomes magnetic flux in the D7

126 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

127 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

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

129 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

130 Top-Down Model x r = N c A AdS 3 F tr f 2 AdS 2 D 2 +V ( ) AdS 2 r =0 N, k, R imp

131 Top-Down Model x r = U(k) N N c A AdS 3 F tr f 2 AdS 2 D 2 +V ( ) AdS 2 r =0 N, k, R imp

132 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 ( )?

133 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.

134 Top-Down Model What is V ( )? Calculation in R 9,1 Gava, Narain, Samadi hep-th/ Aganagic, Gopakumar, Minwalla, Strominger hep-th/ Difficult to calculate in AdS 5 S 5

135 Top-Down Model What is V ( )? Calculation in R 9,1 Gava, Narain, Samadi hep-th/ Aganagic, Gopakumar, Minwalla, Strominger hep-th/ Switch to bottom-up model!

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

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

138 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 ( )

139 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

140 Bottom-Up Model S = S CS + S AdS2 S CS = N 4 tr A da A A A S AdS2 = d 3 x (x) g 1 4 trf 2 + D 2 + V ( ) D = +ia ia V ( ) = m 2

141 Bottom-Up Model S = S CS + S AdS2 S CS = N 4 tr A da A A A S AdS2 = d 3 x (x) g 1 4 trf 2 + D 2 + V ( ) Kondo model specified by N, R imp,k

142 Bottom-Up Model U(k) N S CS = N 4 tr A da A A A S AdS2 = d 3 x (x) g 1 4 trf 2 + D 2 + V ( ) Kondo model specified by N, R imp,k

143 Probe limit U(1) gauge fields Chern-Simons AdS 2 F = da f = da Single channel R imp =. U(1) Nc

144 Equations of Motion =e i µ, = r, t, x m, n = r, t mµ F µ = 4 N (x)j m n gg nq g mp f qp = J m mj m =0 m gg mn n = gg mn (A m a m + m )(A n a n + n ) g V J m 2 gg mn (A n a n + n ) 2

145 Equations of Motion Ansatz: Static solution After gauge fixing, only non-zero fields: (r) a t (r) A x (r) f rt = a t (r) F rx = A x (r) J t (r) = 2 gg tt a t 2

146 Equations of Motion J t (r) = 2 gg tt a t 2 trx F rx = 4 N (x)j t (r) r gg rr g tt f rt = J t (r) r gg rr r gg tt a 2 t gm 2 =0

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

148 AdS-Schwarzschild black hole Hawking temperature = T 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

149 AdS-Schwarzschild black hole Hawking temperature = T 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???

150 AdS-Schwarzschild black hole Hawking temperature = T 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!

151 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)

152 Large-N Approach to the Kondo Effect Spin SU(N) R imp = anti-symm. k =1 N with Ng K fixed S = O( ) c (0, ) ( ) SU(N) U(1) U(1) singlet bi-fundamental

153 Large-N Approach to the Kondo Effect Spin SU(N) R imp = anti-symm. k =1 N with Ng K fixed Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, (2003) T>T c T<T c O =0 O =0 T c T K

154 Large-N Approach to the Kondo Effect Spin SU(N) R imp = anti-symm. k =1 N with Ng K fixed Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, (2003) T>T O =0 c T<T c O =0 Represents the binding of an electron to the impurity

155 Large-N Approach to the Kondo Effect Spin SU(N) R imp = anti-symm. k =1 N with Ng K fixed Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, (2003) T>T c O =0 T<T c O =0 U(1) U(1) U(1) (0+1)-DIMENSIONAL SUPERCONDUCTIVITY

156 Large-N Approach to the Kondo Effect Spin SU(N) R imp = anti-symm. k =1 N with Ng K fixed Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, (2003) T>T c O =0 T<T c O =0 The phase transition is an ARTIFACT of the large-n limit! The actual Kondo effect is a crossover

157 AdS-Schwarzschild black hole Hawking temperature = T 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!

158 The Phase Shift J t (r) = 2 gg tt a t 2 trx F rx = 4 N (x)j t (r) r gg rr g tt f rt = J t (r) r gg rr r gg tt a 2 t gm 2 =0

159 The Phase Shift J t (r) = 2 gg tt a t 2 trx F rx = 4 N (x)j t (r) r gg rr g tt f rt = J t (r) T>T c (r) =0 J t (r) =0

160 The Phase Shift T>T c (r) =0 J t (r) =0 UV gf rt = Q IR gf rt = Q

161 The Phase Shift J t (r) = 2 gg tt a t 2 trx F rx = 4 N (x)j t (r) r gg rr g tt f rt = J t (r) T>T c T<T c (r) =0 J t (r) =0

162 Screening of the Impurity gf rt AdS 2 = Q =1/2 gf rt horizon

163 The Phase Shift J t (r) = 2 gg tt a t 2 trx F rx = 4 N (x)j t (r) r gg rr g tt f rt = J t (r) T>T c T<T c (r) =0 J t (r) =0

164 The Phase Shift J t (r) = 2 gg tt a t 2 trx F rx = 4 N (x)j t (r) magnetic flux electric charge density T>T c T<T c (r) =0 J t (r) =0

165 The Phase Shift trx F rx =2 r A x (r) = 4 N (x)j t (r) Integrate up to some r A x r A x AdS = 2 N (x) drj t (r) x Compactify, integrate over x dx A x r dx A x AdS = 2 N drj t (r)

166 The Phase Shift T>T c T<T c (r) =0 J t (r) =0 dxa x =0 UV dxa x =0 IR e i R dxa x Kraus and Larsen hep-th/

167 The Phase Shift T>T c T<T c (r) =0 J t (r) =0 UV gf rt = Q IR e i R dxa x gf rt <Q

168 The Resistivity (What we know so far) T irr.????? T T c

169 The Resistivity (What we know so far)? Minimum???? T irr.? Absent at large N No logarithm! T c T

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

171 Summary What is the holographic dual of the Kondo effect? Holographic superconductor in AdS 2 with a special boundary condition on the scalar coupled as a defect to a Chern-Simons gauge field in AdS 3

172 Outlook Multi-channel? Other impurity representations? Spin as global symmetry? Entanglement entropy? Quantum Quenches? Multiple impurities? Kondo lattice? Suggestions welcome!

173 Thank You.

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