Holographic Kondo and Fano Resonances

Transcription

1 Holographic Kondo and Fano Resonances Andy O Bannon Disorder in Condensed Matter and Black Holes Lorentz Center, Leiden, the Netherlands January 13, 2017

2 Credits Johanna Erdmenger Würzburg Carlos Hoyos Oviedo Ioannis Papadimitriou Trieste Jonas Probst Oxford Jackson Wu Alabama

3 Credits Based on Erdmenger, Hoyos, A.O B., Papadimitriou, Probst, Wu Erdmenger, Flory, Hoyos, Newrzella, A.O B., Wu Erdmenger, Flory, Hoyos, Newrzella, Wu O B., Papadimitriou, Probst Erdmenger, Hoyos, O B., Wu

4 Outline: Kondo Effect Holographic Kondo Model Holographic Two-Point Functions Holography vs. Quantum Dots Summary and Outlook

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

6 µ gs

7 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

8 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

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

10 Kondo Problem What is the ground state? What is the spectrum of excitations about the ground state? The coupling diverges at low energy! We know the answer!

11 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)

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

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

14 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

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

16 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,

17 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)

18 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

19 Open Problems Non-Equilibrium Physics Latta et al Entanglement Entropy Affleck, Laflorencie, Sørensen Kondo Lattice Kondo S i S j Form singlets with electrons Form singlets with each other Does the competition produce a quantum phase transition? What if we replace the Fermi liquid with STRONGLY INTERACTING degrees of freedom?

20 Open Problems Non-Equilibrium Physics Latta et al Entanglement Entropy Affleck, Laflorencie, Sørensen Let s try AdS/CFT! Kondo Lattice Kondo S i S j Form singlets with electrons Form singlets with each other Does the competition produce a quantum phase transition? What if we replace the Fermi liquid with STRONGLY INTERACTING degrees of freedom?

21 Open Problems Non-Equilibrium Physics THIS TALK Entanglement Entropy Erdmenger, Flory, Hoyos, Newrzella, Wu Kondo Lattice A. O B., Papadimitriou, Probst What if we replace the Fermi liquid with STRONGLY INTERACTING degrees of freedom? Erdmenger, Hoyos, O B., Wu

22 GOAL Compute 2-point functions in a holographic single-impurity Kondo model. Retarded Green s Function G(!) Spectral Function (!) 2ImG(!)

23 GOAL Compute 2-point functions in a holographic single-impurity Kondo model. Retarded Green s Function G(!) Z!! p pole Spectral Function (!) 2ImG(!) peak

24 Advertisement Compute 2-point functions in a holographic single-impurity Kondo model. Quantum Quenches in a Holographic Kondo Model Erdmenger, Flory, Newrzella, Strydom, Wu

25 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)

26 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 linearize around k F c(x ) 1 r e ik F r L (r) e +ik F r R (r)

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

28 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! Single left-moving Virasoro algebra

29 Spin SU(N) k>1 J = L L U(1) J = L L SU(N) J A = L ta L SU(k) Kac-Moody Algebra SU(N) k SU(k) N U(1) kn

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: single Virasoro algebra 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 classically marginal

32 UV SU(N) k SU(k) N U(1) Nk Eigenstates are representations of the Kac-Moody algebra Determine mapping between UV and IR eigenstates 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 classically marginal

34 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)

35 Large-N Approach to the Kondo Effect Spin SU(N) N with Ng K fixed R imp = anti-symm. R imp =. } Q

36 Large-N Approach to the Kondo Effect Spin SU(N) N with Ng K fixed R imp = anti-symm. S = Abrikosov pseudo-fermions Abrikosov, Physics 2, p.5 (1965)

37 Large-N Approach to the Kondo Effect Spin SU(N) N with Ng K fixed R imp = anti-symm. S = U(1) : c! c! e i t j =0

38 Large-N Approach to the Kondo Effect Spin SU(N) N with Ng K fixed R imp = anti-symm. } R imp =. Q = hji

39 Large-N Approach to the Kondo Effect R imp = anti-symm. S = ~S ~J = ~ c ~ c ~ ij ~ kl = il jk ij kl /N = 1 2 c 2 + O(1/N ) O c = 1 2 O O + O(1/N )

40 Large-N Approach to the Kondo Effect R imp = anti-symm. S = O(t) c (0,t) (t) SU(N) {z } singlet SU(k) U(1) {z } U(1) {z } bi-fundamental

41 Large-N Approach to the Kondo Effect R imp = anti-symm. S = O(t) c (0,t) (t) Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, (2003) T>T c O =0 T<T c Tc TK O =0

42 Large-N Approach to the Kondo Effect R imp = anti-symm. S = O(t) c (0,t) (t) Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, (2003) T>T c O =0 T<T c O =0 hoi /(T c T ) 1/2

43 Large-N Approach to the Kondo Effect R imp = anti-symm. S = O(t) c (0,t) (t) Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, (2003) T>T c O =0 T<T c O =0 Represents the screening of the impurity spin

44 Large-N Approach to the Kondo Effect R imp = anti-symm. S = O(t) c (0,t) (t) Coleman PRB 35, 5072 (1987) Senthil, Sachdev, Vojta PRL 90, (2003) T>T c O =0 T<T c O =0 SU(k) U(1) U(1)! SU(k) U(1) (0+1)-DIMENSIONAL SYMMETRY BREAKING

45 Large-N Approach to the Kondo Effect R imp = anti-symm. S = O(t) c (0,t) (t) 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

46 Large-N Approach to the Kondo Effect Take-Away Messages S = O(t) c (0,t) (t) Kondo coupling: S J = 1 2 O O + O(1/N ) Kondo Effect: Condensation of O

47 Outline: Kondo Effect Holographic Kondo Model Holographic Two-Point Functions Holography vs. Quantum Dots Summary and Outlook

48 Holography CFT + large-n + Additional Degrees of Freedom Kac-Moody Symmetry Kondo coupling S J = 1 2 O O + O(1/N ) Kondo effect: Condensation of O

49 Holography H = 1 2 Z +1 dx L i@ x L 1 single Virasoro algebra SU(N) k SU(k) N U(1) kn no impurity yet

50 Holography H = 1 2 Z +1 dx 1 h i L id x L + (gauge fields) single Virasoro algebra SU(N) k SU(k) N U(1) kn Gauge SU(N) k Gauge Anomaly! Probe limit k N

51 Holography H = 1 2 Z +1 dx 1 h i L id x L + (gauge fields) single Virasoro algebra SU(N) k SU(k) N U(1) kn Gauge SU(N) k restricted to gauge singlets (zero net spin)

52 Holography H = 1 2 Z +1 dx 1 h i L id x L + (gauge fields) single Virasoro algebra SU(N) k SU(k) N U(1) kn Gauge SU(N) k D x iga x gauge coupling g

53 Holography H = 1 2 Z +1 dx 1 h i L id x L + (gauge fields) single Virasoro algebra SU(N) k SU(k) N U(1) kn Gauge SU(N) k large-n limit N!1 Ng 2 fixed

54 Holography H = 1 2 Z +1 dx 1 h i L id x L + (gauge fields) single Virasoro algebra SU(N) k SU(k) N U(1) kn Gauge SU(N) k Breaks (chiral) conformal symmetry

55 Holography H CFT = 1 2 Z 1 dx 1 h i L id x L + (gauge fields) +... two Virasoro algebras SU(N) k SU(k) N U(1) kn ADD degrees of freedom (1 + 1)d CFT Maldacena limit N!1!1

56 Holography H CFT = 1 2 Z 1 dx 1 h i L id x L + (gauge fields) +... two Virasoro algebras SU(N) k SU(k) N U(1) kn ADD degrees of freedom (1 + 1)d CFT (1 + 1)d CFT = AdS 3

57 ds 2 = d 2 2 f( ) + 2 f( )dt 2 + dx 2 f( ) =1 2 H/ 2 T = H /(2 ) x = 1 boundary = H horizon

58 Holography H CFT = 1 2 Z 1 dx 1 h i L id x L + (gauge fields) +... two Virasoro algebras SU(N) k SU(k) N U(1) kn Current = J Gauge field A Kac-Moody current = Chern-Simons gauge field rank and level = rank and level

59 k =1 SU(k) N U(1) kn! U(1) N S = N 4 Z A ^ da x = 1 boundary = H horizon

60 Holography H CFT! H CFT Z +1 dx 1 (x) K ~ S ~ J R imp = anti-symm. S = U(1) : c! c! e i t j =0 Current j = Gauge field a

61 f = da S = N 4 Z A ^ da N Z d 2 x p g 1 4 f mn f mn x =0 = 1 boundary = H horizon

62 hji = Q p gf 2 = Q x =0 = 1 boundary = H horizon

63 Holography H CFT! H CFT Z +1 dx (x) 1 K ~ S ~ J R imp = anti-symm. S = O(t) (0,t) (t) L ~S ~J = 1 2 O O + O(1/N )

64 Holography H CFT! H CFT Z +1 dx (x) 1 K ~ S ~ J R imp = anti-symm. S = O(t) (0,t) (t) L SU(N) {z k=1 } U(1) U(1) N {z } singlet bi-fundamental

65 Holography H CFT! H CFT Z +1 dx (x) 1 K ~ S ~ J R imp = anti-symm. S = O(t) (0,t) (t) L bi-fundamental scalar O = bi-fundamental scalar

66 S = N Z N 4 Z d 2 x p A ^ da g Z N d 2 x p g 1 4 f mn f mn h i (D m ) (D m )+V D m =(@ m + ia m ia m ) x =0 = 1 boundary = H horizon

67 S = N Z N 4 Z d 2 x p A ^ da g Z N d 2 x p g 1 4 f mn f mn h i (D m ) (D m )+V V = M 2 x =0 = 1 boundary = H horizon

68 Holography M 2 = Breitenlohner-Freedman bound of AdS 2 O O is classically marginal ( ) = c 1/2 + c 1/2 log +... Kondo coupling c = K c Witten hep-th/ Berkooz, Sever, Shomer hep-th/

69 Phase Transition Erdmenger, Hoyos, O B., Wu T>T c O =0 T<T c O =0 T c T K hoi /(T c T ) 1/2

70 Outline: Kondo Effect Holographic Kondo Model Holographic Two-Point Functions Holography vs. Quantum Dots Summary and Outlook

71 Two-point functions G(!) = retarded Green s function (!) i G(!) G (!) = 2ImG(!) poles in G(!)! ) peaks in (!) (complex ) (real! )

72 G(!) Z Two-point functions )!! p (!) 2Z! I (!! R ) 2 +! 2 I (!)! p =! R + i! I 2Z/! I 2! I! R!

73 Kondo Resonance electron spectral function in energy: at Fermi energy e (!) in real space: at the impurity as T! T + K rises logarithmically height saturates at T K (Friedel sum rule)! T =0 width / T K " F many-body resonance

74 Holography G O O(!) =ho (!)O(!)i ret. hoi =0 D(!) H G O O(!) = N K apple iq i! 2 T 1 H 1 KD(!) apple iq + ln T T c Harmonic Numbers H[x] E + 0 [x + 1] [x + 1]

75 Holography G O O(!) =ho (!)O(!)i ret. hoi 6=0 Only numerical results Q & N/2

76 Poles hoi =0 T =5.5 T c 1!/(2 T ) G OO (!) -3 G O O(!) -4

77 Poles hoi =0 T =1.5 T c 1!/(2 T )

78 Poles hoi =0 T = T c 1!/(2 T )

79 Poles hoi =0 T = T c 1!/(2 T ) ! p =!-2 R + i! I! R / T T c -3! I / T T c -4

80 Poles hoi =0 T =0.75 T c 1!/(2 T )

81 Poles hoi =0 T =0.75 T c 1!/(2 T ) INSTABILITY

82 Poles hoi =0 T =0.50 T c 1!/(2 T ) INSTABILITY

83 Spectral Function O O(!)/N hoi =0 60 T/T c = T/T c = T/T c = T/T c =1.15 T/T c = !/T K

84 Spectral Function O O(!)/N hoi = peak / T 1 T c NOT a Kondo resonance power law, not logarithm !/T K

85 Poles hoi 6=0 T = T c!/(2 T )

86 Poles hoi 6=0 T<T c 1!/(2 T ) ! p / ihoi

87 Poles hoi 6=0 T<T c 1!/(2 T ) ! p / ihoi large-n Kondo resonance! Coleman Intro. to Many-Body -4 Physics CUP 2015

88 Spectral Function hoi 6=0 OO (!)/N T/T c = T/T c = T/T c = ! 2 T

89 Spectral Function hoi 6=0 OO (!)/N T/T c = T/T c = T/T c = Fano Resonance ! 2 T

90 Fano Resonances!

91 Fano Resonances resonance(s) interacts with a continuum Example: light scattering off an atom Fano (!) =!! q 2 (!! 0 ) q = Fano or asymmetry parameter probability of resonant scattering q 2 / probability of non-resonant scattering

92 Fano Resonances!! 0 + q 2 2 (!! 0 ) (!! 0 ) 2 = + q (!! 0 ) (!! 0 ) q 2 (!! 0 ) (!! 0 ) (!! 0 ) = (q 2 1) 2 (!! 0 ) q 2 (!! 0 ) (!! 0 ) = + + interference ( mixing )

93 Fano Resonances 1!

94 Fano Resonances q 2 1! 0!

95 Fano Resonances q 2 +1 q 2 1!! 0! q 2

96 Fano Resonances q 2 +1 q 2 1! 0 q 2!! 0! q 2

97 Fano Resonances q =0 q =1 q = 1 probability of resonant scattering q 2 / probability of non-resonant scattering

98 Spectral Function hoi =0 T =1.5 T c O O(!)/N 10! T K T K 5 q !/T K

99 Spectral Function hoi =0 T =1.5 T c O O(!)/N! p =! R + i! I 10! 0! R 2! I 5 q = q(t,q) !/T K

100 Spectral Function O O(!)/N hoi =0 60 T/T c = T/T c = T/T c = T/T c =1.15 T/T c = !/T K

101 Spectral Function hoi =0 T just above q T c Q

102 Spectral Function hoi 6=0 OO (!)/N T/T c = T/T c = T/T c = ! 2 T -100 q =1-200! p / ihoi 2

103 Outline: Kondo Effect Holographic Kondo Model Holographic Two-Point Functions Holography vs. Quantum Dots Summary and Outlook

104 Quantum Dots Fano Resonances observed in quantum dots Göres et al. PRB (2000) q = 1 serial q finite side-coupled

105 Quantum Dots vs. Holography O O(!)/N 10 q 3.31 K 5 ~S !/T K q finite side-coupled

106 Holography AdS 2 ) (0 + 1) scale invariance OO /! 2 1 continuum! Break scale invariance (relevant deformation, RG flow) + Resonance = Fano Generic to RG flows between (0+1)-dimensional fixed points

107 Outline: Kondo Effect Holographic Kondo Model Holographic Two-Point Functions Holography vs. Quantum Dots Summary and Outlook

108 Summary and Outlook Holographic single-impurity Kondo model Green s and spectral functions Kondo resonance T<T c! p / ihoi 2 Fano resonances Generic to (0+1)-dimensional RG flows SYK models?

109 Summary and Outlook Non-Equilibrium Physics Latta et al Entanglement Entropy Affleck, Laflorencie, Sørensen Kondo Lattice Kondo Form singlets with electrons S i S j Form singlets with each other Does the competition between these produce a quantum phase transition?

110 Thank You.