A Holographic model of the Kondo effect Carlos Hoyos Tel Aviv - PowerPoint PPT Presentation

A Holographic model of the Kondo effect Carlos Hoyos Tel Aviv University Crete Center for Theoretical Physics, March 21, 2013 Johanna Erdmenger, C.H., Andy O’Bannon, Jackson Wu C. Hoyos (TAU) Kondo model Crete 2013 1 / 36

Outline Introduction Stringy model Bottom-up model Future directions C. Hoyos (TAU) Kondo model Crete 2013 2 / 36

Kondo effect Metals: Fermi liquid+phonons+impurities: ρ ∼ ρ 0 + T 2 But in some metals at low temperatures ρ ∼ − log( T ) C. Hoyos (TAU) Kondo model Crete 2013 3 / 36

Kondo effect Scattering with magnetic impurities � � T ρ ∼ ρ 0 1 + κ log | ǫ − ǫ F | Antiferromagnetic coupling κ < 0 C. Hoyos (TAU) Kondo model Crete 2013 4 / 36

Kondo effect Perturbation theory breaks down at T K = | ǫ − ǫ F | e 1 /κ C. Hoyos (TAU) Kondo model Crete 2013 5 / 36

Kondo effect Single impurity problem solved Wilson’s RG, Nozi` eres’ Fermi liquid description, the Bethe Ansatz, large- N limits, conformal field theory... Multiple impurities: Heavy fermion compounds with strange metal behaviour ρ ∼ T may be described by a Kondo lattice Holography: non-perturbative, large- N Goal today: construct a holographic model for a single impurity C. Hoyos (TAU) Kondo model Crete 2013 6 / 36

CFT approach Kondo model [Kondo’ 66; Affleck & Ludwig ’90s] : H = v F 1 2 π ψ † L i ∂ x ψ L + v F λ K δ ( x ) � S · ψ † 2 � τ ψ L , L s-wave reduction: 1+1 dimensions Kondo coupling marginal classically (CFT) Asymptotic freedom: UV fixed point C. Hoyos (TAU) Kondo model Crete 2013 7 / 36

CFT approach UV CFT Symmetries: Spin SU ( N ), k channels SU ( k ), Charge U (1) Kac-Moody algebra: SU ( N ) k × SU ( k ) N × U (1) n + m + k n 2 δ ab δ n , − m [ J a n , J b m ] = if abc J c Finite number of highest weight states. SU (2) k : spin ≤ k / 2 Sugawara construction: 1 1 1 2 π ( N + k ) J a J a + 2 π ( k + N ) J A J A + 4 π Nk J 2 + λ K δ ( x ) � S · � H = J C. Hoyos (TAU) Kondo model Crete 2013 8 / 36

CFT approach IR CFT Redefinition of spin current: J a ≡ J a + π ( N + k ) λ K δ ( x ) S a Critical coupling 2 λ K = N + k Hamiltonian: 1 1 1 2 π ( N + k ) J a J a + 2 π ( k + N ) J A J A + 4 π Nk J 2 H = No impurity! [Affleck & Ludwig ’95] C. Hoyos (TAU) Kondo model Crete 2013 9 / 36

CFT approach IR CFT = UV CFT with shifted spectrum IR spin representations = UV spin representations + impurity spin Example: one channel, spin N = 2, SU (2) 1 × U (1), s imp = 1 / 2 UV: Neveu-Schwarz boundary conditions (spin, charge)= (0 , 0), ( ± 1 / 2 , 1) IR: Ramond boundary condition (spin, charge)= ( ± 1 / 2 , 2), (0 , 1) Phase of electron wavefunction on a circle changes by π/ 2 C. Hoyos (TAU) Kondo model Crete 2013 10 / 36

CFT approach Possible phases in SU (2) k : Underscreening: 2 s imp > k Fermi liquid + impurity of spin | s imp − k / 2 | Critical screening: 2 s imp = k IR fixed point: k free left-movers Overscreening: 2 s imp < k Non-trivial IR fixed point: non-Fermi liquid behavior Qualitatively similar for higher spin C. Hoyos (TAU) Kondo model Crete 2013 11 / 36

Large- N N → ∞ , λ K → 0, λ K N fixed Spin of impurity: Young tableaux with Q boxes Totally antisymmetric representation: S a = χ † T a χ Slave fermions, dimension [ χ ] = 0 χ † χ = Q Additional U ( N f ) symmetry Critical ( k = 1) or overscreening ( k ≥ 2) C. Hoyos (TAU) Kondo model Crete 2013 12 / 36

Large- N Kondo coupling as double-trace deformation: λ K δ ( x ) J a S a = λ K δ ( x ) � � � � ψ † L T a ψ L χ † T a χ = 1 � � � � ψ † χ † ψ L 2 λ K δ ( x ) L χ = 1 2 λ K δ ( x ) OO † O SU ( N ) singlet, charged under U ( N f ) × SU ( k ) × U (1) Dimensions: [ ψ L ] = [ O ] = 1 / 2 Mean field transition: T > T K , �O� = 0 , SU ( k ) × U ( N f ) × U (1) T < T K , �O� � = 0 , SU ( k ) × U ( N f ) × U (1) → U (1) D [Senthil, Sachdev, Vojta] C. Hoyos (TAU) Kondo model Crete 2013 13 / 36

Large- N Summary of Kondo effect at large N s-wave reduction: 1+1 chiral CFT + impurity double-trace coupling 0+1 superconductor These will be the main ingredients to construct a holographic model C. Hoyos (TAU) Kondo model Crete 2013 14 / 36

Impurities in stringy models Supersymmetric defects with localized fermions D5/D3 AdS 2 ⊂ AdS 5 [Kachru, Karch, Yaida] [Harrison, Kachru, Torroba] M2/D2 in ABJM AdS 2 ⊂ AdS 4 [Jensen, Kachru, Karch, Polchinski, Silverstein] D6 in ABJM AdS 2 ⊂ AdS 4 , [Benincasa, Ramallo] with backreaction [Itsios, Sfetsos, Zoakos] D (8 − p ) in Dp background S 7 − p ⊂ S 8 − p [Benincasa, Ramallo] other sphere wrappings [Karaiskos, Sfetsos, Tsatis] Spectrum of Wilson loops [Mueck] [Faraggi, Pando Zayas] [Faraggi, Mueck, Pando Zayas] C. Hoyos (TAU) Kondo model Crete 2013 15 / 36

Stringy model x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 N c D3 • • • • – – – – – – N 7 D7 • • – – • • • • • • N 5 D5 • – – – • • • • • – 3-7 strings = chiral fermions (current algebra) 3-5 strings = slave fermions 5-7 strings = bifundamental scalar (tachyon) D5 branes become magnetic flux on D7 branes � S D 7 ⊂ P [ C 6 ] ∧ F D 7 C. Hoyos (TAU) Kondo model Crete 2013 16 / 36

Stringy model N c → ∞ : AdS 5 × S 5 background � S 5 F 5 = g s (2 π ) 2 (2 πα ′ ) 2 N c 7-7 strings: gauge field A µ dual to J µ � � S D 7 ⊃ − N c � A ∧ dA + 2 3 A ∧ A ∧ A tr 4 π AdS 3 Global symmetry: U ( N 7 ) N c C. Hoyos (TAU) Kondo model Crete 2013 17 / 36

Stringy model 5-5 strings: gauge field a m dual to Q � S D 5 ⊃ N 5 T D 5 P [ C 4 ] ∧ f Dual to completely antisymmetric Wilson loop [Yamaguchi ’06; Gomis, Passerini ’06] Charge Q = number of fundamental strings Size of D5 on S 5 [Camino,Paredes, Ramallo ’01] θ Q = Q S 5 = d θ 2 + sin 2 θ ds 2 ds 2 S 4 , π N c Maximal charge Q = N c − 1 C. Hoyos (TAU) Kondo model Crete 2013 18 / 36

Stringy model 5-7 strings: bifundamental scalar Φ dual to O Double-trace coupling for O analogous to Kondo coupling Double-trace coupling can lead to condensation [Pomoni, Rastelli ’08,’10] Holographic dual: boundary condition for Φ [Witten ’01] C. Hoyos (TAU) Kondo model Crete 2013 19 / 36

Bottom-up model s-wave reduction: 1+1 CFT − → AdS 3 � dz 2 ds 2 = g µν dx µ dx ν = 1 � h ( z ) − h ( z ) dt 2 + dx 2 , h ( z ) = 1 − z 2 / z 2 H , z 2 Temperature: T = 1 / (2 π z H ) SU ( N )-spin, k = 1 channel of chiral fermions, U (1) charge S CS = − N � A ∧ dA 4 π Impurity AdS 2 : U (1) symmetry, operator O d 3 x δ ( x ) √− g � 1 � � 4 f mn f mn + g mn ( D m Φ) † D n Φ + M 2 Φ † Φ S AdS 2 = − D m Φ = ∂ m Φ + iA m Φ − ia m Φ C. Hoyos (TAU) Kondo model Crete 2013 20 / 36

Asymptotics Gauge field: charge Q determines the spin representation of impurity a t ( z ) = Q z + µ Charge is an irrelevant operator: UV behavior is modified Scalar field effective mass fixed at the BF bound eff = M 2 − Q 2 = − 1 M 2 4 Scalar field at the BF bound: Φ = z 1 / 2 ( α log( z ) + β ) UV conformal dimensions ∆ = 1 2 We set M 2 = 0, Q = − 1 / 2 C. Hoyos (TAU) Kondo model Crete 2013 21 / 36

Kondo coupling Double-trace deformation = boundary condition [Witten ’01] α = κβ Renormalization: Φ = z 1 / 2 β 0 ( κ 0 log(Λ z ) + 1) = z 1 / 2 β ( κ log( µ z ) + 1) Running coupling κ 0 κ = � � Λ 1 + κ 0 ln µ Dynamical scale: Λ K = Λ e 1 /κ 0 κ < 0 “antiferromagnetic”: UV asymptotic freedom C. Hoyos (TAU) Kondo model Crete 2013 22 / 36

Phases Normal phase (Φ = 0): Background charge Q = − 1 / 2: a t ( z ) = Q µ = − Q z + µ, z H Broken phase (Φ � = 0): Background charge Q = − 1 / 2: a t ( z ) ≃ Q z + µ T + O ((log z ) 3 ) Background scalar field: Φ ≃ ( z / z H ) 1 / 2 β T ( κ T log( z / z H ) + 1) Values of µ T , κ T and β T determined numerically C. Hoyos (TAU) Kondo model Crete 2013 23 / 36

Kondo coupling Finite temperature solution: Φ = ( z / z H ) 1 / 2 β T ( κ T log( z / z H ) + 1) = β 0 ( κ 0 log(Λ z ) + 1) Temperature-dependent coupling κ 0 κ T = Λ � � 1 + κ 0 ln 2 π T High-temperatures T ≫ Λ K : 1 κ T ≃ � < 0 Λ � ln 2 π T Low temperatures T ≪ Λ K 1 κ T ≃ � > 0 Λ � ln 2 π T C. Hoyos (TAU) Kondo model Crete 2013 24 / 36

Instabilities in the normal phase Scalar field @ BF bound Φ = e − i ω t φ : 1 h ∂ z ( h ∂ z φ ) + ω 2 φ + 4 z 2 φ = 0 Tachyonic modes ω = − i Ω exist when κ = α 1 β = H 1 √ 4Ω 2 − 1 − 1 ) − log 4 2 ( This is possible ∀ κ except κ c < κ ≤ 0, 1 κ c = 2 − log 4 ≃ − 0 . 360674 H − 1 Stable at high temperatures, unstable at low temperatures C. Hoyos (TAU) Kondo model Crete 2013 25 / 36

Free energy Free energy = Euclidean action = - Action Counterterms scalar field: dt √− γ � 1 � � 1 � Φ † Φ + κ dt β 2 S Φ = − 2 + log ε Counterterms gauge field: dt √− γ γ tt a 2 S a t = +1 � t 2 F − F 0 = − αβ − 1 2 Q ( µ + Q ) − 2 Q 2 α 2 + 2 Q 2 αβ − Q 2 β 2 − κβ 2 + finite bulk integral C. Hoyos (TAU) Kondo model Crete 2013 26 / 36

Thermodynamics of the broken phase Free energy vs T / T K Condensate vs T / T K C. Hoyos (TAU) Kondo model Crete 2013 27 / 36