Strange metals from holography David Vegh Simons Center for - PowerPoint PPT Presentation

Introduction Holographic non-Fermi liquids Charge transport Strange metals from holography David Vegh Simons Center for Geometry and Physics Hong Liu, John McGreevy, DV arXiv:0903.2477 Thomas Faulkner, Hong Liu, John McGreevy, DV arXiv:0907.2694 Thomas Faulkner, Gary Horowitz, John McGreevy, Matthew Roberts, DV arXiv:0911.3402 Thomas Faulkner, Nabil Iqbal, Hong Liu, John McGreevy, DV arXiv:1003.1728 and in progress (see also: Sung-Sik Lee, arXiv:0809.3402 ; Cubrovic, Zaanen, Schalm, arXiv:0904.1933) The Galileo Galilei Institute – September 29, 2010 David Vegh Strange metals from holography

Introduction Holographic non-Fermi liquids Charge transport Introduction David Vegh Strange metals from holography

Introduction Holographic non-Fermi liquids Charge transport Introduction Holographic non-Fermi liquids AdS 4 − BH geometry Fermi surfaces Near-horizon AdS 2 and emergent IR CFT David Vegh Strange metals from holography

Introduction Holographic non-Fermi liquids Charge transport Introduction Holographic non-Fermi liquids AdS 4 − BH geometry Fermi surfaces Near-horizon AdS 2 and emergent IR CFT Charge transport Conductivity Summary David Vegh Strange metals from holography

Introduction Holographic non-Fermi liquids Charge transport Fermions at finite density ◮ Landau Fermi liquid theory [Landau, 1957] [Abrikosov-Khalatnikov, 1963] stable RG fixed point [Polchinski, Shankar] (modulo BCS instability) [Benfatto-Gallivotti] ◮ (i) ∃ Fermi surface (ii) (weakly) interacting quasiparticles ⇒ thermodynamics, transport properties ◮ appear as poles in the single-particle Green’s function: x ) = i θ ( t ) · �{ ψ † ( t ,� x ) , ψ (0 ,� G R ( t ,� 0) }� µ, T Z G R ( ω,� k ⊥ ≡ | � Σ ∼ i ω 2 k ) = ω − v F k ⊥ − Σ+ . . . , k |− k F ⋆ k ⊥ → 0 ρ ( ω,� k ) ≡ Im G R ( ω,� k ) − → Z δ ( ω − v F k ⊥ ) with Z finite David Vegh Strange metals from holography

Introduction Holographic non-Fermi liquids Charge transport Non-Fermi liquids do exist ◮ sharp Fermi surface still present � ◮ no long-lived quasiparticles × pole residue can vanish ◮ anomalous thermodynamic and transport properties Examples ◮ 1+1d Luttinger liquid: interacting fermions → free bosons ◮ Mott metal-insulator transition ◮ heavy fermion compounds ◮ high-temperature superconductors [M¨ uller, Bednorz, 1986] pseudogap, Fermi arcs and pockets, density waves optimally doped cuprates: ρ ∼ T Organizing principle for non-Fermi liquids? David Vegh Strange metals from holography

Introduction AdS 4 − BH geometry Holographic non-Fermi liquids Fermi surfaces Charge transport Near-horizon AdS 2 and emergent IR CFT Strategy ◮ set up finite density state ◮ look for a Fermi surface ◮ study low-energy excitations near FS ◮ transport properties David Vegh Strange metals from holography

Introduction AdS 4 − BH geometry Holographic non-Fermi liquids Fermi surfaces Charge transport Near-horizon AdS 2 and emergent IR CFT AdS 4 − BH geometry Relativistic CFT 3 with gravity dual and conserved U (1) global symmetry 1 d 4 x √− g � R + 6 R 2 − 1 � � 4 F µν F µν + . . . S = 2 κ 2 Charged black hole solution ds 2 = r 2 + R 2 dr 2 − f ( r ) dt 2 + d � x 2 � � R 2 f ( r ) r 2 f ( r ) = 1 + Q 2 r 4 − M 1 − r 0 � � A = µ dt r 3 r where µ = chemical potential, horizon at r = r 0 . David Vegh Strange metals from holography

Introduction AdS 4 − BH geometry Holographic non-Fermi liquids Fermi surfaces Charge transport Near-horizon AdS 2 and emergent IR CFT Retarded spinor Green’s function Introduce ψ spinor field in the AdS − BH background d 4 x √− g � ¯ � ψ (/ � S probe = D − m ) ψ + interactions 4 ω ab µ Γ ab − iqA µ . with D µ = ∂ µ + 1 Universality: for two-point functions, the interaction terms do not matter. ∆ = 3 Results only depend on ( q , ∆) 2 ± mR Prescription [Henningson-Sfetsos] [M¨ uck-Viswanathan] [Son-Starinets] [Iqbal-Liu] ◮ Solve the Dirac equation for the bulk spinor in AdS − BH ◮ Impose infalling boundary conditions at the horizon ◮ Expand the solution at the boundary ψ = ( − gg rr ) − 1 / 4 e − i ω t + ikx Ψ Φ α = 1 2 (1 − ( − 1) α Γ r Γ t Γ x )Ψ � � � � 0 1 G α ( ω, k ) = b α r →∞ a α r m + b α r − m Φ α ≈ α = 1 , 2 1 0 a α David Vegh Strange metals from holography

Introduction AdS 4 − BH geometry Holographic non-Fermi liquids Fermi surfaces Charge transport Near-horizon AdS 2 and emergent IR CFT Fermi surfaces At q = 1 , ∆ = 3 / 2 the numerical computation gives [Liu-McGreevy-DV] k = 0 . 9 k = 0 . 925 k F ≈ 0 . 9185284990530 ◮ Quasiparticle-like peaks for k < k F ω ∼ k z ⊥ with z ≈ 2 . 09 ◮ Bumps for k > k F ◮ Scaling behavior: G R ( λ k ⊥ , λ z ω ) = λ − α G R ( k ⊥ , ω ) with α = 1 ◮ non-Fermi liquid! David Vegh Strange metals from holography

Introduction AdS 4 − BH geometry Holographic non-Fermi liquids Fermi surfaces Charge transport Near-horizon AdS 2 and emergent IR CFT Fermi surfaces dispersion from ‘photoemission’ results q = 1 . 0 k F = 0 . 53 q = 1 . 56 k F = 0 . 95 q = 2 . 0 k F = 1 . 32 David Vegh Strange metals from holography

Introduction AdS 4 − BH geometry Holographic non-Fermi liquids Fermi surfaces Charge transport Near-horizon AdS 2 and emergent IR CFT Black hole geometry ds 2 = r 2 dr 2 f ( r ) = 1 + 3 r 4 − 4 − f ( r ) dt 2 + d � � x 2 � + R 2 R 2 f ( r ) r 2 r 3 ◮ Emergent “IR CFT” AdS 2 ⇔ “(0+1)-d CFT” ◮ At low frequencies, the parent theory is controlled by IR CFT. ◮ Each UV operator O is associated to a family of operators O � k in IR. ◮ Conformal dimensions in IR � 2 + k 2 − q 2 �� k = 1 1 ∆ − 3 G k ( ω ) = c ( k ) ω 2 ν k δ � 2 + ν � ν � k = √ k 6 2 2 David Vegh Strange metals from holography

Introduction AdS 4 − BH geometry Holographic non-Fermi liquids Fermi surfaces Charge transport Near-horizon AdS 2 and emergent IR CFT Small ω expansion ◮ To understand scaling around FS, study low- ω behavior of correlators. ◮ Not straightforward: ω -dependent terms in DE are singular near the horizon. Strategy [Faulkner-Liu-McGreevy-DV] ◮ Separate spacetime into UV region and the AdS 2 × R 2 IR region. ◮ Perform small ω expansions separately. ◮ Match them at the overlapping region. Result � � b (0) + + ω b (1) + + O ( ω 2 ) + G k ( ω ) b (0) − + ω b (1) − + O ( ω 2 ) G R ( ω, k ) = � � a (0) + + ω a (1) a (0) − + ω a (1) + + O ( ω 2 ) + G k ( ω ) − + O ( ω 2 ) G k ( ω ) ∈ C retarded correlator for O � k in IR CFT a (0) ± , a (1) ± , b (0) ± , b (1) ± ∈ R are k -dependent functions (from UV region) David Vegh Strange metals from holography

Introduction AdS 4 − BH geometry Holographic non-Fermi liquids Fermi surfaces Charge transport Near-horizon AdS 2 and emergent IR CFT Fermi surfaces � � b (0) + + ω b (1) + + O ( ω 2 ) + G k ( ω ) b (0) − + ω b (1) − + O ( ω 2 ) G R ( ω, k ) = � � a (0) + + ω a (1) a (0) − + ω a (1) + + O ( ω 2 ) + G k ( ω ) − + O ( ω 2 ) a (0) Suppose that for some k F : + ( k F ) = 0 Then, at small ω, k ⊥ we have h 1 G R ( k , ω ) = v F ω − h 2 G k F ( ω ) + . . . 1 k ⊥ − � m 2 + k 2 − q 2 G k ( ω ) = c ( k ) ω 2 ν k 1 ν � k = h 1 , h 2 , v F ∈ R √ 2 6 This is the quasiparticle peak we saw earlier. David Vegh Strange metals from holography

Introduction AdS 4 − BH geometry Holographic non-Fermi liquids Fermi surfaces Charge transport Near-horizon AdS 2 and emergent IR CFT Fermi surfaces: singular and non-singular Suppose ν k < 1 2 h 1 G R ( k , ω ) = k ⊥ − h 2 G kF ( ω ) + . . . 1 − 2 ν kF 2 ν kF Γ( k ) ω ⋆ ( k ) ∼ k z 1 z = 2 ν kF > 1 ω ⋆ ( k ) = const Z ∼ k → 0 , k ⊥ → 0 ⊥ ⊥ Suppose ν k > 1 2 h 1 G R ( k , ω ) = 2 ν kF + . . . k ⊥ − 1 vF ω − h 2 c ( k F ) ω 2 ν kF − 1 Γ( k ) ω ⋆ ( k ) ∼ v F k ⊥ ω ⋆ ( k ) = k → 0 Z = h 1 v F ⊥ David Vegh Strange metals from holography

Introduction AdS 4 − BH geometry Holographic non-Fermi liquids Fermi surfaces Charge transport Near-horizon AdS 2 and emergent IR CFT Fermi surfaces: Marginal Fermi liquid h 1 G R ( k , ω ) = v F ω − h 2 c ( k F ) ω 2 ν kF + . . . k ⊥ − 1 Suppose ν k = 1 2 v F goes to zero, c ( k F ) has a pole h 1 G R ( k , ω ) = k ⊥ + c 1 ω +˜ c 2 ω log ω + ic 2 ω where c ∈ R . This is the Marginal Fermi liquid Green’s function [Varma, 1989] David Vegh Strange metals from holography

Introduction Conductivity Holographic non-Fermi liquids Summary Charge transport Charge transport ◮ Normal phase of optimally doped high- T c : ρ = ( σ DC ) − 1 ∼ T ◮ impurities: ρ ∼ const. ρ ∼ T 2 e-e scattering: e-phonon scattering: ρ ∼ T 5 ◮ Compute conductivity contribution of the holographic Fermi surfaces [Faulkner-Iqbal-Liu-McGreevy-DV] David Vegh Strange metals from holography