Hydrodynamics of Holographic Superconductors I. Amado, M. - PowerPoint PPT Presentation

Hydrodynamics of Holographic Superconductors I. Amado, M. Kaminski, K.L. [arXiv:0902.2209]

Outline Review of the Model Hydrodynamics Holographic Hydro by Quasinormal Modes Summary and Outlook related work: Kovtun, Herzog, Son [arXiv:0809.4870], Herzog, Pufu [arXiv:0902.0409], Herzog, Yarom [arXiv:0906.4810], Yarom [arXiv:arXiv:0903.1353], Maeda, Nustuume, Okamura [arXiv:0904.1914]

The Model Can we realize spontaneous symmetry breaking as function of temperature in AdS/CFT? Hartnoll, Herzog, Horowitz: YES, we can! [arXiv: 0803.3295] based on Gubser [arXiv:0801.2977] Abelian Higgs model in AdS-Blackhole background decoupling limit charge q-> Infty ds 2 = − ( r 2 dr 2 + r 2 L 2 − M r ) dt 2 + L 2 ( dx 2 + dy 2 ) r 2 L 2 − M r L = − 1 Ψ − ( ∂ µ Ψ − iA µ Ψ )( ∂ µ ¯ Ψ + iA µ ¯ 4 F µ ν F µ ν − m 2 Ψ ¯ Ψ )

The Model ρ ) Ψ ′ + Φ 2 Ψ ′′ + ( f ′ f + 2 2 eoms: f 2 Ψ + =0 L 2 f Ψ ρ Φ ′ − 2 Ψ 2 Φ ′′ + 2 =0 f Φ boundary conditions at Horizon: Φ ( ρ H ) = 0 Ψ ( ρ H ) , why? bulk current finite norm at the J µ = ψ 2 A µ Horizon values at boundary 3 L µ ¯ = 4 π T µ , µ − ¯ ρ + O ( 1 n 9 L n 16 π 2 T 2 n , ¯ = = ¯ ρ 2 ) Φ 3 4 π TL 2 � O 1 � , ψ 1 = ρ 2 + O ( 1 ψ 1 ρ + ψ 2 = ρ 2 ) Ψ 9 16 π 2 T 2 L 4 � O 2 � , ψ 2 =

The Model solve eom with either or ψ 2 = 0 ψ 1 = 0 � O 1 � 2 � O 2 � 2 2 4 T c T c 500 4000 400 3000 300 2000 200 1000 100 T T 0 0 T c 0.0 0.2 0.4 0.6 0.8 1.0 T c 0.0 0.2 0.4 0.6 0.8 1.0 � � 1 − T � O i � 2 ∝ T c

Hydrodynamics Hydrodynamics = slow modes k → 0 ω ( k ) = 0 lim ∂ n conservation law ∂ t + � ∇ � j = 0 constitutive relation with external source j = − D � ∇ n + σ � � E taking time derivative and using the continuity eqn i σω 2 � j L � = k 2 A L ω + iD � σ = − i ω � j L j L � k =0

Hydrodynamics broken phase: take Goldstone mode into account (Chaikin, Lubensky) generic prediction: appearance of sound modes σω 2 ˆ predicts correlator D = v 2 ˆ � j L j L � = S ω 2 − ˆ D � k 2 and conductivity � 1 � − i σ ( ω ) = ω + i ǫ ˆ σ = − i P + π ˆ σδ ( ω ) σ = n S ˆ ω including dissipation: ω = ± v s k − i Γ s k 2

Hydro and QNMs poles of retarded Green functions = Quasinormal Modes “Eigenmodes” Horizon: infalling Ψ H = ( ρ − 1) − i ω / 3 (1 + O ( ρ − 1)) Boundary: Pole of holographic GF � 1 � Ψ B = A ρ + B complex scalar field ρ 2 + O ρ 3 O 2 � = B O 1 � = A theory I theory II � O 2 ¯ � O 1 ¯ A B complex frequencies Ψ ∝ e − i ω R t e − ω I t

Hydro and QNMs Unbroken phase: superconducting Instability O 2 Theory O 1 Theory 0 0 � 1 � 1 � 2 � 2 Im � Ω � Im � Ω � � 3 � 3 � 4 � 4 � 5 � 6 � 5 � 3 � 2 � 1 0 1 2 3 � 3 � 2 � 1 0 1 2 3 Re � Ω � Re � Ω � 3 Vector channel: Diffusion mode D = 4 π T

Hydro and QNMs Broken phase: Second sound and Pseudodiffusion � φ 2 L 2 + ω 2 f − k 2 � � � f ′ + 2 f η − 2 i ωφ σ − i ωψ f a t − ik ψ f + 2 f η ′′ + η ′ + r 2 a x , 0 = ρ 2 ρ f � φ 2 L 2 + ω 2 f − k 2 � � � f ′ + 2 f σ + 2 φψ a t + 2 i ωφ f + 2 f σ ′′ + σ ′ + η , 0 = ρ 2 ρ f f � k 2 � ′′ + 2 f a t − ω k ρ 2 + 2 ψ 2 ′ − fa t ρ a t ρ 2 a x − 2 i ωψ η − 4 ψφ σ , 0 = � ω 2 � a x + ω k ′′ + f ′ a x ′ + f − 2 ψ 2 fa x f a t + 2 ik ψ η . 0 = ′ + k ω constraint: ′ = 2 i ( ψ ′ η − ψ η ′ ) f a t ρ 2 a x local ward identity: ∂ µ � j µ � = 2 � O i � η i 0

Hydro and QNMs How to compute QNMs of coupled system four l.i. solutions (one is pure gauge) η IV = i λψ , σ IV = 0 , a IV a IV = λω , = − λ k . t x rescale scalar fields η ( ρ ) = ρη ( ρ ) ˜ σ ( ρ ) = ρσ ( ρ ) ˜ , general solution is now I + α 2 ϕ i II + α 3 ϕ i III + α 4 ϕ i IV ϕ i = α 1 ϕ i QNM = no-source term -> zero determinant � ϕ η I ϕ η II ϕ η III ϕ η IV � � � � � ϕ σ I ϕ σ II ϕ σ III ϕ σ IV � � 0 = � ϕ I ϕ II ϕ III ϕ IV � � � t t t t � ϕ I ϕ II ϕ III ϕ IV � � � x x x x ρ = Λ

Hydro and QNMs Dispersion relation: ω = v s k − i Γ s k 2 2 v s 2 v s 0.8 0.7 0.6 0.6 0.5 0.4 0.4 0.3 0.2 0.2 0.1 T T 0.0 T c 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T c 0.0 0.2 0.4 0.6 0.8 1.0 � s � s 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 T T 0.0 T c 0.0 0.2 0.4 0.6 0.8 1.0 T c 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Hydro and QNMs Pseudo Diffusion ω = − iDk 2 − i γ Ω Ω k k 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 � 0.1 � 0.2 � 0.2 � 0.3 � 0.4 � 0.4 � 0.6 � 0.5 � 0.8 � 0.6 Γ Γ 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 T T 0.0 T c 0.85 0.90 0.95 1.00 T c 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Hydro and QNMs Higher Quasinormal modes 0 � 1 � 2 Im � Ω � � 3 � 4 � 5 � 6 0 2 4 6 8 Re � Ω �

Summary and Outlook Relevant modes of the phase transition unbroken phase: 1 Diffusion mode critical point: 2 massless scalar modes + Diffusion broken phase: 2 modes of sound, Pseudo Diffusion, dynamical scaling z=2 Outlook: study hydro QNMs in the backreacted model (much) more complicated 11 coupled diff eqns two different mechanism of spontaneous symmetry breaking? (2 different QNMs cross the real axes for large and small charges) include fermionic operator