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

ds2 = −( r2 L2 − M r )dt2 + dr2

r2 L2 − M r

+ r2 L2 (dx2 + dy2) L = −1 4FµνF µν − m2Ψ¯ Ψ − (∂µΨ − iAµΨ)(∂µ ¯ Ψ + iAµ ¯ Ψ)

The Model

eoms: boundary conditions at Horizon: why? bulk current finite norm at the Horizon values at boundary

Ψ′′ + (f ′ f + 2 ρ)Ψ′ + Φ2 f 2 Ψ + 2 L2f Ψ =0 Φ′′ + 2 ρΦ′ − 2Ψ2 f Φ =0 Φ(ρH) = 0 , Ψ(ρH) Jµ = ψ2Aµ Φ = ¯ µ − ¯ n ρ + O( 1 ρ2 ) Ψ = ψ1 ρ + ψ2 ρ2 + O( 1 ρ2 )

¯ µ = 3L 4πT µ , ¯ n = 9L 16π2T 2 n , ψ1 = 3 4πTL2 O1 , ψ2 = 9 16π2T 2L4 O2 ,

The Model

solve eom with either or

ψ1 = 0 ψ2 = 0

0.0 0.2 0.4 0.6 0.8 1.0 T Tc 1000 2000 3000 4000 O22 Tc

4

0.0 0.2 0.4 0.6 0.8 1.0 T Tc 100 200 300 400 500 O12 Tc

2

Oi2 ∝

• 1 − T

Tc

Hydrodynamics

Hydrodynamics = slow modes conservation law constitutive relation with external source taking time derivative and using the continuity eqn ∂n ∂t + ∇ j = 0

• j = −D

∇n + σ E

jL = iσω2 ω + iD k2 AL

σ = −i ω jLjLk=0

lim

k→0 ω(k) = 0

Hydrodynamics

broken phase: take Goldstone mode into account (Chaikin, Lubensky) generic prediction: appearance of sound modes predicts correlator and conductivity including dissipation: ˆ D = v2

S

jLjL = ˆ σω2 ω2 − ˆ D k2 σ(ω) = −i ω + iǫ ˆ σ = −iP 1 ω

• + πˆ

σδ(ω) ˆ σ = nS

ω = ±vsk − iΓsk2

Hydro and QNMs

poles of retarded Green functions = Quasinormal Modes “Eigenmodes” Horizon: infalling Boundary: Pole of holographic GF complex scalar field theory I theory II complex frequencies

ΨB = A ρ + B ρ2 + O 1 ρ3

• O1 ¯

O1 = A B O2 ¯ O2 = B A ΨH = (ρ − 1)−iω/3(1 + O(ρ − 1)) Ψ ∝ e−iωRte−ωIt

Hydro and QNMs

Unbroken phase: superconducting Instability

3 2 1 1 2 3 5 4 3 2 1 ReΩ ImΩ

O1Theory

3 2 1 1 2 3 6 5 4 3 2 1 ReΩ ImΩ

O2 Theory

Vector channel: Diffusion mode D =

3 4πT

Hydro and QNMs

Broken phase: Second sound and Pseudodiffusion

= fη′′ +

• f ′ + 2f

ρ

• η′ +

φ2 f + 2 L2 + ω2 f − k2 ρ2

• η − 2iωφ

f σ − iωψ f at − ikψ r2 ax , = fσ′′ +

• f ′ + 2f

ρ

• σ′ +

φ2 f + 2 L2 + ω2 f − k2 ρ2

• σ + 2φψ

f at + 2iωφ f η , = fat

′′ + 2f

ρ at

′ −

k2 ρ2 + 2ψ2

• at − ωk

ρ2 ax − 2iωψ η − 4ψφ σ , = fax

′′ + f ′ax ′ +

ω2 f − 2ψ2

• ax + ωk

f at + 2ikψ η . ω f at

′ + k

ρ2 ax

′ = 2i (ψ′ η − ψ η′)

constraint: local ward identity:

∂µjµ = 2Oiηi

Hydro and QNMs

How to compute QNMs of coupled system four l.i. solutions (one is pure gauge) rescale scalar fields general solution is now QNM = no-source term -> zero determinant

ηIV = iλψ , σIV = 0 , aIV

t

= λω , aIV

x

= −λk . ˜ η(ρ) = ρη(ρ) , ˜ σ(ρ) = ρσ(ρ) ϕi = α1ϕi

I + α2ϕi II + α3ϕi III + α4ϕi IV

=

• ϕηI

ϕηII ϕηIII ϕηIV ϕσI ϕσII ϕσIII ϕσIV ϕI

t

ϕII

t

ϕIII

t

ϕIV

t

ϕI

x

ϕII

x

ϕIII

x

ϕIV

x

• ρ=Λ

Hydro and QNMs

Dispersion relation:

ω = vsk − iΓsk2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T Tc 0.2 0.4 0.6 0.8 vs

2

0.0 0.2 0.4 0.6 0.8 1.0 T Tc 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 vs

2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T Tc 0.5 1.0 1.5 2.0 2.5 s 0.0 0.2 0.4 0.6 0.8 1.0 T Tc 0.0 0.5 1.0 1.5 2.0 s

Hydro and QNMs

Pseudo Diffusion

ω = −iDk2 − iγ

0.1 0.2 0.3 0.4 0.5 k 0.8 0.6 0.4 0.2 Ω 0.1 0.2 0.3 0.4 0.5 k 0.6 0.5 0.4 0.3 0.2 0.1 Ω 0.85 0.90 0.95 1.00 T Tc 0.5 1.0 1.5 2.0 2.5 3.0 Γ 0.70 0.75 0.80 0.85 0.90 0.95 1.00 T Tc 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Γ

Hydro and QNMs

Higher Quasinormal modes

2 4 6 8 6 5 4 3 2 1 ReΩ ImΩ

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