Universal response in anomalous cold holographic superfluids Irene - PowerPoint PPT Presentation

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Universal response in anomalous cold holographic superfluids Irene Amado Technion, Haifa University of Crete, March 27, 2014 Based on collaboration with Amos Yarom and Nir Lisker arXiv:1401.5795

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Goal: understand the role of anomalies in superfluids • In particular: response to magnetic field and vorticity • Quantum anomalies ⇒ Chiral Magnetic and Chiral Vortical Effects • Normal fluids: transport fixed by anomalies • Superfluids: generically unconstrained ⇐ extra d.o.f.’s • Can we make any prediction for the superfluid case?

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Outline Introduction Holographic superfluids Low Temperature Numerical results Conclusions

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Anomalous normal fluid in 3+1 • Charge current in Landau frame J µ = ρ u µ + κ T ( E µ − TP µν ∂ ν µ κ ω ω µ + ˜ κ B B µ T ) + ˜ • Anomaly ∂ µ J µ = − c 8 ǫ µνρσ F µν F ρσ • Chiral conductivities � µ 2 − 2 ρ � � µ − 1 ρ � ǫ + P µ 3 ǫ + P µ 2 κ ω = c ˜ , κ B = c ˜ 3 2 • Entropy current s = su µ − µ T ( J µ − ρ u µ ) + σ ω ω µ + σ B B µ J µ σ ω = c µ 3 σ B = c µ 2 3 T , 2 T [Banerjee et al., Son et al., Bhattacharya et al., ...]

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Anomalous superfluid • Extra hydro dof from Goldstone boson: ξ µ = − ∂ µ φ + A µ • Charge current: J µ = κ ω ω µ + ˜ � � κ B B µ + ˜ parity preserving terms • Entropy current: σ ω − µ σ B − µ � � � � � � J µ ω µ + B µ s = parity preserving terms + T ˜ κ ω T ˜ κ B • Only constraint 2 σ ω − µσ B = − c µ 3 1 3 T [Bhattacharya et al., Chapman et al.]

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Here: • If parity broken only by an anomaly • Generic isotropic holographic superfluids • T → 0 ⇒ Universal chiral response: fixed by the anomaly κ B = c µ 2 κ B = c σ B = µ ˜ κ ω = 0 ˜ 3 µ σ ω = 0 T ˜ 3 T

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Anomalous holographic superfluid • Minimal superfluid model in 3+1 dim: • Einstein-Maxwell-Higgs on asymptotically AdS 5 • Abelian gauge field A M and charged scalar field ψ • SSB of U(1) by condensation of the charged scalar • U(1) 3 anomaly: parity odd topological term

Introduction Holographic superfluids Low Temperature Numerical results Conclusions S = S EH + S matter + S CS 1 d 5 x √− g ( R + 12) � S EH = 2 κ 2 d 5 x √− g 1 � � − 1 � 4 V F ( | ψ | ) F MN F MN − V ψ ( | ψ | )( D M ψ )( D M ψ ) ∗ − V ( | ψ | ) S matter = 2 κ 2 d 5 x √− g ǫ MNPQR A M F NP F QR S CS = c � 24 • D M = ∂ M − iqA M • V F (0) = V ψ (0) = 1 and V (0) = 0 • c ≡ anomaly strenght [Bhattacharya et al. 1105.3733]

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Stationary superfluid • Ansatz ds 2 = − r 2 f ( r ) dt 2 + r 2 d � x 2 + 2 h ( r ) dtdr , ψ = ̺ ( r ) e iq ϕ ( r ) A M = ( A 0 ( r ) , 0 , 0 , 0 , A 4 ( r )) , G M = A M − ∂ M ϕ • Thermodynamic properties from asymptotics T = r 2 s = 2 π r 3 h f ′ ( r h ) h 4 π h ( r h ) , κ 2 f = 1 − 2 κ 2 P h = 1 − ∆ C 2 ∆ | � O ψ � | 2 + O ( r − 5 ) , + O ( r − 2∆ − 2 ) r 4 6 r 2∆ G 0 = µ − κ 2 ρ t ̺ = C ∆ | � O ψ � | + O ( r ∆ − 2 ) , + O ( r − 3 ) r ∆ r 2 • Noether charge and Gibbs Duhem relation Q 1 = r 5 f ′ − r 3 V F G 0 G ′ 0 = sT = 4 P − µρ t 2 κ 2 h

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Small superfluid velocity • Linear perturbations g ti = − r 2 γ ( r ) ∂ i φ G i = − g ( r ) ∂ i φ • Asymptotics ( ρ t − ρ ) κ 2 g = 1 − ( ρ t − ρ ) κ 2 γ = 1 + O ( r − 5 ) , + O ( r − 3 ) 2 r 4 µ r 2 • Conserved charges Q 2 = r 5 γ ′ + r 3 V F gG ′ 0 = ρ 2 κ 2 h f Q 2 = γ Q 1 + fr 3 V F ( gG ′ 0 − g ′ G 0 ) 2 κ 2 h

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Transport coefficients from fluid-gravity correspondence • Boost solution ⇒ constant normal component velocity • Allow for space-time dependence of thermo variables • Add corrections to metric and matter fields to satisfy eoms • Solve order by order in gradient expansion of thermo variables • Apply AdS/CFT dictionary to compute the (covariant) current • Read off the transport coefficients

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Chiral conductivities � ∞ g 2 G ′ κ B = c ˜ 0 + R ( G 0 − g µ ) gG ′ 0 dr r h � ∞ 0 + R ( G 0 − µ g ) 2 G ′ ( G 0 − µ g ) gG ′ κ ω = − 2 c ˜ 0 dr r h � ∞ σ B = c gG 0 G ′ 0 dr T r h � ∞ σ ω = − 2 c ( G 0 − µ g ) G 0 G ′ 0 dr T r h with ρ R = 4 P − µ ( ρ t − ρ )

Introduction Holographic superfluids Low Temperature Numerical results Conclusions • For T > T c : no condensate ⇒ ψ = 0 and ρ t = ρ • RN background + g = 1, γ = 0 ⇒ Exact chiral conductivities: µ 2 − ρ µ − ρ � 6 P µ 3 � � 8 P µ 2 � ˜ κ ω = c ˜ κ B = c σ ω = c µ 3 σ B = c µ 2 3 T 2 T • For T < T c : condensate ⇒ ρ t > ρ ⇒ In general ˜ κ , σ model dependent • But! for T → 0 : ρ → 0 ⇒ Universal ˜ κ , σ

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Low temperature • At low T (small Q 1 ) � ∞ 2 κ 2 Q 2 h µ g 0 r ′ 3 dr ′ + O ( Q 1 ) G 0 = 1 + µ V F ( ψ ) G 2 r • Finite g / G 0 at horizon implies Q 2 → 0 ⇒ g = G 0 /µ • Zero temperature chiral conductivities κ B = c µ 2 κ B = c σ B = µ ˜ κ ω = 0 ˜ 3 µ σ ω = 0 T ˜ 3 T • Zero temperature ≡ zero normal charge density ρ = 0

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Ground state of isotropic superfluids following [Gubser-Nellore, Horowitz-Roberts] • Zero temperature limit of the BH dual to the condensed phase ? • Domain wall between aAdS in the UV and an IR stationary configuration ds 2 = r 2 ( − dt 2 + d � x 2 ) + 2 dtdr • UV asymptotic AdS: • Two different possible IR emergent behaviors • If ψ sits at a minimum of V ( ψ ) ⇒ AdS geometry • If ψ sits at a different constant value ⇒ Lifshitz like geometry

Introduction Holographic superfluids Low Temperature Numerical results Conclusions AdS to AdS domain wall • IR: ψ = ψ IR minimum of V ⇒ IR AdS solution: √ f 0 ds 2 = r 2 ( − f 0 dt 2 + d � x 2 ) + 2 L IR dtdr L 2 IR G 0 = p 0 r ∆ G − 3 , g = g 0 r ∆ G − 3 , γ = 0 � 1 + 2 q 2 | ψ IR | 2 V ψ ( ψ IR ) ∆ G = 2 + L 2 IR V F ( ψ IR ) • AdS to AdS stable if current operator irrelevant, i.e. ∆ G > 4 • g / G 0 finite ⇒ normal charge density has to vanish

Introduction Holographic superfluids Low Temperature Numerical results Conclusions AdS to Lifshitz domain wall • IR: ψ = ψ 0 ⇒ IR Lifshitz solution: √ ds 2 = − zp 2 0 V F ( ψ 0 ) 3 zp 0 V F ( ψ 0 ) 2( z − 1) r 2 z dt 2 + r 2 d � x 2 + r z − 1 drdt � q ψ 0 ( z − 1) V ψ ( ψ 0 ) γ = − zg 0 p 0 V F ( ψ 0 ) G 0 = p 0 r z , g = g 0 r z , r 2 z − 2 2( z − 1) • z is fixed given the couplings and potential: V ψ , V F and V • Reality of the solution demands z > 1 • again g / G 0 finite ⇒ normal charge density vanishes

Introduction Holographic superfluids Low Temperature Numerical results Conclusions Numerics • Study temperature dependence of ˜ κ ’s and σ ’s • Construct explicit AdS to AdS and AdS to Lifshitz DW. • Choose particular potential and couplings V ( | ψ | ) = m 2 | ψ | 2 + u 2 | ψ | 4 V ψ = 1 , V F = 1 with m 2 < 0 and u > 0. • Admits both AdS and Lifshitz ground states depending on { q , m , u }

Introduction Holographic superfluids Low Temperature Numerical results Conclusions • Conformal fixed point: � � − m 2 24 u ψ IR = L IR = m 4 + 24 u u • Lifshitz fixed point: � 0 + m 2 + 2 q 2 (9 + z (2 + z )) − m 2 u + 2 q 2 ( z − 1) u 2 ψ 4 ψ 0 = − 12 = 0 zu 3 z • Might be that both solutions are possible ⇒ Stability

Introduction Holographic superfluids Low Temperature Numerical results Conclusions AdS to AdS: m 2 = − 15 / 4, q = 2 and u = 6 f h - 1 T = 0 T C 1.05 1.00 0.95 T = 0.01 T C T = 0.1 T C T = 0.5 T C T = 1. T C 2 Log @ r ê m D - 6 - 4 - 2 0

Introduction Holographic superfluids Low Temperature Numerical results Conclusions AdS to AdS: m 2 = − 15 / 4, q = 2 and u = 6 é é w ê m 2 B ê m , k k 0.8 é B ê m 0.335 k 1 ê 3 0.6 é w ê m 2 0.331 k 0.4 0.2 T ê m 0.05 0.10 0.15

Introduction Holographic superfluids Low Temperature Numerical results Conclusions AdS to Lifshitz: m 2 = − 15 / 4, q = 3 / 2 and u = 7 f h - 1 1.0 0.8 T = 0 T C 0.6 T = 2. ¥ 10 - 8 T C T = 2. ¥ 10 - 5 T C 0.4 T = 1. ¥ 10 - 2 T C T = 6. ¥ 10 - 1 T C 2 Log @ r ê m D - 6 - 4 - 2 0