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

Jµ = ρuµ + κ T (E µ − TPµν∂ν µ T ) + ˜ κωωµ + ˜ κBBµ

• Anomaly

∂µJµ = −c 8ǫµνρσFµνFρσ

• Chiral conductivities

˜ κω = c

• µ2 − 2

3 ρ ǫ + P µ3

• ,

˜ κB = c

• µ − 1

2 ρ ǫ + P µ2

• Entropy current

Jµ

s = suµ − µ

T (Jµ − ρuµ) + σωωµ + σBBµ σω = c µ3 3T , σB = c µ2 2T

[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: ξµ = −∂µφ + Aµ
• Charge current:

Jµ =

• parity

preserving terms

• + ˜

κωωµ + ˜ κBBµ

• Entropy current:

Jµ

s =

• parity

preserving terms

• +
• σω − µ

T ˜ κω

• ωµ+
• σB − µ

T ˜ κB

• Bµ
• Only constraint

1 2σω − µσB = −c µ3 3T

[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

˜ κω = 0 ˜ κB = c 3µ σω = 0 σB = µ T ˜ κB = c µ2 3T

Introduction Holographic superfluids Low Temperature Numerical results Conclusions

Anomalous holographic superfluid

• Minimal superfluid model in 3+1 dim:
• Einstein-Maxwell-Higgs on asymptotically AdS5
• Abelian gauge field AM 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 = SEH + Smatter + SCS

SEH = 1 2κ2

• d5x√−g (R + 12)

Smatter = 1 2κ2

• d5x√−g
• − 1

4 VF (|ψ|)FMNF MN − Vψ(|ψ|)(DMψ)(DMψ)∗ − V (|ψ|)

• SCS = c

24

• d5x√−gǫMNPQRAMFNPFQR
• DM = ∂M − iqAM
• VF(0) = Vψ(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

ds2 = −r 2f (r)dt2 + r 2d x2 + 2h(r)dtdr , ψ = ̺(r)eiqϕ(r) AM = (A0(r), 0, 0, 0, A4(r)) , GM = AM − ∂Mϕ

• Thermodynamic properties from asymptotics

T = r 2

h f ′(rh)

4πh(rh) , s = 2πr 3

h

κ2 f = 1 − 2κ2P r 4 + O(r −5) , h = 1 − ∆C 2

∆| Oψ |2

6r 2∆ + O(r −2∆−2) ̺ = C∆| Oψ | r ∆ + O(r ∆−2) , G0 = µ − κ2ρt r 2 + O(r −3)

• Noether charge and Gibbs Duhem relation

Q1 = r 5f ′ − r 3VFG0G ′ 2κ2h = sT = 4P − µρt

Introduction Holographic superfluids Low Temperature Numerical results Conclusions

Small superfluid velocity

• Linear perturbations

Gi = −g(r)∂iφ gti = −r 2γ(r)∂iφ

• Asymptotics

γ = 1 2 (ρt − ρ)κ2 r 4 + O(r −5) , g = 1 − (ρt − ρ)κ2 µr 2 + O(r −3)

• Conserved charges

Q2 = r 5γ′ + r 3VFgG ′ 2κ2h = ρ f Q2 = γQ1 + fr 3VF(gG ′

0 − g ′G0)

2κ2h

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

˜ κB = c ∞

rh

g 2G ′

0 + R(G0 − gµ)gG ′ 0dr

˜ κω = −2c ∞

rh

(G0 − µg)gG ′

0 + R(G0 − µg)2G ′ 0dr

σB = c T ∞

rh

gG0G ′

0dr

σω = −2c T ∞

rh

(G0 − µg)G0G ′

0dr

with R = ρ 4P − µ(ρt − ρ)

Introduction Holographic superfluids Low Temperature Numerical results Conclusions

• For T > Tc: no condensate ⇒ ψ = 0 and ρt = ρ
• RN background + g = 1, γ = 0 ⇒ Exact chiral conductivities:

˜ κω = c

• µ2 − ρ

6P µ3 ˜ κB = c

• µ − ρ

8P µ2 σω = c µ3 3T σB = c µ2 2T

• For T < Tc: condensate ⇒ ρt > 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 Q1)

µg G0 = 1 + µ ∞

r

2κ2Q2h VF(ψ)G 2

0 r ′3 dr ′ + O(Q1)

• Finite g/G0 at horizon implies Q2 → 0 ⇒ g = G0/µ
• Zero temperature chiral conductivities

˜ κω = 0 ˜ κB = c 3µ σω = 0 σB = µ T ˜ κB = c µ2 3T

• 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
• UV asymptotic AdS:

ds2 = r 2(−dt2 + d x2) + 2dtdr

• 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 = ψIR minimum of V ⇒ IR AdS solution:

ds2 = r 2 L2

IR

(−f0dt2 + d x2) + 2 √f0 LIR dtdr G0 = p0r ∆G −3 , g = g0r ∆G −3 , γ = 0 ∆G = 2 +

• 1 + 2q2|ψIR|2Vψ(ψIR)

VF(ψIR) L2

IR

• AdS to AdS stable if current operator irrelevant, i.e. ∆G > 4
• g/G0 finite ⇒ normal charge density has to vanish

Introduction Holographic superfluids Low Temperature Numerical results Conclusions

AdS to Lifshitz domain wall

• IR: ψ = ψ0 ⇒ IR Lifshitz solution:

ds2 = −zp2

0VF(ψ0)

2(z − 1) r 2zdt2 + r 2d x2 + √ 3zp0VF(ψ0) qψ0

• (z − 1)Vψ(ψ0)

r z−1drdt G0 = p0r z , g = g0r z , γ = −zg0p0VF(ψ0) 2(z − 1) r 2z−2

• z is fixed given the couplings and potential: Vψ, VF and V
• Reality of the solution demands z > 1
• again g/G0 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|) = m2|ψ|2 + u 2 |ψ|4 Vψ = 1 , VF = 1 with m2 < 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:

ψIR =

• −m2

u LIR =

• 24u

m4 + 24u

• Lifshitz fixed point:

ψ0 =

• −m2

u + 2q2(z − 1) zu u 2 ψ4

0 +m2 + 2q2(9 + z(2 + z))

3z −12 = 0

• Might be that both solutions are possible ⇒ Stability

Introduction Holographic superfluids Low Temperature Numerical results Conclusions

AdS to AdS: m2 = −15/4, q = 2 and u = 6

• 6
• 4
• 2

2 Log@rêmD 0.95 1.00 1.05

f h-1

T=0TC T=0.01TC T=0.1TC T=0.5TC T=1.TC

Introduction Holographic superfluids Low Temperature Numerical results Conclusions

AdS to AdS: m2 = −15/4, q = 2 and u = 6

0.05 0.10 0.15

Têm

0.2 0.4 0.6 0.8

k é

Bêm, k

é

wêm2

k é

Bêm

k é

wêm2

0.331 1ê3 0.335

Introduction Holographic superfluids Low Temperature Numerical results Conclusions

AdS to Lifshitz: m2 = −15/4, q = 3/2 and u = 7

• 6
• 4
• 2

2 Log@rêmD 0.4 0.6 0.8 1.0

f h-1

T=0TC T=2.¥ 10-8TC T=2.¥ 10-5TC T=1.¥ 10-2TC T=6.¥ 10-1TC

Introduction Holographic superfluids Low Temperature Numerical results Conclusions

AdS to AdS: m2 = −15/4, q = 3/2 and u = 7

k é

Bêm

k é

wêm2

1ê3 0.35 0.02 0.04 0.06 0.08 0.10

Têm

0.2 0.4 0.6 0.8

k é

Bêm, k

é

wêm2

Convengerce gets worse the larger the z

Introduction Holographic superfluids Low Temperature Numerical results Conclusions

Conclusions and outlook

• At T → 0: the chiral conductivities are universal ⇐ ρ → 0
• The chiral vortical parameters vanish: no normal component to support

the vorticity

• General validity ?
• Other dimensions
• Other anomalies
• Other parity breaking effects
• Other ground states
• Tip from C. Hoyos:

possible to fix the coefficients in effective action in [Chapman et al.] Thanks