Translational Symmetry Breaking in Holographic Zero Sound and - - PowerPoint PPT Presentation

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Translational Symmetry Breaking in Holographic Zero Sound and Conductivity

K.B. Fadafan1, A. O’Bannon2 and M.J. Russell2

1Shahrood University of Technology, Iran 2SHEP Group, University of Southampton

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Gauge/Gravity Duality Applications of Gauge/Gravity Duality QCD/CMP in AdS/CFT

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Gauge/Gravity Duality Applications of Gauge/Gravity Duality QCD/CMP in AdS/CFT Dispersion Relations Holographic Zero Sound TSB in HZS

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Outline

Taster Sound Modes and Conducitivty from Green’s Functions AdS/CFT and Green’s Functions Landau Zero Sound Translational SB in AdS/CFT Massive Gravity Conclusions

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Taster

In a classic Drude model the conductivity without translational symmetry breaking (TSB) is given by: σ(ω) ∝ 1 iω (1) This is remedied by introducing TSB that allows momentum to relax at a timescale τ: σ(ω) ∝ 1 1 − iωτ (2)

( D.Tong)

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Taster

ω = vsk − iΓk2 + O(k3) (3) ω ≈ −ivsτk2 = −iDk2 (4)

(1710.08425, A. Lucas, K. Fong) ◮ TSB is important for sound modes and conductivity. ◮ What is the effect of TSB for dispersion relations and conductivity

in holography?

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Dispersion Relations from Green’s Functions

Z ⊃ e

• d4xφO

→ Gµν

R (ω, k) = δ2logZ

δφaδφb = δOa δφb (5) Example: a global U(1) external field Aµ we would have: Z ⊃ e

• d4xAµJµ

→ Gµν

R (ω, k) = δ2logZ

δAµδAν = δJµ δAν (6)

◮ If we have some sort of propagating mode via the perturbation, it

isn’t too surprising to expect it to show up in the Greens function.

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Dispersion Relations from Green’s Functions

If we calculate this Greens function for a given action under some perturbation we get: Gµν

R (ω, k) ∝

1 ω − ǫ(k) (7) The pole of this Greens Function will give a dispersion relation that might look like: ω = vsk − iΓk2 + O(k3) (8) ω = −iDk2 + ... (9)

◮ The pole of the Greens function gives a dispersion relation

related to some propagating mode.

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Conducitivty from Green’s Functions

To relate the conductivity to Greens functions we look at Kubo’s

• Formula. Conducitivty is defined by:

Jx = σEx (10) Where we can write the electric field in terms of the gauge field: Ex = ∂tδAx → −iωδ ˜ Ax i.e. Jx = −iωσδ ˜ Ax (11) But we have: ˜ Gxx

R (ω, k) = δJx

δ ˜ Ax = −iωσ (12) So the Kubo formula for conducitivty is: σ(ω) = Im ˜ Gxx

R

ω (13)

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AdS/CFT and Green’s Functions

N = 4 SY M with SU(N) ⇐ ⇒ Type IIB string theory on AdS5 × S5 Strong ⇐ ⇒ Weak e

• d4xφ(0)O ⇐

⇒ e−SSUGRA[φ|∂]

◮ A field in the bulk sources an operator at the boundary. Therefore

we can write the Green’s functions as: Gµν

R = δ2SSUGRA[φ|∂]

δφa

(0)δφb (0)

→ δ2SSUGRA[Aµ|∂] δAµ

(0)δAν (0)

(14)

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The D3/D7 Probe Brane Model

So far we only have adjoint degrees of freedom on both sides. To introduce fundamental, i.e quarks, we do the following: Nf D7 branes in bulk ⇐ ⇒ N = 2 hypermultiplets under U(Nf) SD7 = −T

• d8ξ
• −(det(gab + Fab))

(15)

(1009.5678, N. Evans et al.)

Probe Limit: S = SBulk + SD7 ≈ SBulk Nf ≪ NAdjoint

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AdS/CFT and Green’s Functions

◮ Can we get sound modes and conductivity? Yes.

Gµν

R = δ2SD7[Aµ|∂]

δAµ

(0)δAν (0)

(16) We get a pole in the Greens Function at low temperature ω = ±vk − iΓk2 + O(k3) (17)

◮ Holographic Zero Sound

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Landau Zero Sound

How do we describe a system of interacting fermions? Take help from non-interacting Fermi gas: Ground state non-interacting = ⇒ Ground state interacting Excitation = ⇒ Quasiparticle Fermi surface of particles = ⇒ Fermi surface of quasiparticles

◮ What type of collective excitations do we get from this

quasiparticle description? n(t, x, p) = n0(p) + δn(t, x, p) (18) dn dt = I(n) (19)

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Landau Zero Sound

Lmfp ≪ λ → Collisions dominate → hydrodynamic sound Lmfp ≫ λ → QP interactions dominate → zero sound = ⇒ Zero sound is a low temperature collective vibration of the quasiparticle ground state around the Fermi surface with dispersion relationship ω = vk − iΓk2 + ... But In AdS/CFT we cannot rely on a Fermi surface argument. HZS is not well described by LFL nor like a hydrodynamic sound - it is something ...

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Translational SB in AdS/CFT

◮ So how do we go about incorporating TSB of the CFT? Well, lets

look at the symmetries of the theories: Superconformal of SYM ⇐ ⇒ Isometry of gravity theory SO(4, 2) + SO(6) ⇐ ⇒ SO(4, 2) + SO(6)

◮ Example: Breaking the AdS5 diffeomorphisms along the spatial

directions only results in breaking conservation of momentum, but not energy.

◮ This modifies our conserved currents:

∂aT ai = 0 ∽ τ −1

rel

(20)

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Translational SB in AdS/CFT

How to incorporate into a model?

◮ Add a potential to bulk action that gives the graviton a mass and

breaks diffeomorphisms. (1301.0537, D. Vegh)

◮ Introduce spatially dependant scalar sources into action. These

will then couple to a operator in the CFT and give a non-zero contribution to the Ward identity. (1311.5157, T. Andrade, B. Withers)

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Massive Gravity

SBulk = 1 2κ2

• d4x√−g
• R − 2Λ − 1

4F2 + ˜ m2

4

• i=1

ciUi

• (21)

SDBI = −T

• d8ξV [(∂ψ)2]
• det(gab + W[(∂ψ)2]Fab)

(22) ds2 = 1 z2 dz2 f(z) − f(z)dt2 + dx2 + dy2

• (23)

f(z) = 1 + α1z + α2z2 − mz3 + µ2z4 4z2 (24)

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Holographic Renormalisation

As z(or r) → 0 we get IR/UV divergences in SDBI: Sdivergent = −NV

• d3x 1

3ǫ3 (25)

( D.Tong)

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Holographic Renormalisation

As z → 0 we get IR/UV divergences in SDBI: Sdivergent = −NV

• d3x 1

3ǫ3 (26) We can write the counter-term in terms of invariants: SCT = 1 2κ2

• d3x√γ

1 3 + 1 12 + α2

1

16α2

• T z

z −

α2

1

16α2 [T]

• (27)

For this model: SCT = NV

• d3x√γ

1 3 − α1U1 12 − α2U2 12 + α2

1U2

16

• (28)

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Thermodynamics

TD quantities characterise the model: Ω = −SDBI (29) Small temperature expansion: s = −∂Ω ∂T = #T 0 + #T + #T 2 + ... cv = ∂s ∂T = # + #T + ... (30)

◮ The linearity in the heat capacity is reminiscent of a Fermi liquid.

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Translational SB and conductivity

The TSB conductivity has already been studied in the models above and gives a result of: σ(ω) ∽ σDC 1 − iωτrel + corrections (31)

(1306.5792, R. Davison) ◮ These corrections are in terms of the additional parameters

added by the TSB terms above and vanish when TS is restored.

◮ The effect of the corrections is to pull spectral weight away from

the Drude peak to higher frequency.

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Outlook

As outlined above to compute the holographic sound mode we vary the gauge field on the DBI action and compute the Greens function: Aµ(r) → Aµ(r) + aµ(r, t, x) (32) Gµν

R (ω, k) = δ2SDBI

a0,µa0,µ (33)

◮ The pole of this Greens Function will give the translationally

broken holographic sound mode.

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Conclusions and Outlook

◮ Gauge/Gravity duality is useful in probing strongly coupled field

theories.

◮ We can calcualte conductivity, sound modes and other aspects

• f CMP using the duality via Greens functions.

◮ These calculations could be pivatol in understanding how

systems behave at strong coupling.

◮ This project will focus on the response of HZS when TS is broken. ◮ Can also look at what happens to plasmons under probe TSB.