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

Translational Symmetry Breaking in Holographic Zero Sound and Conductivity K.B. Fadafan 1 , A. O’Bannon 2 and M.J. Russell 2 1 Shahrood University of Technology, Iran 2 SHEP Group, University of Southampton 1/23

Gauge/Gravity Duality Applications of Gauge/Gravity Duality QCD/CMP in AdS/CFT 2/23

Gauge/Gravity Duality Applications of Gauge/Gravity Duality QCD/CMP in AdS/CFT Dispersion Relations Holographic Zero Sound TSB in HZS 3/23

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 4/23

Taster In a classic Drude model the conductivity without translational symmetry breaking (TSB) is given by: σ ( ω ) ∝ 1 (1) iω This is remedied by introducing TSB that allows momentum to relax at a timescale τ : 1 σ ( ω ) ∝ (2) 1 − iωτ ( D.Tong) 5/23

Taster ω = v s k − i Γ k 2 + O ( k 3 ) (3) ω ≈ − iv s τk 2 = − iDk 2 (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? 6/23

Dispersion Relations from Green’s Functions R ( ω, k ) = δ 2 logZ δφ a δφ b = δ �O a � d 4 xφ O � G µν Z ⊃ e → (5) δφ b Example: a global U (1) external field A µ we would have: R ( ω, k ) = δ 2 logZ = δ � J µ � d 4 xA µ J µ � G µν Z ⊃ e → (6) δA µ δA ν δA ν ◮ 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. 7/23

Dispersion Relations from Green’s Functions If we calculate this Greens function for a given action under some perturbation we get: 1 G µν R ( ω, k ) ∝ (7) ω − ǫ ( k ) The pole of this Greens Function will give a dispersion relation that might look like: ω = v s k − i Γ k 2 + O ( k 3 ) (8) ω = − iDk 2 + ... (9) ◮ The pole of the Greens function gives a dispersion relation related to some propagating mode. 8/23

Conducitivty from Green’s Functions To relate the conductivity to Greens functions we look at Kubo’s Formula . Conducitivty is defined by: � J x � = σE x (10) Where we can write the electric field in terms of the gauge field: E x = ∂ t δA x → − iωδ ˜ � J x � = − iωσδ ˜ (11) A x i.e. A x But we have: R ( ω, k ) = δ � J x � ˜ G xx = − iωσ (12) δ ˜ A x So the Kubo formula for conducitivty is: σ ( ω ) = Im ˜ G xx R (13) ω 9/23

AdS/CFT and Green’s Functions ⇒ Type IIB string theory on AdS 5 × S 5 N = 4 SY M with SU ( N ) ⇐ Strong ⇐ ⇒ Weak d 4 xφ (0) O ⇐ � ⇒ e − S SUGRA [ φ | ∂ ] e ◮ A field in the bulk sources an operator at the boundary. Therefore we can write the Green’s functions as: R = δ 2 S SUGRA [ φ | ∂ ] → δ 2 S SUGRA [ A µ | ∂ ] G µν (14) δA µ δφ a (0) δφ b (0) δA ν (0) (0) 10/23

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: N f D7 branes in bulk ⇐ ⇒ N = 2 hypermultiplets under U ( N f ) � d 8 ξ � S D 7 = − T − ( det ( g ab + F ab )) (15) Probe Limit: S = S Bulk + S D 7 ≈ S Bulk N f ≪ N Adjoint (1009.5678, N. Evans et al.) 11/23

AdS/CFT and Green’s Functions ◮ Can we get sound modes and conductivity? Yes. R = δ 2 S D 7 [ A µ | ∂ ] G µν (16) δA µ (0) δA ν (0) We get a pole in the Greens Function at low temperature ω = ± vk − i Γ k 2 + O ( k 3 ) (17) ◮ Holographic Zero Sound 12/23

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 ) = n 0 ( p ) + δn ( t, x, p ) (18) dn dt = I ( n ) (19) 13/23

Landau Zero Sound L mfp ≪ λ → Collisions dominate → hydrodynamic sound L mfp ≫ 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 − i Γ k 2 + ... 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 ... 14/23

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 AdS 5 diffeomorphisms along the spatial directions only results in breaking conservation of momentum, but not energy. ◮ This modifies our conserved currents: ∂ a T ai � = 0 ∽ τ − 1 (20) rel 15/23

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) 16/23

Massive Gravity 4 d 4 x √− g � � 1 � R − 2Λ − 1 4 F 2 + ˜ m 2 � S Bulk = c i U i (21) 2 κ 2 i =1 � d 8 ξV [( ∂ψ ) 2 ] � S DBI = − T det ( g ab + W [( ∂ψ ) 2 ] F ab ) (22) � dz 2 ds 2 = 1 � f ( z ) − f ( z ) dt 2 + dx 2 + dy 2 (23) z 2 f ( z ) = 1 + α 1 z + α 2 z 2 − mz 3 + µ 2 z 4 (24) 4 z 2 0 17/23

Holographic Renormalisation As z ( or r ) → 0 we get IR/UV divergences in S DBI : d 3 x 1 � S divergent = −N V (25) 3 ǫ 3 ( D.Tong) 18/23

Holographic Renormalisation As z → 0 we get IR/UV divergences in S DBI : d 3 x 1 � S divergent = −N V (26) 3 ǫ 3 We can write the counter-term in terms of invariants: � 1 α 2 α 2 1 � d 3 x √ γ � 1 � � 1 T z 1 z − S CT = 3 + 12 + [ T ] (27) 2 κ 2 16 α 2 16 α 2 For this model: + α 2 d 3 x √ γ � 1 3 − α 1 U 1 − α 2 U 2 1 U 2 � � S CT = N V (28) 12 12 16 19/23

Thermodynamics TD quantities characterise the model: Ω = − S DBI (29) Small temperature expansion: s = − ∂ Ω ∂T = # T 0 + # T + # T 2 + ... (30) c v = ∂s ∂T = # + # T + ... ◮ The linearity in the heat capacity is reminiscent of a Fermi liquid. 20/23

Translational SB and conductivity The TSB conductivity has already been studied in the models above and gives a result of: σ DC σ ( ω ) ∽ + corrections (31) 1 − iωτ rel (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. 21/23

Outlook As outlined above to compute the holographic sound mode we vary the gauge field on the DBI action and compute the Greens function: A µ ( r ) → A µ ( r ) + a µ ( r, t, x ) (32) R ( ω, k ) = δ 2 S DBI G µν (33) a 0 ,µ a 0 ,µ ◮ The pole of this Greens Function will give the translationally broken holographic sound mode. 22/23

Conclusions and Outlook ◮ Gauge/Gravity duality is useful in probing strongly coupled field theories. ◮ We can calcualte conductivity, sound modes and other aspects of 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. 23/23