Zero Temperature Dissipation & Holography

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1 Zero Temperature Dissipation & Holography Pinaki Banerjee National Strings Meeting PB & B. Sathiapalan (on going)

2 Outline 1 Introduction 2 Langevin Dynamics From Holography 3 Dissipation 4 Conclusions

3 Introduction Motivation AdS/CFT has been serving as a theoretician s best tool in studying strongly coupled systems analytically. Its predictions are mostly qualitative in nature, but they can be quantitative too (e.g, η s = 1 4π ). The duality has glued many phenomena appearing in apparently different branches of physics together. Studying Brownian motion of a heavy particle using classical gravity technique is one such example.

4 Introduction Langevin Dynamics The Langevin equation M d 2 x dt + γ dx = ξ(t) 2 dt (1) with ξ(t)ξ(t ) = Γ δ(t t ) (2)

5 Introduction Langevin Dynamics The Langevin equation M d 2 x dt + γ dx = ξ(t) 2 dt (1) with ξ(t)ξ(t ) = Γ δ(t t ) (2) The generalized Langevin equation for a heavy particle under noise ξ d 2 x(t) t M 0 + dt G dt 2 R (t, t )x(t ) = ξ(t) ξ(t)ξ(t ) = ig sym(t, t ) (3)

6 Introduction Langevin Dynamics The Langevin equation M d 2 x dt + γ dx = ξ(t) 2 dt (1) with ξ(t)ξ(t ) = Γ δ(t t ) (2) The generalized Langevin equation for a heavy particle under noise ξ [ ] M 0ω 2 + G R (ω) x(ω) = ξ(ω) ξ( ω)ξ(ω) = ig sym(ω) (3)

7 Introduction Langevin Dynamics The Langevin equation M d 2 x dt + γ dx = ξ(t) 2 dt (1) with ξ(t)ξ(t ) = Γ δ(t t ) (2) The generalized Langevin equation for a heavy particle under noise ξ [ ] M 0ω 2 + G R (ω) x(ω) = ξ(ω) ξ( ω)ξ(ω) = ig sym(ω) (3) Expanding G R (ω) for small frequencies G R (ω) = Mω 2 iγω +...

8 Introduction Langevin Dynamics The Langevin equation M d 2 x dt + γ dx = ξ(t) 2 dt (1) with ξ(t)ξ(t ) = Γ δ(t t ) (2) The generalized Langevin equation for a heavy particle under noise ξ [ ] M 0ω 2 + G R (ω) x(ω) = ξ(ω) ξ( ω)ξ(ω) = ig sym(ω) (3) Expanding G R (ω) for small frequencies G R (ω) = Mω 2 iγω +... Fluctuation-Dissipation relation ig sym(ω) = (1 + 2n B ) Im G R (ω) (4)

9 Introduction Retarded Green s Function For small frequency G R (ω) = iγ ω M ω 2 iρ ω Dimensional analysis G R (ω) [M] 3 { γ T 2 M T ρ T 0 At zero temperature : G R (ω) = iρ ω 3 Dissipative!

10 Langevin Dynamics From Holography Idea & Set-up Boundary Brownian Particle Horizon J. de Boer et al; Son & Teaney (2009)

11 Langevin Dynamics From Holography Retarded Green s function : Holographic prescription Strongly coupled field theory Weakly coupled gravity. ( ) exp φ i 0O i = Z QG (φ i 0) (5) S d CFT

12 Langevin Dynamics From Holography Retarded Green s function : Holographic prescription Strongly coupled field theory Weakly coupled gravity. ( ) exp φ i 0O i = Z QG (φ i 0) (5) S d Real time retarded Green s function for scalar field theory can be obtained by choosing ingoing boundary condition at the horizon. G R (k) = K gg rr f k (r) r f k (r) (6) Boundary CFT Son & Starinets (2002)

13 Langevin Dynamics From Holography Retarded Green s function : Holographic prescription Strongly coupled field theory Weakly coupled gravity. ( ) exp φ i 0O i = Z QG (φ i 0) S d Real time retarded Green s function for our case can be obtained by choosing ingoing boundary condition at the horizon. G R (ω) = T 0 (r)f ω (r) r f ω (r) (7) Boundary CFT

14 Langevin Dynamics From Holography Computing G R (ω) : 4 Steps 1 Solve the EOM for the string in that non-trivial background f ω(r) = C 1f (1) ω (r) + C 2f (2) ω (r) (8) 2 Impose ingoing wave boundary condition at the horizon f R ω (r) := C 1f (1) ω (r) + C 2f (2) ω (r) (9) 3 Properly normalize it at the boundary F R ω (r) := f R ω (r) f R ω (r B ) (10) 4 Compute the boundary action and take functional derivative w.r.t source to obtain the retarded Green s function G 0 R(ω).

15 Langevin Dynamics From Holography An Example : Brownian Motion in 1+1 Dimensions String EOM String in BTZ f ω (r) + 2(2r 2 4π 2 T 2 L 4 ) r(r 2 4π 2 T 2 L 4 ) f ω(r) L 4 ω 2 + fω(r) = 0 (11) (r 2 4π 2 T 2 L 4 ) 2 Two independent solutions f ω(r) = C 1 P iω 2πT r 1 ( ) 2πTL 2 r + C 2 Q iω 2πT r 1 ( ) 2πTL 2 r (12) Ingoing & normalized solution F R ω (r) = iω P 2πT 1 ( r 2πTL 2 ) r iω P 2πT r 1 ( B 2πTL 2 ) r B (13)

16 Langevin Dynamics From Holography An Example : Brownian Motion in 1+1 Dimensions String in BTZ (cont.) Retarded propagator can be read off from the on-shell action G R (ω) GR(ω) 0 + µ ω2 2π = µ ω (ω 2 + 4π 2 T 2 ) µ 2π (ω + i λ ) PB & B. Sathiapalan (2014) Viscous drag γ = 2 λπt 2 Higher order dissipation coefficient ρ = λ 2π 2( λ) 3 πt 2 T 0 λ === µ 2 2π

17 Dissipation Zero temperature dissipation is physical It is finite and therefore no need to renormalize by adding counterterms. It cannot be renormalized away in the boundary theory by any hermitian counter term. Quark moving at constant velocity doesn t feel any drag at T= 0. Explanation : This zero temperature dissipation is due to radiation of accelerated charged particle. Remarkably the dissipation in this highly non-linear boundary theory is given by simple Abraham-Lorentz -like formula for radiation reaction in classical electrodynamics! λ F rad = 2π a

18 Dissipation Dissipation at T = 0 String in pure AdS d+1 String EOM Ingoing & normalized solution f ω (r) + 4 r f ω(r) + L4 ω 2 r 4 f ω(r) = 0 (14) Retarded Green s function F R ω (r) = r B r e +i L2 ω r (r il 2 ω) e +i L2 ω rb (r B il 2 ω) (15) G R (ω) := GR(ω) 0 + µω2 2π = µω3 2π 1 (ω + i µ λ ) (16) Higher order dissipation coefficient ρ = λ 2π (Identical!) (17)

19 Dissipation Is the zero temperature dissipation universal?

20 Dissipation Is the zero temperature dissipation universal? String in AdS 5-BH String EOM can not be solved exactly! f ω (r) + Ingoing & normalized ansatz 4r 3 (r 4 π 4 T 4 L 8 ) f ω(r) + ω 2 L 4 r 4 fω(r) = 0 (18) (r 4 π 4 T 4 L 8 ) 2 ) Fω R (r) = (1 π4 T 4 L 8 i Ω 4 (1 iωf1(r) Ω 2 f 2(r) + iω 3 f 3(r) +...) (19) r 4 The limit we are interested in ω, T 0 and Ω := ω πt = fixed (20) Solving f 1(r), f 2(r) and f 3(r) perturbatively and following the same procedure ( ) G R (ω) GR 0 + µω2 λ π Log4 2π = i ω 3 (21) 2π 4

21 Dissipation A phase transition at T= 0? Dissipation as T 0 Calculating G R (ω) in AdS 5 -Schwarzschild black hole bulk geometry ρ = (π log 4) λ 4 2π Dissipation at T= 0 Calculating G R (ω) in pure AdS 5 bulk geometry ρ = λ 2π

22 Dissipation A phase transition at T= 0? Dissipation as T 0 Calculating G R (ω) in AdS 5 -Schwarzschild black hole bulk geometry ρ = (π log 4) λ 4 2π Dissipation at T= 0 Calculating G R (ω) in pure AdS 5 bulk geometry ρ = λ 2π Conclusion : Possibly due to deconfinement transition at T= 0.

23 Dissipation Dissipation at finite density Reissner-Nordström (RN) black hole in asymptotically AdS space time ds 2 = L2 d+1 z 2 ( f (z)dt2 + d x 2 ) + L2 d+1 z 2 dz 2 f (z) (22) where, f (z) = 1 + Q 2 z 2d 2 Mz d A t (z) = µ(1 zd 2 ) z d 2 0 We ll define a new length scale z where Q := There are two possibilities : Extremal BH (T = 0) Non-extremal BH (T 0) d d 2 1 z d 1

24 Dissipation Dissipation at finite density The string EOM Finite density and zero temperature x ω(z) + d dz ( f (z) z ) 2 x f (z) ω(z) + ω2 [f (z)] 2 x ω(z) = 0 (23) z 2 Subtlety : At zero temperature the f (z) has a double zero at the horizon. Thus this singular term dominates at the horizon irrespective of however small ω we choose. Matching technique : Isolate the singular near horizon region and treat ω perturbatively outside. Inner/IR region : AdS 2 R d 1 Outer/UV region : Full RN-AdS background Hong Liu et al. (2009)

25 Dissipation Dissipation at finite density Matching the solutions in two regions near z = z we obtain G R (ω) := b + + G R (ω)z b a + + G R (ω)z a The leading-order Green s function = (b(0) + + ω 2 b (2) ) iω(b (0) + ω 2 b (2) +...)z (24) (a (0) + + ω 2 a (2) ) iω(a (0) + ω 2 a (2) +...)z G (0) R (ω) = i ωz (1 + i ωz a (0) ) (25) For small frequency G (0) R (ω) i ωz (1 i ωz a (0) ) (26)

26 Dissipation Dissipation at finite density Choosing a (0) = 1 Numerical analytic is nice a straight line in each case. Therefore the Green s functions match well up to some overall normalization.

27 Dissipation Dissipation at finite density Choosing a (0) = Re G R (ω) ω Numerical analytic is not a straight line!

28 Dissipation Dissipation at finite density Finite density and small temperature Inner Region changes to BH in AdS 2 R d 1 The story is same with two possible modifications The G R (ω) may change and can be T -dependent. a ±, b ± will be T -dependent. Actually the retarded Green s function at small T becomes G T R (ω) = b +(ω, T ) + G R (ω, T )z b (ω, T ) a + (ω, T ) + G R (ω, T )z a (ω, T ) = b +(ω, T ) iωz b (ω, T ) a + (ω, T ) iωz a (ω, T ) (27) Leading order dissipation is same as zero temperature.

29 Conclusions The temperature independent dissipation is identical for all dimensions as long as the systems are in zero temperature (bulk is pure AdS). For higher dimensions T 0 and T = 0 (e.g, AdS 5-BH and pure AdS 5 bulk, say) the coefficients don t match! Retarded Green s function at T = 0 is computed at finite density. Zero temperature dissipation shows up as leading term. The form of the Retarded Green s function at finite density and small (but finite) temperature is also obtained. The leading dissipative part remains the same. The leading order Green s function is matched (or rather compared) with numerical results up to some overall normalization.

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31 Back up slides I Perturbative solutions Functional forms of f 1(z), f 2(z), f 3(z) f 1(z) = 1 2 tan 1 (πtz) 1 2 Log(1 + πtz) Log(1 + π2 T 2 z 2 ) (28) f 2(z) = 1 32 [4{ 4 + tan 1 (πtz) Log(1 + πtz)}{tan 1 (πtz) Log(1 + πtz)} 4{2 + tan 1 (πtz) Log(1 + πtz)}log(1 + π 2 T 2 z 2 ) + Log(1 + π 2 T 2 z 2 ) 2 ] (29) f 3(z) =... (30)

32 Back up slides II Black hole in AdS 2 R d 1 The near horizon geometry for Near-extremal (T µ) RN Black hole ) ds 2 = L2 2 ( g(ζ)dt 2 ζ 2 + dζ2 + µ 2 g(ζ) L 2 d+1d x 2 (31) 1 1 A t (ζ) = 2d(d 1) ζ (1 ζ ) ζ 0 (32) where g(ζ) := (1 ζ2 ), ζ ζ0 2 0 := z 2 d(d 1)(z z 0) The corresponding temperature, T = 1 2πζ 0 This nice structure breaks down for large temperature (T µ).