Holographic zero sound at finite temperature

Transcription

1 Holographic zero sound at finite temperature Richard Davison and Andrei Starinets Rudolf Peierls Centre for Theoretical Physics Oxford University arxiv: 1109.XXXX [hep-th] Holographic Duality and Condensed Matter Physics Program KITP 20 September 2011

2 In a system of interacting fermions at low (or zero) temperature there exists a collective excitation known as the zero sound (Landau, 1957; observed in liquid He-3 in 1960s) Temperature dependence of the zero sound mode is described (at sufficiently low temperatures) by Landau Fermi-liquid theory A similar mode (holographic zero sound) was shown to exist in holographic models at zero temperature We would like to understand its temperature dependence

3 Outline Zero sound in Landau Fermi-liquid theory Zero sound in He-3 Holographic zero sound in the D3/D7 system

4 Zero sound in Landau Fermi-liquid theory Original Landau papers L. D. Landau, The theory of a Fermi liquid, Zh. Eksp. Teor. Fiz. 30, 1058 (1956) [Soviet Phys. JETP 3, 920 (1957)]. L. D. Landau, Oscillations in a Fermi liquid, Zh. Eksp. Teor. Fiz. 32, 59 (1957) [Soviet Phys. - JETP 5, 101 (1959)]. Relativistic version G.Baym and S.A.Chin, Landau theory of relativistic Fermi liquids, Nucl Phys A262 (1976), 527 Microscopic (effective field theory) version R.Shankar, Renormalization group approach to interacting fermions, Rev Mod Phys, 66 (1994), 129 J.Polchinski, Effective Field Theory and the Fermi Surface, hep-th/

5 Zero sound in Landau Fermi-liquid theory In LFL, the dominant low energy excitations are quasiparticles carrying the same quantum numbers as fundamental particles Quasiparticle energy: Width: Equilibrium distribution function: At low temperature: Specific heat:

6 Zero sound in Landau Fermi-liquid theory Landau interaction function can be expanded in Legendre polynomials yielding Landau parameters In a weakly-interacting theory, can be computed perturbatively

7 Zero sound in Landau Fermi-liquid theory For small deviations from equilibrium we have where the function obeys the Landau-Silin transport equation whose third term involves the interaction function Solutions in the zero-temperature limit (with vanishing r.h.s.) are known as zero sound (Landau, 1957)

8 Zero sound in Landau Fermi-liquid theory Zero sound dispersion relation appears as a pole in the density-density correlation function and thus as the lowest quasinormal frequency in the spectrum of a dual gravity background e.g. in the D3/D7 system (in the probe brane limit) holographically describing

9 Zero sound at finite temperature Zero sound mode is affected by quasiparticle interactions Let be the mean time between quasiparticle collisions Three regimes can be distinguished for the collective mode with frequency : Collisionless quantum regime: Collisionless thermal regime: Hydrodynamic regime: In LFL kinetic theory:

10 Zero sound at finite temperature LFL applicability conditions

11 quantum limit ω T, ω ν lies outside of the LFL applicability range (1)) Zero sound at finite temperature ation of the hydrodynamic (first) sound is determined by viscosity and is propor /T 2 [13, 14, 16]. e dimensionless variables which are most convenient for identifying the three re /µ and q/µ. In these variables, the regions I, II and III corresponding to the hyd, collisionless thermal and collisionless quantum regimes respectively, are separat ales (T/µ) 2 and T/µ. Alternatively, in the language of the traditional hydrody les ω/t and q/t, the relevant scales are T/µ,1and µ /T (see Table I and Fig. TABLE II. Sound attenuation coefficients in a Landau Fermi-liquid Γ ω Γ q Arg q Hydrodynamic regime Collisionless thermal regime T 2 µ Collisionless quantum regime µ T q 2 µ 2 q 2 µ µ ω 2 T 2 µ T T 2 µ ω 2 µ T µ ω µ 2 ω µ 2 µ ω

12 Zero sound at finite temperature arg q Μ I II III T Μ 2 T Μ Ω Μ FIG. 2. A sketch of the dependence of the sound mode damping on frequency in the hydrodynamic (I), collisionless thermal (II) and collisionless quantum (III) regimes of a Landau Fermi-liquid. First sound propagates in region I while the zero sound mode exists in regions II and III. istic frequency ν 1/τ,where τ is the mean time between quasiparticle collisions. One distinguishes three regimes: the hydrodynamic regime, characterized by ω ν, the col-

13 Zero sound at finite temperature

14 s as the low-energy theory on the worldvolume ofaset o D3/D7 zero sound at finite temperature rsecting along (3+1)-dimensions. Taking N c wi and subsequently taking λ and N f /N c 0, w to this field theory [10, 35]: S = S adjoint + S fundamental, = S adjoint N f T D7 d 8 ξ det (g ab + F ab ), he ten-dimensional supergravity action and T D7 is th obe brane limit, the metric is fixed and it is S fundam

15 hen nmbedding at the r field = rcoordinate H theory ) times θ(u) is at a a five-sphere: (which determines which S non-zero temperature, the background 3 section of the backgr spacetime is s). The gauge S-Schwarzschild D3/D7 black zero brane sound (with a at horizon finite attemperature r = r H ) times a five-sphere: 1 r4 1 field H R 2 on the brane is dual to a global flavor current in the fiel us turning r 4 on the r 2 time dr2 component + R 2 ds 2 S 5 of. a U(1) (6) U(N f ) gauge field A ds 2 10 = r2 1 r4 H dt 2 + dx r4 1 t (u) on t H R 2 onds to introducing R 2 a finite r 4 density d of the U(1) r 4 baryon r 2 dr2 charge + R 2 indsthe 2 S. 5 fiel S 3 section of the metric, with the horizon case, it corresponds to a net density of fundamental fermions and scalars. 7-brane wraps an asymptotically AdS 5 S 3 section of the metric, with the ure equations of the of field motion theory for the via background T = r H /πr fields 2. are obtained from the DBI a s of the background related to the temperature of the field theory via T = r H / N = N f T D7 V S 3 S fundamental = Nr4 H dud 4 x9 cos3 θ u 3 1+4u 2 fθ 2 4 u3 A 2 rh 2 t, is a normalization constant determined by the gauge-gravity ary [33] and primes denote derivatives with respect to u. One of the equ derived from the action (8) reduces to Parameters: A t (u) = r H d 2 1+4u 2 f(u)θ (u) 2 cos 6 θ(u)+ d 2 u 3,

16 rethe the m diffusion = 0 case, constant the coupled is given equations by [27, 34] of motion for bulk fluctuations imply th field theory operators mix and the method to determine the retarded Green s fun D3/D7 re involved (see Appendix D( zero d) = sound d 1 3 A for the 1+ at finite temperature 2 details). d 3 2 2F 1 2, 1 3 ; 4 3 ; d 2. t zero mass, high densities ω,q d 1 3 een these two extreme temperature limits, and strictly the poles zero of temperature, the Green s the functions dominan ar e correlators G R Zero J t J and temperature, G R t J z J has finite the density, n analytically. dispersion zero hypermultiplet relation [11, mass 46] z Γ 1 2 hese results were generalized ω = ± to q the i case of a massive 3 Γ 1 Γ q 2 + O hypermultiplet q 3. in [20, 34]. sound mode (27) persists when the hypermultiplet has a finite mass m, althoug rresponds ersion relation to ais Zero collective altered temperature, toexcitation finite density, of 13 the finite system hypermultiplet - the holographic mass zero sound This corresponds to a collective excitation of the system - the holographic zero sound mode. d is Its equal speedto is equal ω = ± 1 thattoof that of 1 m 2 hydrodynamic 1/2 sound, Γ 1 and and it has it an has imaginary 2 (1 m 2 an ) 4/3 imaginary part q 2. part In the hydrodynamic limit ω,q 3 1 m 2 T the q dominant i /3 Γ 1 pole Γ 1 is a purely imaginary (1 m 2 /3) q2 pole + Owith q 3 the, rodynamic limit ω,q T the pole is a purely imaginary pole w Fickian diffusion re m = m/ diffusion dispersion relation dispersion relation 2 µ High = Mtemperature, q /µ. The real finite part density, of zero the hypermultiplet dispersion relation mass (30) was obtain and the attenuation is derived ω= in Appendix id q 2 + B. O( q In the 3 ), hydrodynamic limit, there is where the diffusion constant is given by [27, 34] ffusion pole (28) whose diffusion constant can be derived via an Einstein relation he diffusion constant is D( given d d) = 1 1 by d 1 3 [27, 34] 3 2 G(u) ω = id q 2 + O( q 3 ), (28) 1+ d 3 2 2F 1 2, 1 3 ; 4 3 ; d 2. (29) 4f(u)u 2 θ (u)

17 D3/D7 zero sound at finite temperature Χzz Ω Ω 4. The holographic zero sound peak in the collisionless quantum regime. The longitudina Holographic zero sound peak in the collisionless quantum regime ral function is shown at m =0, d =10 6, q =0.4.

18 D3/D7 zero sound at finite temperature Zero sound dispersion relation in the collisionless quantum regime

19 Log Im Ω T Im Ω 0 D3/D7 zero sound at finite temperature FIG. 7. The holographic zero sound dispersion relation for m =0.76, d =10 5.D results at low T and the solid lines show the analytic result (30) at T = Log d 1 3 Log T d 1 3 Log T Μ FIG. 8. The imaginary part of the holographic zero sound dispersion relation at showing the transition between the collisionless quantum (left) and collisionle Collisionless quantum collisionless thermal crossover regimes. The points are the numerical data, the dashed line denotes q = d 1 solid line is the best-fit straight line for the T/µ q =0.01 points showing t

20 D3/D7 zero sound at finite temperature

21 D3/D7 zero sound at finite temperature arg q Μ I II III T Μ 2 T Μ Ω Μ A sketch of the dependence of the sound mode damping on frequency in the hydrodynamic isionless thermal (II) and collisionless quantum (III) regimes of a Landau Fermi-liquid. und propagates in region I while the zero sound mode exists in regions II and III. Collisionless quantum collisionless thermal crossover equency ν 1/τ,whereτ is the mean time between quasiparticle collisions. One uishes three regimes: the hydrodynamic regime, characterized by ω ν, the colss thermal (classical) regime, with ω ν, ω T, and the collisionless quantum

22 D3/D7 zero sound at finite temperature Collisionless thermal hydrodynamic crossover

23 D3/D7 zero sound at finite temperature Poles of the density-density correlator: The collisionless thermal hydrodynamic crossover

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25 best-fit straight lines which both have gradient D3/D7 zero sound at finite temperature in the familiar language of the Landau theory by introducing an e 0 0 l mfp =1/q 2 cross, and computing their temperature dependence. A 2 4 dependence of the form Log q cross. rature dependence of the collisionless-hydrodynamic crossover value of fre- 8 8 m for m =0. The points are our numerical results and the solid lines are the 10 d 1/3 10 which both have gradient we expect a plot of log( q cross ) or log( ω cross ) versus log( Log d cross. 1 3 Log 1 T cross. Log Ωcross. 4 6 l mfp τ d 1/3 Log d cross. 1 3 Log 1 T cross. FIG. 13. The temperature dependence of the collisionless-hydrodynamic crossover value of frequency Landau momentum theoryfor by m =0. introducing The points are our annumerical effective results τ and =1/ ω the solid cross lines and are age of the the are clearly consistent with a simple power law dependence. The sl best-fit straight lines which both have gradient computing their temperature dependence. Assuming a simple powerorm line is -2.0 in each case, which leads to the result in the familiar language of the Landau theory by introducing an effective τ =1/ ω cross and α l mfp =1/q cross, and computing their temperature dependence. Assuming a simple powerdependence of the form d 1/3 α l mfp τ d 1/3 T l mfp τ d 1/3 T 1/3, (35) d d 1/3, l mfp τ d 1/3 T 2 (35) µt 2, log( q cross ) asor inlog( ω a Landau cross ) versus Fermi-liquid log( cross) [16] to- yield recall a table straight I. For line non-zero of m T α, d 1/3 cross) gradient α. These plots are shown in Fig. 13 for the massless ca

26 Unusual features of D3/D7 thermodynamics Density-density correlator apparently shows no singularity at

27 Conclusions D3/D7 zero sound at finite temperature behaves exactly as LFL zero sound D3/D7 thermodynamics (in the probe brane limit) appears to be incompatible with LFL It would be interesting to: a) Understand this discrepancy b) Consider other holographic and field-theoretic systems c) Understand this from the (holographic) RG perspective

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