SYMMETRIES AND DISSIPATIVE MAGNETOHYDRODYNAMICS

Transcription

1 SAŠO GROZDANOV INSTITUUT-LORENTZ FOR THEORETICAL PHYSICS LEIDEN UNIVERSITY GENERALISED GLOBAL SYMMETRIES AND DISSIPATIVE MAGNETOHYDRODYNAMICS OXFORD,

2 2 OUTLINE hydrodynamics generalised global symmetries and magnetohydrodynamics plasma and holography outlook

3 FRAGMENT I with D. Hofman and N. Iqbal FRAGMENT II with N. Poovuttikul 3

4 HYDRODYNAMICS

5 5 HYDRODYNAMICS QCD and quark-gluon plasma low-energy limit of QFTs (effective field theory) T µ u,t,µ =(" + P ) u µ u + Pg µ µ r u µ +... J µ (u,t,µ)=nu µ T µ r (µ/t )+... r µ T µ =0 r µ J µ =0 tensor structures (phenomenological gradient expansions) with transport coefficients (microscopic)

6 6 HYDRODYNAMICS conformal (Weyl-covariant) hydrodynamics T µ µ =0 g µ! e 2!(x) g µ T µ! e 6!(x) T µ infinite-order asymptotic expansion T µ = 1X n=0 T µ (n)! = O H X n=0 n k n+1 classification of tensors beyond Navier-Stokes first order: 2 (1 in CFT) - shear and bulk viscosities second order: 15 (5 in CFT) - relaxation time, [Israel-Stewart and extensions] third order: 68 (20 in CFT) - [S. G., Kaplis, PRD 93 (2016) 6, , arxiv: ]

7 7 HYDRODYNAMICS diffusion and sound dispersion relations in CFT shear:! = i " " + P k2 i 2 1 (" + P ) " + P sound:! = ±c s k i c k 2 c 2c s c 2c 2 s k 3 i # k 4 + O k 5 " (" + P ) " + P # k 4 + O k 5 loop corrections break analyticity of the gradient expansion (long-time tails), but are 1/N suppressed [Kovtun, Yaffe (2003)] entropy current, constraints on transport and new transport coefficients (anomalies, broken parity) non-relativistic hydrodynamics hydrodynamics from effective Schwinger-Keldysh field theory with dissipation [Nicolis, et. al.; S. G., Polonyi; Haehl, Loganayagam, Rangamani; de Boer, Heller, Pinzani-Fokeeva; Crossley, Glorioso, Liu]

8 MAGNETOHYDRODYNAMICS

9 9 STATES OF MATTER solids liquids gases plasma

10 10 PLASMA AND MHD ionised gas, uncharged at large distances (screening) the theory: magnetohydrodynamics + r ( @t + ~v r ~v = rp + J ~ B ~ r E ~ B r B ~ = µ 0 J ~ r ~B =0 r ~E =0 continuity + Euler (or Navier-Stokes) Maxwell ~E + ~v ~ B =0, (!1) Ohm s law + ~v r =0 Equation of state

11 11 PLAN AND WISHES generalise MHD to bring it to the effective field theory level of (charged) relativistic hydrodynamics: symmetries and field content is there a way to avoid assumptions of the matter content? what if a plasma is strongly coupled, contains strong magnetic fields, B/T 2 1 its equation of state depends on magnetic properties and the matter sector is strongly coupled to the charged sector, p(t,b) quantum effects: pair-creation (Euler-Heisenberg), Landau levels,? systematic derivative expansions: transport coefficients look at an example of such a plasma and learn something new about real plasmas (fusion, magnetars, quark-gluon plasma, )?

12 12 GENERALISED GLOBAL SYMMETRIES symmetry based formulation of MHD [S. G., Hofman, Iqbal, PRD (2017) 9, , arxiv: ] gauge vs. global U(1) symmetry in electromagnetism 1 g 2 r µf µ = j el J µ = 1 2 µ F 0-form symmetry (1-form conserved current) O(x)! e iq O(x) 1-form symmetry (2-form conserved current) [Gaiotto, Kapustin, Seiberg, Willet] Z W (C)! exp iq µ dx W (C) matter-less electromagnetism has two U(1) 1-form symmetries C spontaneous breaking of the symmetry gives a photon

13 13 GENERALISED GLOBAL SYMMETRIES such a symmetry can be source or gauged like any other symmetry Z S[b] S 0 + S[b], S[b] d 4 x p gb µ J µ related to one-form current by a dualisation Z S[b] = d 4 x p ga j ext, j ext b µ our theory has the following conserved global symmetries: energymomentum and a U(1) number of magnetic flux lines (in the presence of charged matter) r µ T µ =0 r µ J µ = 1 2 r µ (" µ F )=0 external source r µ T µ = H J

14 14 IDEAL MHD AND WAVES construct a general hydrodynamic gradient expansion with fields: u µ,t,h µ,µ u 2 = 1, h 2 =1,h µ u µ =0 ideal MHD (string hydro) T µ (0) =(" + p) uµ u + pg µ µ h µ h J µ (0) =2 u[µ h ] " + p = Ts+ µ dp = sdt + dµ thermodynamics r µ T µ =0 Alfvén and (slow and fast) magnetosonic waves from r µ J µ =0! = ±v A k! = ±v M k v 2 A = V 2 A cos 2 v 2 M = 1 2 V 2 A + V 2 0 cos 2 + V 2 S sin 2 ± q (V 2 A V 2 0 ) cos2 + V 2 S sin2 2 +4V4 cos 2 sin 2

15 15 IDEAL MHD AND WAVES Alfvén waves have been observed in solar corona [McIntosh, et. al (2011)]

16 16 SPEEDS OF MHD WAVES assume weak field limit of the equation of state to reproduce all of standard MHD results (Alfvén and magnetosonic waves) [Hernandez, Kovtun] µ T 2 : p weak (µ, T )= a 4 T 4 + g2 2 µ2 + 4 µ 4 T 4 + strong field limit T 2 µ : p strong (µ, T )= g02 2 µ2 + a0 4 T T 8 µ 2 +

17 17 DISSIPATION construct first-order (anisotropic) dissipative corrections 7 terms consistent with positive entropy production and chargeconjugation invariant MHD (2 shear, 3 bulk viscosities, 2 resistivities); 11 in general MHD [Hernandez, Kovtun] S µ = su µ 1 T T µ (1) u µ T J µ (1) h, r µ S µ 0 T µ (1) = f µ + h µ h +2`(µ h ) + t µ J µ (1) =2m[µ h ] + s µ? 0 k 0 r? 0 r k 0? 0? k 2 f =? µ r µ u (1) h µ h r µ u = (2) µ r µ u 2 k h µ h r µ u `µ = 2 µ k h r ( u ) t µ µ 1 = 2? 2 m µ = 2r? T µ h r [ h ] µ T µ r ( u ) s µ = 2r k µ µ r [ h ]

18 18 DISSIPATION corrections to the dispersion relations of MHD waves limits of small momentum and large angle do not commute expand in small k, e.g. the Alfvén wave is! = ±k cos s 1 1+ Ts µ 1 sin 2 2 i Ts+ µ? + cos2 Ts+ µ k + µ cos2 r? + µ sin2 r k k 2 for non-infinitesimal k, there is a critical angle c (k) at which propagating modes (slow-ms and Alfvén) cease to exist and become diffusive (weak/strong B)

19 19 DISSIPATION in the regime of a strong magnetic field, when anisotropic corrections become large, critical angle decreases from Kubo formulae, e.g. resistivities r k =lim JJ (!) i!!!0 G xy,xy r? =lim JJ (!) i!!!0 G xz,xz normally hee(!)i ( i!) 2 g 2 G (!) 1 hj el j el (!)i g 2 G (!) r ( i!) 1 hj el j el (!)i 1 g 2 G (!) + hjj(!)i + + now, new diagrams: e.g. r 6= 1

20 20 MHD AT ZERO TEMPERATURE take the limit of zero temperature a sector of the theory without entropy production, and T =0 symmetry breaking patterns finite T : zero T : SO(3, 1)! SO(2) SO(3, 1)! SO(1, 1) SO(2) degrees of freedom µ, u µ s.t. u µ u µ = 2, u µ u u = u µ ideal hydrodynamics (no 1st order correction) T µ (0) = " µ + p µ J µ (0) = uµ µ u µ u µ g µ µ entropy conservation vs. closure of equations (reduces 8 (7) to 5) (r µ T µ ) + µ (r µ J µ ) u =0

21 21 MHD AT ZERO TEMPERATURE this theory has no slow MS waves due to vanishing fluctuation of the temperature universal dispersion relation for Alfvén waves (to all orders)! = ±k cos fast MS waves receive corrections from 2nd order hydrodynamics and travel at the! M = ± v M k + M k 3 + v M = cos 2 + 1/2 µ in absence of any scales beyond B, speed of fast MS waves is the speed of light

22 HOLOGRAPHIC PLASMA

23 23 HOLOGRAPHIC MHD example of a strongly coupled plasma [S. G., Poovuttikul, arxiv: ] fields at the boundary source dual conserved operators generating functional W [g µ,b µ ]= apple Z exp i d 4 x p g 1 2 T µ g µ + J µ b µ construct a theory of a metric and a two-form gauge field in 5D S = 1 2apple 2 5 Z d 5 x p G R + 12 L 2 1 3e 2 H H abc H abc H = db naively, it is dual to the bulk Einstein-Maxwell theory with F ab F ab H =? F F ^? F! H ^? H if we write :

24 24 HOLOGRAPHIC MHD the connection to Einstein-Maxwell theory is more subtle dualise by writing the path integral over F Z Z DF ab DB ab exp i N 2 c 8 2 Z d 5 x p G F ab F ab + e 1 H B ab abcde r c F de hj µ R i = N 2 c 2 2 lim u!0 F uµ = N 2 c 2 2 e H lim u!0 B g ab, B ab g ab, A ab Dirichlet BC von Neumann BC (standard quantisation) Z (alternative quantisation) Z d 4 xj µ B µ d 4 xa µ J µ the gauge field on the boundary is dynamical (photons) [Hofman, Iqbal]

25 25 HOLOGRAPHIC MHD full holographic action S = N 2 c 8 2 " Z d 5 x p G R e 2 H H abc H abc + d 4 x p g 2trK 6+ 1 e 2 H # H µ H µ ln C D Hoker-Kraus black brane on-shell action at a cut-off (FG coordinates) source ds 2 = G ab dx a dx b = r 2 h H = Br2 h e 2V+W 2u 3/2p w dt ^ dz ^ du S on shell = N 2 c 4 2 F (u)dt 2 + e2v(u) v Z (dx 2 + dy 2 )+ e2w(u) w dz2 d 4 x p g H µ B (0) µ + B (1) µ ln C 2 B µ (0) + B µ (1) ln C 2 = 4 2 Nc 2 b µ H µ = n a H aµ + du2 4u 3 F (u)

26 26 HOLOGRAPHIC MHD expectation values of dual conserved operators ht µ i = hj µ i = N 2 c r 2 h K µ µ K 3 µ R[ ] µ ln(c ) 4 2 lim!0 1 H µ H 4 µ H H lim!0 H µ = = 1 4 µ R[ ] matter Maxwell stress-energy tensor has two contributions, more precisely [Fuini, Yaffe] ht µ i = lim!1/ 2 N 2 c 2 2 g µ (2) g µ (0) (g (2) ) h µ + h µ ln C ) = N 2 c 2 h µ ln( /C) = N 2 c 2 h µ ln( /M )+ h µ 2 e 2 r Nc 2 2 ln( /M )

27 27 HOLOGRAPHIC MHD htµ matter i = N c ht EM µ i = 2 ht µ i = ht matter µ i + ht EM µ i g (2) µ g (0) µ (g (2) ) h µ N 2 c e 2 r Nc 2 ln ( /M ) 2 h µ 2 h µ ln( /M ) electromagnetic term comes from the Maxwell action ht EM µ i = 1 e( /M ) 2 hf µ F i S EM = 1 4e( /M ) µ hf F i Z = d 4 x p gf µ F µ 1 e( /M ) 2 hf µ ihf i 1 4 µ hf ihf i trace anomaly and N=4 NSVZ beta function of U(1) R (from SU(4) R ) 1/e 2 = µ de 2 ht µ µi = dµ = N 2 c 2 2 " (e) e 3 B2 = N c X =1 4 2 B2 (q f ) # 3X (qs a ) 2 a=1 = N 2 c 2 2

28 28 HOLOGRAPHIC MHD final renormalised expectation values ht µ i = N 2 c 2 2 hj µ i =2B (1) µ g (2) µ g (0) µ (g (2) ) h µ RG-condition-dependent electromagnetic coupling = N 2 c 2 2 = N 2 c 2 2 e 2 r 4 = Ward identities from MHD are satisfied 2 e 2 r h µ r µ T µ = H J r µ J µ =0 interpretation: not quite gauged N=4 SYM (gauge anomaly of U(1) R, no Chern-Simons term, B-term not in IIB supergravity)

29 THERMODYNAMICS 29

30 30 TRANSPORT COEFFICIENTS maximal resistivity bulk viscosities? k = 2

31 31 SPEEDS OF SOUND! = ±vk idk 2

32 32 SOUND ATTENUATION! = ±vk idk 2

33 33 TRANSMUTATION OF MODES transition from propagating to non-propagating modes (from sound to diffusion) at a k- dependent critical angle c (k)

34 34 ELECTRIC CHARGE behaviour of waves (e.g. Alfvén) strongly dependent on the renormalisation condition for the boundary electric charge

35 35 SUMMARY new, symmetry-based formulation of MHD, with new transport coefficients which agrees with (complete) standard MHD at low magnetic field strength [Hernandez, Kovtun] we claim this theory can be used for any plasma MHD may survive the zero-t limit and become non-dissipative the theory can be studied via a holographic toy model, which includes dynamical electromagnetism (other applications?) MHD modes could be examined at any strength of B transmutation of propagating to non-propagating modes, leastconductive plasma, saturation of bulk viscosity inequality at strong coupling

36 36 OUTLOOK is any of this useful for realistic plasmas?

37 THANK YOU!