Non-Abelian holographic superfluids at finite isospin density. Johanna Erdmenger

Transcription

1 .. Non-Abelian holographic superfluids at finite isospin density Johanna Erdmenger Max Planck-Institut für Physik, München work in collaboration with M. Ammon, V. Graß, M. Kaminski, P. Kerner, A. O Bannon 1

2 Motivation Use generalizations of the AdS/CFT correspondence to describe strongly coupled systems

3 Motivation Use generalizations of the AdS/CFT correspondence to describe strongly coupled systems Prime example: Quark-gluon plasma

4 Motivation Use generalizations of the AdS/CFT correspondence to describe strongly coupled systems Prime example: Quark-gluon plasma Explore further possibilities 2

5 Motivation Holographic Superconductor from charged scalar in Einstein-Maxwell gravity Gubser; Hartnoll, Herzog, Horowitz

6 Motivation Holographic Superconductor from charged scalar in Einstein-Maxwell gravity Gubser; Hartnoll, Herzog, Horowitz p-wave superconductor (current dual to gauge field condensing) Gubser, Pufu

7 Motivation Holographic Superconductor from charged scalar in Einstein-Maxwell gravity Gubser; Hartnoll, Herzog, Horowitz p-wave superconductor (current dual to gauge field condensing) Gubser, Pufu (AdS 4 examples) 3

8 Follow-up questions: Motivation

9 Motivation Follow-up questions: Is there a holographic superconductor...

10 Motivation Follow-up questions: Is there a holographic superconductor... 1) for which the field theory is explicitly known?

11 Motivation Follow-up questions: Is there a holographic superconductor... 1) for which the field theory is explicitly known? 2) within ten-dimensional type IIB supergravity? 4

12 Motivation A holographic superconductor with field theory in 3+1 dimensions for which

13 Motivation A holographic superconductor with field theory in 3+1 dimensions for which 1. the dual field theory is explicitly known

14 Motivation A holographic superconductor with field theory in 3+1 dimensions for which 1. the dual field theory is explicitly known 2. there is a qualitative ten-dimensional string theory picture of condensation p-wave superconductor Ammon, J.E., Kaminski, Kerner ,

15 Motivation A holographic superconductor with field theory in 3+1 dimensions for which 1. the dual field theory is explicitly known 2. there is a qualitative ten-dimensional string theory picture of condensation p-wave superconductor Ammon, J.E., Kaminski, Kerner , Other examples: Embedding of s-wave superconductors in 10d supergravity or M theory Gauntlett et al, Gubser et al 5

16 Motivation This is achieved in the context of adding flavor to gauge/gravity duality

17 Motivation This is achieved in the context of adding flavor to gauge/gravity duality N = 4 theory: all fields in adjoint rep of gauge group φ e iλ φ e iλ

18 Motivation This is achieved in the context of adding flavor to gauge/gravity duality N = 4 theory: all fields in adjoint rep of gauge group φ e iλ φ e iλ QCD: quarks transform in fundamental rep of gauge group q e iλ q

19 Motivation This is achieved in the context of adding flavor to gauge/gravity duality N = 4 theory: all fields in adjoint rep of gauge group φ e iλ φ e iλ QCD: quarks transform in fundamental rep of gauge group q e iλ q Brane probes added on gravity side fundamental d.o.f. in the dual field theory Additional hyperplanes within AdS 5 S 5 or deformed version thereof 6

20 Motivation Fluctuations of brane probes Mesons

21 Motivation Fluctuations of brane probes Mesons Brane embeddings in confining 10d backgrounds Chiral symmetry breaking

22 Motivation Fluctuations of brane probes Mesons Brane embeddings in confining 10d backgrounds Chiral symmetry breaking Brane probes in AdS black hole geometry Quarks added to finite temperature field theory

23 Motivation Fluctuations of brane probes Mesons Brane embeddings in confining 10d backgrounds Chiral symmetry breaking Brane probes in AdS black hole geometry Quarks added to finite temperature field theory Chemical potentials for baryon, isospin density: From non-trivial A t on gravity side

24 Motivation Fluctuations of brane probes Mesons Brane embeddings in confining 10d backgrounds Chiral symmetry breaking Brane probes in AdS black hole geometry Quarks added to finite temperature field theory Chemical potentials for baryon, isospin density: From non-trivial A t on gravity side Rich phase structure 7

25 Outline 1. Holographic Quarks at finite Temperature and Density 2. Superfluidity and Superconductivity 3. Including the backreaction in 5d Einstein-Yang-Mills theory 8

26 Quarks (fundamental fields) within the AdS/CFT correspondence Adding D7 brane probe: D3 X X X X D7 X X X X X X X X 9

27 N probe D7 f Quarks (fundamental fields) from brane probes N D SYM quarks conventional open/closed string duality flavour open/open string duality AdS R N (standard Maldacena limit), N f small (probe approximation) duality acts twice: AdS 5 brane N = 4 SU(N) Super Yang-Mills theory coupled to N = 2 fundamental hypermultiplet Karch, Katz 2002 IIB supergravity on AdS 5 S 5 + Probe brane action on AdS 5 S 3 Dirac-Born-Infeld action 10

28 Gauge/Gravity Duality with Flavor DBI (Dirac-Born-Infeld) action: S DBI = T 7 d 8 ξ tr det(p [G] + 2πα F ) Contributions of order N f /N c

29 Gauge/Gravity Duality with Flavor DBI (Dirac-Born-Infeld) action: S DBI = T 7 d 8 ξ tr det(p [G] + 2πα F ) Contributions of order N f /N c Field theory involves fundamental fermions and scalars 11

30 Gauge/Gravity Duality at Finite Temperature N = 4 Super Yang-Mills theory at finite temperature is dual to AdS black hole Witten 1998 ds 2 = 1 2 ( ϱ R) 2 ( f 2 f dt 2 + fd x 2 ) + ( ) 2 R (dϱ 2 + ϱ 2 dω 2 ϱ 5) f(ϱ) = 1 ϱ4 H ϱ 4, f(ϱ) = 1 + ϱ 4 H ϱ 4 Temperature and horizon related by T = ϱ H πr 2 R: AdS radius For ϱ H 0, metric of AdS 5 S 5 is recovered. 12

31 D7 brane embedding in black hole background Minkowski phase Black hole phase First order phase transition Babington, J.E., Evans, Guralnik, Kirsch Mateos, Myers, Thomson 13

32 Masses and decay widths of mesons Standard procedure in D3/D7: Mateos, Myers et al 2003 Meson masses calculated from linearized fluctuations of D7 embedding Fluctuations: δw(x, ρ) = f(ρ)e i( k x ωt), M 2 = k 2

33 Masses and decay widths of mesons Standard procedure in D3/D7: Mateos, Myers et al 2003 Meson masses calculated from linearized fluctuations of D7 embedding Fluctuations: δw(x, ρ) = f(ρ)e i( k x ωt), M 2 = k 2 For black hole embeddings, ω develops negative imaginary part damping decay width

34 Masses and decay widths of mesons Standard procedure in D3/D7: Mateos, Myers et al 2003 Meson masses calculated from linearized fluctuations of D7 embedding Fluctuations: δw(x, ρ) = f(ρ)e i( k x ωt), M 2 = k 2 For black hole embeddings, ω develops negative imaginary part damping decay width Quasinormal modes of D7 brane fluctuations determine meson mass and width Landsteiner, Hoyos, Montero

35 Finite U(1) baryon density Baryon density n B and U(1) chemical potential µ from non-trivial profile for gauge field time component: Ā 0 (ρ) µ + d ρ 2, d = 2 5/2 N f λt 3 n B Mateos, Myers, Matsuura et al At finite baryon density, all embeddings are black hole embeddings w r 15

36 Phase diagram with finite U(1) baryon density Phase diagram: 1 d = grey region: n B = 0 white region: n B 0 µq/mq d = 0 d = 0.25 d = T/ M Sin, Yogendran et al; Mateos, Myers et al; Karch, O Bannon;... 16

37 Isospin chemical potential and density Embed two coincident D7-branes into AdS-Schwarzschild gauge fields A µ = A a µ σ a u(2) = u(1) B su(2) I Finite isospin density: A Explicit breaking to u(1) 3 Dynamics of Flavour degrees is described by non-abelian DBI action Field theory described: N = 4 Super Yang-Mills plus two flavors of fundamental matter at finite temperature and finite isospin density 17

38 ρ meson condensation Above a critical isospin density, a new phase forms J.E., Kaminski, Kerner, Rust

39 ρ meson condensation Above a critical isospin density, a new phase forms J.E., Kaminski, Kerner, Rust New phase is unstable 18

40 Quasinormal modes Instability: Imw Rew 19

41 A new ground state forms There is a new solution to the equations of motion with non-zero vev for A 1 3σ 1 in addition to the non-zero A 3 0σ 3

42 A new ground state forms There is a new solution to the equations of motion with non-zero vev for A 1 3σ 1 in addition to the non-zero A 3 0σ 3 A 3 0 = µ d 3 0 2πα ρ H ρ , A1 3 = d 3 1 2πα ρ H ρ

43 A new ground state forms There is a new solution to the equations of motion with non-zero vev for A 1 3σ 1 in addition to the non-zero A 3 0σ 3 A 3 0 = µ d 3 0 2πα ρ H ρ , A1 3 = d 3 1 2πα ρ H ρ Pole structure: ÈË Ö Ö ÔÐ Ñ ÒØ ÁÑÛ Ê Û Ò ½ Ò ½ Ò ¼ Ò ¼ Ò ¼ 20

44 Superconductivity Ammon, J.E., Kaminski, Kerner , The new ground state has properties known from superconductors: infinite DC conductivity, gap in the AC conductivity second order phase transition, critical exponent of 1/2 (mean field) a remnant of the Meissner Ochsenfeld effect 21

45 Superfluidity and Superconductivity Order parameter d 1 3 ψ u γ 3 ψ d + ψ d γ 3 ψ u + bosons 0 Dual to A 1 3σ 1 in gravity theory

46 Superfluidity and Superconductivity Order parameter d 1 3 ψ u γ 3 ψ d + ψ d γ 3 ψ u + bosons 0 Dual to A 1 3σ 1 in gravity theory Spontaneous breaking of (global) U(1) 3 Flavor superfluid

47 Superfluidity and Superconductivity Order parameter d 1 3 ψ u γ 3 ψ d + ψ d γ 3 ψ u + bosons 0 Dual to A 1 3σ 1 in gravity theory Spontaneous breaking of (global) U(1) 3 Flavor superfluid Condensate corresponds to ρ meson superfluid discussed in QCD literature Son, Stephanov; Splittorff; Sannino... ρ condensation in Sakai-Sugimoto model: Aharony, Peeters, Sonnenschein, Zamaklar 22

48 Order parameter: p wave condensate d T/T c Red: Vanishing quark mass; Black: Finite quark mass, µ/m q = 3 Blue: Fit displaying critical exponent 1/2 23

49 Thermodynamics Flavor contribution to Grand potential vs. temperature W T T c 24

50 Heat Capacity Flavor contribution to heat capacity c v T T T c 25

51 Conductivity Frequency-dependent conductivity σ(ω) = i ω GR (ω) G R retarded Green function for fluctuation a Re σ w w = ω/(2πt ) T/T c : Black:, Red: 1, Orange: 0.5, Brown: (Vanishing quark mass) Interpretation: Frictionless motion of mesons through plasma 26

52 Meissner effect À ¼ Ñ ÒØ Ì ½ ¼ Lower phase: magnetic field and condensate coexist Upper phase: condensate vanishes 27

53 Non-abelian DBI action Evaluation non-trivial in presence of both σ 0, σ 1 Two evaluation methods: 1) Expansion to fourth order 2) Simplification: Modified prescription for symmetrized trace Allows for all-order calculation of the non-abelian DBI Error of order 1/N f cf. Myers, Constable, Tafjord

54 String picture Strings stretched between D7 branes and horizon induce a charge near the horizon System unstable above a critical charge density Horizon strings recombine to D7 D7 strings D7 D7 strings propagate into the bulk, balancing flavorelectric and gravitational forces D7 D7 strings distribute isospin charge into the bulk condensate 29

55 Charge distributions 3.0 Ã 3 0 Ã ρ ρ p p ρ ρ 30

56 More recently... Turn on both isospin and baryon chemical potential  ½ Ü ¼ Õ ¾ Ì Á  ½ Ü ¼ Á Figure by Patrick Kerner 31

57 More recently... 32

58 Including the backreaction Ammon, J.E., Graß, Kerner, O Bannon Simple gravity model: Einstein-Yang-Mills-Theory with SU(2) gauge group S = d 5 x g [ ] 1 1 (R Λ) 2κ2 4ĝ 2 F µνf a aµν α = κ 5 ĝ α 2 number of charged d.o.f./all d.o.f. In presence of isospin chemical potential, same condensation process as before 33

59 Including the backreaction Solution to Einstein-Yang-Mills equations of motion: Reissner-Nordström black hole

60 Including the backreaction Solution to Einstein-Yang-Mills equations of motion: Reissner-Nordström black hole At low temperatures, RN black hole unstable against fluctuations in A 1 3 condensation

61 Including the backreaction Solution to Einstein-Yang-Mills equations of motion: Reissner-Nordström black hole At low temperatures, RN black hole unstable against fluctuations in A 1 3 condensation S µ A 1 3 µ A m eff (A 1 3) 2, m eff = 2g tt g xx (A 3 0) 2 m eff lower than BF bound for certain values of radial coordinate r Instability 34

62 Solutions in the broken phase Numerical solution of EOM with ansatz ds 2 = N(r)σ(r) 2 dt N(r) dr2 + r 2 f(r) 4 dx 2 + r 2 f(r) 2 ( dy 2 + dz 2) A = φ(r)τ 3 dt + w(r)τ 1 dx, N(r) = 2m(r) r 2 + r2 L Öµ Öµ Ö Öµ Ñ Öµ ½¼ Û Öµ Ö 35

63 Phase transition ¾ ÂÜ ½ ¾ «¾ Ì Ì Ì Phase transition becomes first order above α crit 36

64 Phase diagram Numerics not trustable RN metastable Α RN unstable Μ 37

65 Conclusion and Outlook

66 Conclusion and Outlook A holographic superconductor for which the field theory is explicitly known First explicit example of superconductivity (superfluidity) from 10d action Embedding of two coincident D7 branes Finite isospin density Simple model with backreaction: Phase transition becomes first order for certain values of κ 5 /ĝ

67 Conclusion and Outlook A holographic superconductor for which the field theory is explicitly known First explicit example of superconductivity (superfluidity) from 10d action Embedding of two coincident D7 branes Finite isospin density Simple model with backreaction: Phase transition becomes first order for certain values of κ 5 /ĝ Fermions Talk by Matthias Kaminski at 2pm 38