Vortex dynamics in finite temperature two-dimensional superfluid turbulence. Andrew Lucas

Transcription

1 Vortex dynamics in finite temperature two-dimensional superfluid turbulence Andrew Lucas Harvard Physics King s College London, Condensed Matter Theory Special Seminar August 15, 2014

2 Collaborators 2 Paul Chesler Harvard Physics based on: Chesler, Lucas, arxiv:1408.xxxx

3 Two-Dimensional Superfluids 3 Superfluid Vortices Gross-Pitaevskii action: low energy EFT of a superfluid. L = i ψ t ψ 2 ψ 2 2m + µ ψ 2 λ 2 ψ 4.

4 Two-Dimensional Superfluids 3 Superfluid Vortices Gross-Pitaevskii action: low energy EFT of a superfluid. L = i ψ t ψ 2 ψ 2 2m + µ ψ 2 λ 2 ψ 4. µ > 0: ψ 2 > 0. U(1) symmetry ψ e iθ ψ spontaneously broken.

5 Two-Dimensional Superfluids 3 Superfluid Vortices Gross-Pitaevskii action: low energy EFT of a superfluid. L = i ψ t ψ 2 ψ 2 2m + µ ψ 2 λ 2 ψ 4. µ > 0: ψ 2 > 0. U(1) symmetry ψ e iθ ψ spontaneously broken. vortices: topological point defects near which ψ e iw θ

6 Two-Dimensional Superfluids 3 Superfluid Vortices Gross-Pitaevskii action: low energy EFT of a superfluid. L = i ψ t ψ 2 ψ 2 2m + µ ψ 2 λ 2 ψ 4. µ > 0: ψ 2 > 0. U(1) symmetry ψ e iθ ψ spontaneously broken. vortices: topological point defects near which ψ e iw θ SF velocity field: v = W ˆφ θ m m r

7 Classical Turbulence 4 d = 3: Direct Cascade what is turbulence?

8 Classical Turbulence 4 d = 3: Direct Cascade what is turbulence? strongly nonlinear, chaotic motion of vortices (incompressible approximation)

9 Classical Turbulence 4 d = 3: Direct Cascade what is turbulence? strongly nonlinear, chaotic motion of vortices (incompressible approximation) scale invariant in inertial range

10 Classical Turbulence 4 d = 3: Direct Cascade what is turbulence? strongly nonlinear, chaotic motion of vortices (incompressible approximation) scale invariant in inertial range d = 3 spatial dimensions: (approximate) flux of energy (ε) from IR to UV dissipative scale: direct cascade

11 Classical Turbulence 4 d = 3: Direct Cascade what is turbulence? strongly nonlinear, chaotic motion of vortices (incompressible approximation) scale invariant in inertial range d = 3 spatial dimensions: (approximate) flux of energy (ε) from IR to UV dissipative scale: direct cascade Kolmogorov s 5/3 law (dimensional analysis): energy stored at wave vector k (in inertial range) E(k) ε 2/3 k 5/3 Kolmogorov, Proceedings of the USSR Academy of Sciences (1941)

12 Classical Turbulence 5 d = 2: Inverse Cascade d = 2 normal turbulence instead has inverse cascade: energy convected UV IR Andrew

13 Classical Turbulence 5 d = 2: Inverse Cascade d = 2 normal turbulence instead has inverse cascade: energy convected UV IR enstrophy conservation: 0 = d dt Andrew d 2 x ω 2. ω = ɛ ij i v j. Kraichnan, Physics of Fluids (1967)

14 Classical Turbulence 6 A (Holographic) Inverse Cascade Adams, Chesler, Liu, Physical Review Letters (2014)

15 Superfluid Turbulence 7 Principles vortex annihilation = enstrophy no longer conserved

16 Superfluid Turbulence 7 Principles vortex annihilation = enstrophy no longer conserved what causes annihilation?

17 Superfluid Turbulence 7 Principles vortex annihilation = enstrophy no longer conserved what causes annihilation? is a direct cascade now possible?

18 Superfluid Turbulence 7 Principles vortex annihilation = enstrophy no longer conserved what causes annihilation? is a direct cascade now possible? is SF turbulence with annihilation characterized by Kolmogorov scaling?

19 Superfluid Turbulence 8 Holography AdS/CFT calculation: energy UV; driven by vortex annihilation Chesler, Liu, Adams, Science (2013)

20 Superfluid Turbulence 9 Holography AdS/CFT calculation: energy UV; driven by vortex annihilation Chesler, Liu, Adams, Science (2013)

21 Superfluid Turbulence 10 Gross-Pitaevskii: Inverse Cascade dynamics of T = 0 superfluid using GPE Simula, Davis, Helmerson, arxiv:

22 Superfluid Turbulence 11 Experiments: Cold Atomic Gases vortex annihilation in cold atomic turbulent BEC of 23 Na atoms:

23 Superfluid Turbulence 11 Experiments: Cold Atomic Gases vortex annihilation in cold atomic turbulent BEC of 23 Na atoms: experiment can t distinguish between ±1 vortices. gyroscope proposal to do this Kwon, Moon, Choi, Seo, Shin, arxiv: Powis, Sammut, Simula, arxiv:

24 Effective Theory 12 Our Approach could an effective theory of vortex dynamics capture this phenomenology?

25 Effective Theory 12 Our Approach could an effective theory of vortex dynamics capture this phenomenology? dilute limit: (intervortex spacing) r ξ (vortex core size)

26 Effective Theory 12 Our Approach could an effective theory of vortex dynamics capture this phenomenology? dilute limit: (intervortex spacing) r ξ (vortex core size) leading order dynamics of Gross-Pitaevskii equation (e.g.): point-vortex dynamics: Ẋ n i = V n i = m n κ m m ɛ ij (X n j Xm j ) X n X m 2

27 Effective Theory 12 Our Approach could an effective theory of vortex dynamics capture this phenomenology? dilute limit: (intervortex spacing) r ξ (vortex core size) leading order dynamics of Gross-Pitaevskii equation (e.g.): point-vortex dynamics: Ẋ n i = V n i = m n κ m m ɛ ij (X n j Xm j ) X n X m 2 corrections from sound?

28 Effective Theory 12 Our Approach could an effective theory of vortex dynamics capture this phenomenology? dilute limit: (intervortex spacing) r ξ (vortex core size) leading order dynamics of Gross-Pitaevskii equation (e.g.): point-vortex dynamics: Ẋ n i = V n i = m n κ m m ɛ ij (X n j Xm j ) X n X m 2 corrections from sound? finite temperature?

29 Effective Theory 12 Our Approach could an effective theory of vortex dynamics capture this phenomenology? dilute limit: (intervortex spacing) r ξ (vortex core size) leading order dynamics of Gross-Pitaevskii equation (e.g.): point-vortex dynamics: Ẋ n i = V n i = m n κ m m ɛ ij (X n j Xm j ) X n X m 2 corrections from sound? finite temperature? long, controversial history: HVI equations Sonin, Physical Review B (1997) Thompson, Stamp, Physical Review Letters (2011)

30 Effective Theory 13 T = 0: Corrections from Sound T = 0: start from explicit SF action (e.g. Gross-Pitaevskii), integrate out fluctuations: ψ = ψ PV (X 1,..., X N ) + δψ e is eff[x n] = Dδψ e is GP[ψ PV +δψ]

31 Effective Theory 13 T = 0: Corrections from Sound T = 0: start from explicit SF action (e.g. Gross-Pitaevskii), integrate out fluctuations: ψ = ψ PV (X 1,..., X N ) + δψ e is eff[x n] = Dδψ e is GP[ψ PV +δψ] terms in δs at O( r 2 ) beyond kinetic term: e.g., X m X n 2, 3,4-body terms...

32 Effective Theory 13 T = 0: Corrections from Sound T = 0: start from explicit SF action (e.g. Gross-Pitaevskii), integrate out fluctuations: ψ = ψ PV (X 1,..., X N ) + δψ e is eff[x n] = Dδψ e is GP[ψ PV +δψ] terms in δs at O( r 2 ) beyond kinetic term: e.g., X m X n 2, 3,4-body terms... no log-divergent prefactors in equations of motion. contrast with standard lore: m vortex log(l/ξ)...

33 Effective Theory 13 T = 0: Corrections from Sound T = 0: start from explicit SF action (e.g. Gross-Pitaevskii), integrate out fluctuations: ψ = ψ PV (X 1,..., X N ) + δψ e is eff[x n] = Dδψ e is GP[ψ PV +δψ] terms in δs at O( r 2 ) beyond kinetic term: e.g., X m X n 2, 3,4-body terms... no log-divergent prefactors in equations of motion. contrast with standard lore: m vortex log(l/ξ)... corrections do not induce ± pair annihilation (conservation of energy)

34 Effective Theory 13 T = 0: Corrections from Sound T = 0: start from explicit SF action (e.g. Gross-Pitaevskii), integrate out fluctuations: ψ = ψ PV (X 1,..., X N ) + δψ e is eff[x n] = Dδψ e is GP[ψ PV +δψ] terms in δs at O( r 2 ) beyond kinetic term: e.g., X m X n 2, 3,4-body terms... no log-divergent prefactors in equations of motion. contrast with standard lore: m vortex log(l/ξ)... corrections do not induce ± pair annihilation (conservation of energy) inverse cascade at T = 0. Chesler, Lucas, Surówka, in preparation.

35 Effective Theory 14 T > 0: Normal Fluid exchange of energy/momentum only at vortex core and symmetry = first order HVI equation : ρ ) s m κ nɛ ij (Ẋn j Vj n }{{} Magnus force (κ n = ±1 denotes winding number) = η(ẋn i U n i ) η κ n ɛ ij (Ẋn j U n j ) }{{} vortex drag force

36 Effective Theory 14 T > 0: Normal Fluid exchange of energy/momentum only at vortex core and symmetry = first order HVI equation : ρ ) s m κ nɛ ij (Ẋn j Vj n }{{} Magnus force (κ n = ±1 denotes winding number) = η(ẋn i U n i ) η κ n ɛ ij (Ẋn j U n j ) }{{} vortex drag force η, η microscopic coefficients beyond EFT; vanish as T 0

37 Effective Theory 14 T > 0: Normal Fluid exchange of energy/momentum only at vortex core and symmetry = first order HVI equation : ρ ) s m κ nɛ ij (Ẋn j Vj n }{{} Magnus force (κ n = ±1 denotes winding number) = η(ẋn i U n i ) η κ n ɛ ij (Ẋn j U n j ) }{{} vortex drag force η, η microscopic coefficients beyond EFT; vanish as T 0 (normal) sound radiated from momentum exchange suppressed faster than r 1

38 Effective Theory 14 T > 0: Normal Fluid exchange of energy/momentum only at vortex core and symmetry = first order HVI equation : ρ ) s m κ nɛ ij (Ẋn j Vj n }{{} Magnus force (κ n = ±1 denotes winding number) = η(ẋn i U n i ) η κ n ɛ ij (Ẋn j U n j ) }{{} vortex drag force η, η microscopic coefficients beyond EFT; vanish as T 0 (normal) sound radiated from momentum exchange suppressed faster than r 1 (normal) sound from vortex annihilation suppressed faster than r 4 ; consistently set U n = 0.

39 Effective Theory 14 T > 0: Normal Fluid exchange of energy/momentum only at vortex core and symmetry = first order HVI equation : ρ ) s m κ nɛ ij (Ẋn j Vj n }{{} Magnus force (κ n = ±1 denotes winding number) = η(ẋn i U n i ) η κ n ɛ ij (Ẋn j U n j ) }{{} vortex drag force η, η microscopic coefficients beyond EFT; vanish as T 0 (normal) sound radiated from momentum exchange suppressed faster than r 1 (normal) sound from vortex annihilation suppressed faster than r 4 ; consistently set U n = 0. no single fluid continuum description: η is not viscosity it is κ-dependent friction Ambegaokar, Halperin, Nelson, Siggia, Physical Review Letters (1978)

40 Effective Theory 15 The Magnus Force consider vortex at rest in background flow V: F i = dx j Π ij = dx j (ρ s v i v j + µδ ij ) V R

41 Effective Theory 15 The Magnus Force consider vortex at rest in background flow V: F i = dx j Π ij = dx j (ρ s v i v j + µδ ij ) R V v = V + κ m ˆθ r

42 Effective Theory 15 The Magnus Force consider vortex at rest in background flow V: F i = dx j Π ij = dx j (ρ s v i v j + µδ ij ) R V v = V + κ ˆθ m r Integrate over circle of large radius R: F i = ɛ ij κ ρ s m V j. Galilean invariance: boost to frame with vortex moving Sonin, Physical Review B (1997)

43 Effective Theory 16 The Equations of Motion if l = characteristic length, rescale X n = l ˆX n, t = ml2 ˆt. EOM l-independent: exact scale invariance

44 Effective Theory 16 The Equations of Motion if l = characteristic length, rescale X n = l ˆX n, t = ml2 ˆt. EOM l-independent: exact scale invariance after rescaling t to remove η : ) κ n ɛ ij (Ẋn j Vj n = η eff Ẋi n. a single dimensionless parameter η eff η.

45 Effective Theory 16 The Equations of Motion if l = characteristic length, rescale X n = l ˆX n, t = ml2 ˆt. EOM l-independent: exact scale invariance after rescaling t to remove η : ) κ n ɛ ij (Ẋn j Vj n = η eff Ẋi n. a single dimensionless parameter η eff η. η eff > 0. system dissipates energy into normal fluid. vortex/anti-vortex pairs annihilate in finite time.

46 Effective Theory 16 The Equations of Motion if l = characteristic length, rescale X n = l ˆX n, t = ml2 ˆt. EOM l-independent: exact scale invariance after rescaling t to remove η : ) κ n ɛ ij (Ẋn j Vj n = η eff Ẋi n. a single dimensionless parameter η eff η. η eff > 0. system dissipates energy into normal fluid. vortex/anti-vortex pairs annihilate in finite time. length scales: core size ξ (numerical deletion of vortices); torus size L; initial conditions

47 Inverse vs. Direct Cascade 17 η eff = : Inverse Cascade

48 Inverse vs. Direct Cascade 18 η eff = 0.1: Direct Cascade

49 Inverse vs. Direct Cascade 19 Kolmogorov Scaling: Theory how to properly define energy at length scale 1/k?

50 Inverse vs. Direct Cascade 19 Kolmogorov Scaling: Theory how to properly define energy at length scale 1/k? energy at wavelength k when kζ T 1: project wave function onto k modes, measure H: ψ P k HP k ψ : 1 1 ( E qu (k) = dk θ 2 kψ 2 dk θ ve iθ) (k) 2. 2

51 Inverse vs. Direct Cascade 19 Kolmogorov Scaling: Theory how to properly define energy at length scale 1/k? energy at wavelength k when kζ T 1: project wave function onto k modes, measure H: ψ P k HP k ψ : 1 1 ( E qu (k) = dk θ 2 kψ 2 dk θ ve iθ) (k) 2. 2 T > 0 SF loses phase coherence when kζ T 1. here use classical definition: 1 E cl (k) = dk θ 2 v(k) 2.

52 Inverse vs. Direct Cascade 19 Kolmogorov Scaling: Theory how to properly define energy at length scale 1/k? energy at wavelength k when kζ T 1: project wave function onto k modes, measure H: ψ P k HP k ψ : 1 1 ( E qu (k) = dk θ 2 kψ 2 dk θ ve iθ) (k) 2. 2 T > 0 SF loses phase coherence when kζ T 1. here use classical definition: 1 E cl (k) = dk θ 2 v(k) 2. E cl most common in literature. ignoring quantum phenomena!

53 Inverse vs. Direct Cascade 20 Classical Kolmogorov Scaling: Inverse Cascade N(t) = N(0) = η eff = E k 5/3 k k k

54 Inverse vs. Direct Cascade 21 Quantum Kolmogorov Scaling: Direct Cascade N(t) = N(0) = η eff = k 5/3 k E k

55 Counting Vortices 22 Experimental Results experiment easily counts vortices at time t, N(t) dn dt = Γ 1N Γ 2 N 2. vortices leave imaging region

56 Counting Vortices 22 Experimental Results experiment easily counts vortices at time t, N(t) dn dt = Γ 1N Γ 2 N 2. vortices leave imaging region they measured Γ 2 T 2 /µl 2. Kwon, Moon, Choi, Seo, Shin, arxiv:

57 Counting Vortices 23 Smoke Ring Dynamics our equations can be exactly solved for ± pair of vortices

58 Counting Vortices 23 Smoke Ring Dynamics our equations can be exactly solved for ± pair of vortices given initial separation r 0, annihilation in finite time t ann = r2 0 m η 2 eff η eff

59 Counting Vortices 23 Smoke Ring Dynamics our equations can be exactly solved for ± pair of vortices given initial separation r 0, annihilation in finite time t ann = r2 0 m 1 + ηeff 2 4 η eff uniform gas : r 0 LN 1/2 = global vortex count N(t) obeys dn dt Γ 2N 2, Γ 2 8 η eff ml 2 N(t) = N(0) 1 + N(0)Γ 2 t.

60 Counting Vortices 24 Simulations: Random Mixture η eff = dn dt 10 1 two-body fit 10 2 N N

61 Counting Vortices 25 The Inverse Cascade large same-sign clusters form. annihilation only at edges of clusters (a curve of fractal dimension d f = 4/3)

62 Counting Vortices 25 The Inverse Cascade large same-sign clusters form. annihilation only at edges of clusters (a curve of fractal dimension d f = 4/3) dn dt d 2 x η eff m n N 2 +n Γ 2 fractal area L2

63 Counting Vortices 25 The Inverse Cascade large same-sign clusters form. annihilation only at edges of clusters (a curve of fractal dimension d f = 4/3) dn dt d 2 x η eff m n N 2 +n Γ 2 fractal area L2 finite size: fractal on length scales r < x < L. ( ) L df fractal area r 2 r

64 Counting Vortices 25 The Inverse Cascade large same-sign clusters form. annihilation only at edges of clusters (a curve of fractal dimension d f = 4/3) dn dt d 2 x η eff m n N 2 +n Γ 2 fractal area L2 finite size: fractal on length scales r < x < L. ( ) L df fractal area r 2 r r LN 1/2 : dn dt N 1+d f/2 N 5/3 Bernard, Boffetta, Celani, Falkovich, Nature Physics (2006)

65 Counting Vortices 26 Simulations: Inverse Cascade η eff = 0.02 (hybrid inverse/direct cascade) dn dt N 5/ N

66 Vortex Drag 27 Why Does η T 2? using FDT, η T 2 diffusion constant D T 1

67 Vortex Drag 27 Why Does η T 2? using FDT, η T 2 diffusion constant D T 1 particle-vortex duality : low energy EFT for single vortex equivalent to (relativistic) QED with scalar (of mass M ) L 1 2 ( µ ia µ ) φ M 2 φ 2 1 4e 2 F 2

68 Vortex Drag 27 Why Does η T 2? using FDT, η T 2 diffusion constant D T 1 particle-vortex duality : low energy EFT for single vortex equivalent to (relativistic) QED with scalar (of mass M ) L 1 2 ( µ ia µ ) φ M 2 φ 2 1 4e 2 F 2 T M: kinetic theory computation of D. phase space factors in integral provide D T 1. Lee, Fisher, International Journal of Modern Physics B (1991) Moore, Teaney, Physical Review C (2005)

69 Conclusions 28 Our Model first-order HVI equations are the effective theory of 2d SF turbulence: ) κ n ɛ ij (Ẋj n Vn j = η eff Ẋi n

70 Conclusions 28 Our Model first-order HVI equations are the effective theory of 2d SF turbulence: ) κ n ɛ ij (Ẋj n Vn j = η eff Ẋi n two types of turbulence possible:

71 Conclusions 28 Our Model first-order HVI equations are the effective theory of 2d SF turbulence: ) κ n ɛ ij (Ẋj n Vn j = η eff Ẋi n two types of turbulence possible: classical turbulence: inverse cascade: l r, ζt ; insensitive to vortex quantization

72 Conclusions 28 Our Model first-order HVI equations are the effective theory of 2d SF turbulence: ) κ n ɛ ij (Ẋj n Vn j = η eff Ẋi n two types of turbulence possible: classical turbulence: inverse cascade: l r, ζt ; insensitive to vortex quantization quantum turbulence: direct cascade: l r, ζ T ; driven by vortex drag. phase of order parameter plays a role

73 Conclusions 28 Our Model first-order HVI equations are the effective theory of 2d SF turbulence: ) κ n ɛ ij (Ẋj n Vn j = η eff Ẋi n two types of turbulence possible: classical turbulence: inverse cascade: l r, ζt ; insensitive to vortex quantization quantum turbulence: direct cascade: l r, ζ T ; driven by vortex drag. phase of order parameter plays a role predictions for experiment: dn dt 8 η eff rand ml 2 N 2. dn dt N 5/3. inv

74 Conclusions 29 Further Questions what is the mechanism behind k 5/3 in quantum cascade?

75 Conclusions 29 Further Questions what is the mechanism behind k 5/3 in quantum cascade? quantitative description of values of L/ r, η eff, etc., for when inverse vs. direct cascade arises

76 Conclusions 29 Further Questions what is the mechanism behind k 5/3 in quantum cascade? quantitative description of values of L/ r, η eff, etc., for when inverse vs. direct cascade arises further analogies btwn quantum turbulence and classical turbulence? ( ψ) n r n/3?

77 Conclusions 29 Further Questions what is the mechanism behind k 5/3 in quantum cascade? quantitative description of values of L/ r, η eff, etc., for when inverse vs. direct cascade arises further analogies btwn quantum turbulence and classical turbulence? ( ψ) n r n/3? computation of η eff in microscopic models of 2d SF