Superfluid Helium-3: From very low Temperatures to the Big Bang

Transcription

1 Superfluid Helium-3: From very low Temperatures to the Big Bang Universität Frankfurt; May 30, 2007 Dieter Vollhardt

2 Contents: The quantum liquids 3 He and 4 He Superfluid phases of 3 He Broken symmetries and long-range order Topological defects Big Bang simulation in the low temperature lab

3 Helium Two stable Helium isotopes: Janssen/Lockyer (1868) 4 He: air, oil wells,... Ramsay (1895) 3 He: He Luft Li + n H + α (1939) He + e He 5 10, 1 10 He Luft Research on macroscopic samples of 3 He since 1947

4 Helium Atoms: spherical, hard core diameter 2.5 Å Interaction: hard sphererepulsion van der Waals dipole (+...) attraction Boiling point: 4.2 K, 4 He Kamerlingh Onnes (1908) 3.2 K, 3 He Sydoriak, et al. (1949) Nobel Prize 1913 Dense, simple liquid { isotropic short-range interactions extremely pure nuclear spin S=1/2

5 Helium solid solid 3 He 4 He P (bar) superfluid superfluid λ-line normal fluid 0 0 vapor vapor T(K) Atoms: spherical shape weak attraction light mass strong zero-point motion λ T 0, P < 30 bar: Helium remains liquid kt B T 0 Macroscopic quantum phenomena

6 Helium 4 He 3 He Electron shell: 2 e -, S = 0 Nucleus: Atom(!) is a p n n p S = 0 Boson T λ = 2.2 K ( BEC ) n p p S = 1 2 Fermion T c =??? Fermi liquid theory Quantum liquids

7 Fermi gas: Ground state k z k y k x Fermi sea

8 Fermi gas: Excited states (T>0) k z Particle Hole k y k x Switch on interaction adiabatically

9 Landau Fermi liquid 1-1 correspondence between states k z (Quasi-) Hole (Quasi-) Particle = elementary excitation k y k x Prototype: Helium-3 Large effective mass Strongly enhanced spin susceptibility Strongly reduced compressibility

10 The Nobel Prize in Physics 1962 "for his pioneering theories for condensed matter, especially liquid helium" Lev Davidovich Landau USSR b. 1908, d. 1968

11 Instability of Landau Fermi liquid k z + 2 non-interacting particles k y k x Fermi sea

12 Arbitrarily weak attraction Cooper instability k z k k y k x k Universal fermionic property

13 k,α ( k, α; k, β ) Arbitrarily weak attraction Cooper pair ξ 0 k, β Ψ L= 0,2,4,... = ψ () r S=0 (singlet) Ψ = r + L= 1,3,5,... ψ () + ψ () r ψ () - r L = 0: isotropic wave function L > 0: anisotropic wave function S=1 (triplet) Helium-3: Strongly repulsive interaction L>0 expected

14 BCS theory Bardeen, Cooper, Schrieffer (1957) Generalization to macroscopically many Cooper pairs Energy gap Δ(T) "Pair condensate" with macroscopically coherent wave function E F ε c <<E F Transition temperature T = 1.13 ε exp( 1/ N( 0) V ) c c L "weak coupling theory" ε c, V L : Magnitude? Origin (e - +?)? T c?

15 Thanksgiving 1971: Transition in 3 He at T c = K The Nobel Prize in Physics 1996 "for their discovery of superfluidity in helium-3" Osheroff, Richardson, Lee (1972) David M. Lee USA, Cornell b Douglas D. Osheroff USA, Stanford b Robert C. Richardson USA, Cornell b. 1937

16 Phase diagram of Helium-3 P-T phase diagram Dense, simple liquid { isotropic short-range interactions extremely pure nuclear spin S=1/2 ordered spins disordered spins Solid (bcc) Normal Fermi liquid

17 Phase diagram of Helium-3 P-T-H phase diagram Very low temperatures : T << T boiling ~ 3-4 K << T backgr. rad. ~ 3 K

18 Superfluid phases of 3 He Theory + experiment: L=1, S=1 in all phases Leggett Wölfle Mermin, Pairing mechanism: Spin fluctuations Anderson, Brinkman (1973) anisotropy directions in a 3 He Cooper pair ˆd ˆl orbital part spin part Anisotropy + quantum coherence Long-range order in real and spin space

19 and a mystery! NMR experiment on nuclear spins I= 1 2 Osheroff et al. (1972) ω ω ω L T 2 2 Δ 2 ( ) 3mT?! ω L ω Larmor frequency: L = γ H superfluid T C,A normal T Shift of ω L spin-nonconserving interactions nuclear dipole interaction g 10 D 7 K TC Origin of frequency shift?! Leggett (1973)

20 The superfluid phases of 3 He

21 B-phase Ψ= Δ ( k) =Δ 0 Balian, Werthamer (1963) Vdovin (1963) (pseudo-) isotropic state s-wave superconductor Weak-coupling theory: stable for all T<T c

22 A-phase Ψ= + strong anisotropy Δ ( k ˆ ) =Δ sin( k ˆ, l ˆ ) Anderson, Morel (1961) 0 ˆl Cooper pair orbital angular momentum Axial state has point nodes unconventional pairing in heavy fermion/high-t c superconductors Strong-coupling effect

23 3 He-A: Spectrum near poles Volovik (1987) ^ l E k Energy Egap k Excitations Δ k Fermi sea = Vacuum ( ) 2 E = v k k sin (ˆˆ F +Δ kl, ) e k F = 1 kˆ kˆ A = k l ˆ F p= k ea + lˆ lˆ 2 chiralities ij = g pp i j ij 2 Δ g = v Fll i j+ ( δij ll i j) kf Lorentz invariance: Symmetry enhancement at low energies 2

24 3 He-A: Spectrum near poles Volovik (1987) ^ l E k Energy gap Excitations Fermi sea = Vacuum Δ k ( ) 2 E = v k k sin (ˆˆ F +Δ kl, ) e k F = 1 kˆ kˆ + lˆ lˆ 2 chiralities ij = g pp i j Fermi point: spectral flow ij 2 Δ g = v Fll i j+ ( δij ll i j) kf 2 Massless, chiral leptons, e.g., neutrino E( p) = cp Chiral anomaly of standard model The Universe in a Helium Droplet, Volovik (2003)

25 A 1 -phase finite magnetic field Ψ= Long-range ordered magnetic liquid

26 Broken Symmetries, Long Range Order

27 Broken Symmetries, Long Range Order Normal 3 He 3 He-A, 3 He-B: 2. order phase transition T<T c : higher order, lower symmetry of ground state I. Ferromagnet T>T c Average magnetization: M = 0 Symmetry group: SO(3) T<T c M 0 Order parameter U(1) SO(3) T<T C : SO(3) rotation symmetry in spin space spontaneously broken

28 Broken Symmetries, Long Range Order 2. order phase transition T<T c : higher order, lower symmetry of ground state II. Liquid crystal T>T c Symmetry group: SO(3) T<T c U(1) SO(3) T<T C : SO(3) rotation symmetry in real space spontaneously broken

29 Broken Symmetries, Long Range Order 2. order phase transition T<T c : higher order, lower symmetry of ground state III. Conventional superconductor Pair amplitude Gauge transf. c c T>T c = k k 0 gauge invariant c c e kσ kσ iϕ T<T c i Δe φ not gauge invariant Order parameter

30 Broken Symmetries, Long Range Order 2. order phase transition T<T c : higher order, lower symmetry of ground state III. Conventional superconductor T>T c T<T c Pair amplitude c c = k k 0 i Δe φ Order parameter Gauge transf. c c e kσ kσ iϕ Symmetry group: U(1) ---

31 Broken Symmetries, Long Range Order 2. order phase transition T<T c : higher order, lower symmetry of ground state III. Conventional superconductor T>T c T<T c T<T C : U(1) gauge symmetry spontaneously broken Conventional superconductivity (BCS) Superfluid 4 He (BEC)

32 Cooper pairing of Fermions vs. Bose-Einstein condensation Cooper pair: Quasi-boson Conventional superconductors ξ Å Superfluid 3 He ξ0 150 Å BCS High T C superconductors ξ0 10 Å Continuous crossover? Tightly packed bosons ξ0 1 Å BEC New insights from BEC of cold atoms Leggett (1980)

33 Broken symmetries in superfluid 3 He L=1, S=1 in all phases Cooper pair: ˆd ˆl orbital part spin part Quantum coherence in { phase anisotropy direction for spin anisotropy direction in real space Superfluid, magnetic liquid crystal Characterized by 2 (2L + 1) (2S + 1) = 18 real numbers 3x3 3x3 order parameter matrix A iμ iμ SO(3) S SO(3) L U(1) φ symmetry spontaneously broken Leggett (1975)

34 Broken symmetries in superfluid 3 He Mineev (1980) Bruder, DV (1986) 3He-B SO(3) S SO(3) L U(1) φ symmetry broken SO(3) S+L _ Unconventional" pairing Spontaneously broken spin-orbit Symmetry Leggett (1972) Cooper pairs Fixed relative orientation

35 Broken symmetries in superfluid 3 He Mineev (1980) Bruder, DV (1986) 3He-B SO(3) S SO(3) L U(1) φ symmetry broken SO(3) S+L _ Unconventional" pairing Relation to high energy physics Isodoublet u d, u d L R chiral invariance Global symmetry SU(2) L SU(2) R qq condensation ( Cooper pairing ) SU(2) L+R Goldstone excitations (bosons) 3 pions

36 Broken symmetries in superfluid 3 He Mineev (1980) Bruder, DV (1986) 3He-A SO(3) S SO(3) L U(1) φ symmetry broken U(1) Sz U(1) Lz - φ Unconventional" pairing ˆl heavy fermion/high-t c superconductivity Cooper pairs Fixed absolute orientation

37 Resolution of the NMR puzzle: Macroscopic quantum amplification

38 Superfluid 3 He as quantum amplifier Cooper pairs in 3 He-A Coupling of d ˆ,? ˆl Nuclear dipole interaction iiiiiiiii a iiiiiiiii a anisotropic! spin-orbit coupling Dipole interaction energy of 3 He nuclei: g D 2 μ 10 3 a 7 K T C Unimportant?!

39 Superfluid 3 He as quantum amplifier Cooper pairs in 3 He-A Long-range order in d ˆ, ˆl 7 g 10 K lifts degeneracy of relative orientation D d ˆ, ˆl Quantum coherence locked in all Cooper pairs NMR frequency increases: 2 2 ω = ( γ H ) + g 2 DΔ ( T ) Leggett (1973) Nuclear dipole interaction macroscopically measurable

40 The Nobel Prize in Physics 2003 "for pioneering contributions to the theory of superconductors and superfluids" Alexei A. Abrikosov, USA and Russia b Vitaly L. Ginzburg, Russia b Anthony J. Leggett, UK and USA b. 1938

41 Order parameter textures and topological defects

42 Order parameter textures Orientation of anisotropy directions d ˆ, ˆl in 3 He-A: Magnetic field Walls ˆl ˆd d ˆ, ˆl Textures in d ˆ, ˆl liquid crystals Topologically stable defects

43 Order parameter textures and topological defects D=2: domain walls in ˆd or ˆl Single domain wall Domain wall lattice

44 Order parameter textures and topological defects D=1: Vortices Vortex formation in rotation experiments e.g., Mermin-Ho vortex (non-singular)

45 Order parameter textures and topological defects D=0: Monopoles e.g., boojum in ˆl -texture of 3 He-A Defect formation by, e.g., geometric constraints rotation rapid crossing through phase transition

46 Universality in continuous phase transitions High symmetry, short-range order T>T c Spins: paramagnetic Helium: normal liquid Universe: Unified forces and fields T=T c Phase transition Broken symmetry, long-range order ferromagnetic superfluid elementary particles, fundamental interactions Defects: domain walls vortices, etc. cosmic strings, etc. Kibble (1976) T<T c nucleation of galaxies?

47 Rapid thermal quench through 2. order phase transition Kibble (1976) 1. Local temperature T T C Regions overlap Expansion + rapid cooling Nucleation of independently ordered regions T < T C : Vortex tangle Density of defects Zurek (1985) Kibble-Zurek mechanism : How to test?

48 Big Bang simulation in the low temperature laboratory Grenoble: Bäuerle et al. (1996), Helsinki: Ruutu et al. (1996) 3 He-B Measured vortex tangle density: Quantitative support for Kibble-Zurek mechanism

49 Big Bang simulation in the low temperature laboratory Grenoble: Bäuerle et al. (1996), Helsinki: Ruutu et al. (1996) 3 He-B

50 Present research on superfluid 3 He: Quantum Turbulence = Turbulence in the absence of viscous dissipation Test system: 3 He-B Vinen, Donnelly: Physics Today (April, 2007)

51 Conclusion Superfluid Helium-3: Anisotropic superfluid - 3 different bulk phases - Cooper pairs with internal structure Large symmetry group broken - Close connections to particle theory - Zoo of topological defects - Kibble-Zurek mechanism quantitatively verified