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

Transcription

1 Center for Electronic Correlations and Magnetism University of Augsburg Superfluid Helium-3: From very low Temperatures to the Big Bang Dieter Vollhardt Dvořák Lecture Institute of Physics, Academy of Sciences of the Czech Republic, Praha June 8, 2011

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

3 Helium Two stable Helium isotopes: 4 He: air, oil wells,... Janssen/Lockyer (1868) He air 3 6 He Cleveit (UO 2 ) 4 He air Ramsay (1895) from Jáchymov 3 He: Li n H He e (1939) Research on macroscopic samples of 3 He since 1947

4 Helium Atoms: spherical, hard core diameter 2.5 Å Interaction: hardsphererepulsion van der Waals dipole/multipole attraction Boiling point: 4.2 K, 4 He Kamerlingh Onnes (1908) 3.2 K, 3 He Sydoriak et al. (1949) Dense, simple liquid isotropic short-range interactions extremely pure

5 Helium solid 3 He 4 He P (bar) T c? superfluid superfluid λ-line normal fluid T(K) vapor Atoms: spherical shape weak attraction low mass strong zero-point motion T0, 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: p n n p n p p Atom(!) is a S = 0 Boson S = 1 2 Fermion Quantum liquids Phase transition T λ = 2.2 K ( BEC ) T c =??? Fermi liquid theory

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

8 Fermi gas: Excited states (T>0) k z Particle Fermi surface Hole k y k x Fermi sea Exact k-states ("particles"): infinite life time Switch on interaction adiabatically (d=3)

9 Landau Fermi liquid Standard model of condensed matter physics k z (Quasi-) Hole (Quasi-) Particle Landau (1956/58) 1-1 correspondence between one-particle states (k,σ) = elementary excitation k y k x Prototype: Helium-3 Large effective mass Strongly enhanced spin susceptibility Strongly reduced compressibility

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

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

12 Arbitrarily weak attraction Cooper pair ( k, ; k, ) k, ξ 0 k, L0,2,4,... () r S=0 (singlet) + r L1,3,5,... () () r 0 () - r L = 0: isotropic wave function L > 0: anisotropic wave function S=1 (triplet) Helium-3: Strongly repulsive interaction L > 0 expected

13 BCS theory Bardeen, Cooper, Schrieffer (1957) Generalization to macroscopically many Cooper pairs ε c <<E F E F

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? T c? Thanksgiving 1971: Transition in 3 He at T c = K Osheroff, Richardson, Lee (1972)

15 The Nobel Prize in Physics 1996 "for their discovery of superfluidity in helium-3" David M. Lee Cornell (USA) Douglas D. Osheroff Stanford (USA) Robert C. Richardson Cornell (USA)

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, Attraction due to spin fluctuations Anderson, Brinkman (1973) anisotropy directions in a 3 He Cooper pair ˆd ˆl orbital part spin part

19 and a mystery! NMR experiment on nuclear spins I= 1 2 Osheroff et al. (1972) L T ( ) 3mT?! L Larmor frequency: L 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 ( k) 0 Balian, Werthamer (1963) Vdovin (1963) (pseudo-) isotropic state s-wave superconductor Weak-coupling theory: stable for all T<T c

22 A-phase strong anisotropy ( k ˆ ) 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 Sr 2 RuO 4 Strong-coupling effect

23 3 He-A: Spectrum near poles Volovik (1987) ˆl ^ˆl l E k Energy Egap k Excitations k Fermi sea = Vacuum 2 E v k k sin (ˆˆ F kl, ) k F 0 e 1 1 kˆ kˆ A k l ˆ F pkea lˆ lˆ 2 chiralities

24 3 He-A: Spectrum near poles Volovik (1987) ˆl ^ˆl l E k Energy Egap k Excitations k Fermi sea = Vacuum 2 E v k k sin (ˆˆ F kl, ) e k F kˆ kˆ A k l ˆ F pkea lˆ lˆ 2 chiralities ij = g pp i j ij 2 g v Fll i j ( ijll i j) kf Lorentz invariance: Symmetry enhancement at low energies 2

25 3 He-A: Spectrum near poles ˆl ^ˆl l E k Volovik (1987) Energy gap Excitations Fermi sea = Vacuum k 2 E v k k sin (ˆˆ F kl, ) e k F kˆ kˆ lˆ lˆ 2 chiralities ij = g pp i j Fermi point: spectral flow ij 2 g v Fll i j ( ijll 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)

26 A 1 -phase finite magnetic field Long-range ordered magnetic liquid

27 Broken Symmetries, Long Range Order

28 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 T<T c Average magnetization: M 0 M 0 Order parameter Symmetry group: SO(3) U(1) SO(3) T<T C : SO(3) rotation symmetry in spin space spontaneously broken

29 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 T<T c Symmetry group: SO(3) U(1) SO(3) T<T C : SO(3) rotation symmetry in real space spontaneously broken

30 Broken Symmetries, Long Range Order 2. order phase transition T<T c : higher order, lower symmetry of ground state III. Conventional superconductor Pair amplitude c c T>T c k k 0 T<T c i e i Gauge transf. ck ck e : gauge invariant not gauge invariant Symmetry group U(1) Order parameter

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 U(1) gauge symmetry also broken in BEC

32 Broken symmetries in superfluid 3 He S=1, L=1 in all phases Cooper pair: ˆd ˆl orbital part spin part Quantum coherence in anisotropy direction for spin anisotropy direction in real space phase magnetic liquid crystal superfluid Characterized by (2S 1) ( 2L 1) 2 = 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)

33 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" superfluidity Spontaneously broken spin-orbit symmetry Leggett (1972) Cooper pairs Fixed relative orientation

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" superfluidity 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 pair ) SU(2) L+R Goldstone excitations (bosons) 3 pions

35 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 Cooper pairs Fixed absolute orientation

36 Resolution of the NMR puzzle

37 Superfluid 3 He - a quantum amplifier Cooper pairs in 3 He-A Fixed absolute orientation What determines the actual relative orientation of d ˆ, ˆl? Anisotropic spin-orbit interaction of nuclear dipoles: Dipole-dipole coupling of 3 He nuclei: g D 10 7 K TC Unimportant?!

38 Superfluid 3 He - a quantum amplifier Cooper pairs in 3 He-A Fixed absolute orientation Long-range order in d ˆ, ˆl due to Cooper pairing 7 g 10 K: tiny (but lifts degeneracy of relative orientation) D d ˆ, ˆl Quantum coherence locked in all Cooper pairs in the same way NMR frequency increases: 2 2 ( H ) g 2 D ( T ) Leggett (1973) Nuclear dipole interaction macroscopically measurable

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

40 Order parameter textures and topological defects

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

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

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

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

45 Big bang simulation in the low temperature lab BANG!

46 Universality in continuous phase transitions BANG! 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 3. Expansion + rapid cooling Defects overlap 2. Nucleation of independently ordered regions Clustering of ordered regions Defects 4. T T C : Vortex tangle Estimate of 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 Present research on superfluid 3 He: Quantum Turbulence Classical Turbulence Leonardo da Vinci ( ) Flow through grid Quantum Turbulence = Turbulence in the absence of viscous dissipation (superfluid at T0) What provides dissipation in the absence of friction? Why are quantum and classical turbulence so similar? Test system: 3 He-B Vinen, Donnelly: Physics Today (April, 2007)

50 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