Center for Electronic Correlations and Magnetism University of Augsburg Superfluid Helium-3: Universal Concepts for Condensed Matter and the Big Bang Dieter Vollhardt GSI Kolloquium, Darmstadt; May 9, 2017
Periodic table of the elements Noble gas Helium: after hydrogen the most abundant element in the universe
Helium Two stable Helium isotopes: 4 He, 3 He 4 He: air, oil wells, ... Janssen/Lockyer/Secci (1868) Ramsay (1895) 4 3 He He Cleveit (UO 2 ) − − ≈ × ≈ × 6 6 5 10 1 10 4 air He air + → + α 3 He: Li n H 6 1 3 3 0 1 ↓ Research on macroscopic samples 2 He 3 of 3 He only since 1947 4 He: Coolant, Welding, Balloons 3 He: - Contrast agent in medicine - Neutron detectors - 3 He- 4 He dilution refrigerators (quantum computers!)
Helium spherical, hard core diameter ∼ 2.5 Å Atoms: • hard sphere repulsion Interaction: • 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) { isotropic Dense, simple liquids short-range interactions extremely pure
Helium 40 solid 3 He P (bar) [10 bar = 1 MPa] 4 He 30 normal fluid 20 superfluid superfluid λ-line 10 ? vapor 0 0 1 2 3 4 5 6 T(K) • spherical shape weak attraction Atoms: • light mass strong zero-point motion < T 0, P 3 MPa: Helium remains liquid λ ∝ → → T 0 quantum phenomena on a macroscopic scale k T B
Helium 4 He 3 He 2 e - , S = 0 Electron shell: p p n Nucleus: n n p p 1 S = S = 0 2 Quantum Atom(!) is a Fermion Boson liquids T λ = 2.2 K T c = ??? Phase transition Bose-Einstein condensation superfluid with frictionless flow
Interacting Fermions (Fermi liquid): Ground state k z Landau (1956/57) Fermi surface k y Fermi sphere k x
Instability of Fermi liquid k z + 2 non-interacting fermions k y Fermi sphere k x
⇒ Arbitrarily weak attraction instability Cooper (1956) k z Cooper pair k k y k x − k Universal fermionic property
⇒ α − β Arbitrarily weak attraction Cooper pair k k ( , ; , ) , α k ξ 0 − k , β Antisymmetry Ψ = ψ ↑ ↑ Ψ = ψ ↑↓ − ↓ ↑ + r ( ) ( ) r = = L L 1,3,5,... 0,2,4,... + ψ ↑↓ + ↓ ↑ 0 ( ) r S=0 (singlet) + ψ ↓↓ - r ( ) S=1 (triplet) L = 0 (“s-wave”): isotropic pair wave function L > 0 (“p,d,f,… -wave”): anisotropic pair wave function 3 He: Strongly repulsive interaction L > 0 expected
BCS theory Bardeen, Cooper, Schrieffer (1957) Generalization to macroscopically many Cooper pairs ε c <<E F Energy gap Δ(T) Energy gap Δ(T) here: L=0 (s-wave) here: L=0 (s-wave) E F Pair condensate = − T ε N V with transition temperature 1.13 exp( 1/ ( 0) ) c c L in weak coupling theory ε c , V L : Magnitude ? Origin ? T c ? Thanksgiving 1971: Transition in 3 He at T c = 0.0026 K Osheroff, Richardson, Lee (1972) Osheroff, Gully, Richardson, Lee (1972)
The Nobel Prize in Physics 1996 "for their discovery of superfluidity in helium-3" David M. Lee Douglas D. Osheroff Robert C. Richardson Cornell (USA) Stanford (USA) Cornell (USA)
Phase diagram of Helium-3 { P-T phase diagram isotropic short-range interactions Dense, simple liquid extremely pure nuclear spin S=1/2 ordered disordered Solid (bcc) spins spins http://ltl.tkk.fi/research/theory/helium.html Fermi liquid
Phase diagram of Helium-3 { P-T phase diagram isotropic short-range interactions Dense, simple liquid extremely pure nuclear spin S=1/2 http://ltl.tkk.fi/research/theory/helium.html viscosity high viscosity (machine oil) zero
Phase diagram of Helium-3 P-T-H phase diagram http://ltl.tkk.fi/images/archive/ab.jpg Millikelvin Cryostat WMI Garching “Very (ultra) low temperatures”: T << T boiling ~ 3 K and << T backgr. rad. ~ 3 K
Superfluid phases of 3 He Experiment: Osheroff, Richardson, Lee, Wheatley, ... Theory: Leggett, Wölfle, Mermin, … L=1, S=1 (“p-wave, spin-triplet“) in all 3 phases Attraction due to spin fluctuations Anderson, Brinkman (1973) orbital part ˆ l anisotropy directions in every 3 He Cooper pair ˆ spin part d
… and a mystery! NMR experiment on nuclear spins I = 1 2 Osheroff et al. (1972) ω ω − ω ∝ ∆ T 2 2 2 ( ) L ?! 3mT ω L ω = γ H Larmor frequency: L superfluid normal T T C,A ⇔ Shift of ω L spin-nonconserving interactions − g K T 7 nuclear dipole interaction 10 D C Origin of frequency shift ?! Leggett (1973)
The superfluid phases of 3 He
B-phase ↑↑ ↑↓ + ↓↑ ↓↓ All spin states occur equally , , ∆ = ∆ Balian, Werthamer (1963) ( ) k 0 Vdovin (1963) energy gap Fermi sphere ↔ “(pseudo-) isotropic state“ s-wave superconductor Weak-coupling theory: stable for all T<T c
A-phase Spin states ↑↑ ↓↓ occur equally , strong gap anisotropy ∆ = ∆ ˆ ˆ, ) ˆ k k l Anderson, Morel (1961) ( ) sin( 0 Cooper pair ˆ l orbital angular momentum energy gap Helped to understand unconventional pairing in • heavy fermion superconductors (CeCu 2 Si 2 , UPt 3 , …) Fermi sphere • high-T c (cuprate) superconductors Energy gap with point nodes Strong-coupling effect “axial (chiral) state”
3 He-A: Spectrum near nodes Volovik (1987) ˆ l ^ l ˆ l E k Excitations ∆ k Energy gap E k Fermi sea = Vacuum ( ) = − 2 + ∆ sin ( ˆ ˆ E k k k l 2 2 2 2 p p ij v , ) = g Lorentz invariance F k F 0 i j 2 ∆ + + ˆ ˆ k l chirality “up” = + δ − 1 l l l l ij 2 g v ( ) e = i j ij i j F k − ˆ − ˆ F k l chirality “down” 1 = ˆ k F l A = − e p k A
3 He-A: Spectrum near nodes The Universe in a Helium Droplet, Volovik (2003) ˆ l ^ l ˆ l E k Excitations ∆ k ⇔ Energy gap E k Fermi sea = Vacuum Fermi point: spectral flow of fermionic charge ( ) = − 2 + ∆ sin ( ˆ ˆ E k k k l 2 2 2 2 p p p p ij ij v , ) = = g g Lorentz invariance F k F 0 i i j j 2 ∆ + + ˆ ˆ k l chirality “up” = + δ − 1 l l l l ij 2 g v ( ) e = i j ij i j F k − ˆ − ˆ F k l chirality “down” 1 ⇔ = E cp Massless, chiral leptons, e.g., neutrino ( ) p Chiral (Adler) anomaly measured Bevan et al. (1997)
A 1 -phase In finite magnetic field Only spin state ↑↑ Long-range ordered magnetic liquid
Cooper pairing of Fermions vs. Bose-Einstein condensation ξ ≈ 10 Å } Cooper pair: “Quasi-boson“ ξ ≈ 10000 Å Conventional superconductors 0 BCS ξ ≈ 150 Å Superfluid 3 He 0 High-T C superconductors ? 0 Superfluid 4 He: ξ 1 Å BEC Tightly packed fermions (boson) 0 Leggett (1980) New insights from BEC of cold atoms
Broken Symmetries & Long-Range Order
Broken Symmetries & Long-Range Order ↔ Normal 3 He 3 He-A, 3 He-B: 2 nd order phase transition T<T c : higher order, lower symmetry of ground state I. Ferromagnet T>T T<T c c ≠ Order parameter = Average magnetization: M 0 M 0 ⊂ SO(3) U(1) SO(3) Symmetry group: T<T C : SO(3) rotation symmetry in spin space spontaneously broken
Broken Symmetries & Long-Range Order ↔ Normal 3 He 3 He-A, 3 He-B: 2 nd order phase transition T<T c : higher order, lower symmetry of ground state II. Liquid crystal T>T T<T c c ⊂ SO(3) U(1) SO(3) Symmetry group: T<T C : SO(3) rotation symmetry in real space spontaneously broken
Broken Symmetries & Long-Range Order ↔ Normal 3 He 3 He-A, 3 He-B: 2 nd order phase transition T<T c : higher order, lower symmetry of ground state III. Conventional superconductor . . . . . . . . . . . . . T>T T<T c c ↓ = e φ ∆ c c − i † † 0 Pair amplitude complex order parameter ↑ k k ϕ → i c c e † † Gauge transf. : gauge invariant not gauge invariant σ σ k k Symmetry group U(1) —
Broken Symmetries & Long-Range Order ↔ Normal 3 He 3 He-A, 3 He-B: 2 nd order phase transition T<T c : higher order, lower symmetry of ground state III. Conventional superconductor . . . . . . . . . . . . . T>T T<T c c T<T C : U(1) “gauge symmetry“ spontaneously broken
Broken symmetries in superfluid 3 He L=1, S=1 in all 3 phases orbital part ˆ l Cooper pair: spin part ˆ d { phase (complex order parameter) Superfluid, anisotropy direction in real space liquid crystal Quantum coherence in anisotropy direction in spin space magnet × + × + Characterized by L S = 18 real numbers 2 (2 1) (2 1 ) 3x3 order parameter matrix A iμ SO(3) S ´ SO(3) L ´ U(1) φ symmetry spontaneously broken Leggett (1975) SU(2) L ´ SU(2) R ´ U(1) Y for electroweak interactions Pati, Salam (1974)
Mineev (1980) Broken symmetries in superfluid 3 He Bruder, Vollhardt (1986) SO(3) S ´ SO(3) L ´ U(1) φ symmetry broken 3He-A ´ U(1) L z - φ U(1) S z Unconventional pairing ˆ l Cooper pairs Fixed absolute orientation
… solution of the NMR mystery
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