superfluid helium-3: from very low temperatures to the big ... filethe quantum liquids 3he and 4he...
TRANSCRIPT
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Superfluid Helium-3:
From very low Temperaturesto the Big Bang
Dieter Vollhardt
Group Seminar, Theoretical Physics III November 4, 2009
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The quantum liquids 3He and 4He Superfluid phases of 3He Broken symmetries and long-range order Topologically stable defects Big Bang simulation in the low temperature lab
Contents:
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The Superfluid Phases of Helium 3D. Vollhardt and P. Wölfle(Taylor & Francis, 1990)
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Two stable Helium isotopes:
4He: air, oil wells, ... Janssen/Lockyer (1868)Ramsay (1895)
3He: (1939)6 1 33 0 1Li n H
32He e
Helium
Research on macroscopic samples of 3He since 1947
6 6
air5 10 , 1 10
3
4
He Heair He
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Interaction: • hard sphere repulsion• van der Waals dipole/multipole attraction
spherical, hard core diameter 2.5 ÅAtoms:
4.2 K, 4He Kamerlingh Onnes (1908)
3.2 K, 3He Sydoriak, et al. (1949)
Boiling point:
Helium
Nobel Prize 1913
Dense, simple liquidisotropicshort-range interactionsextremely pure
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0T(K)
P (b
ar)
superfluid10
20
30
40
01 2 3 4 5 6
vapor
Helium
normal fluid
4He
λ-line
solid
superfluid
3He
0T
Bk T
Macroscopic quantum phenomena
T0, P 30 bar: Helium remains liquid
• spherical shape weak attraction• light mass strong zero-point motion
Atoms:
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Tλ = 2.2 K(“BEC“)
Nucleus:
Atom(!) is a
S = 0
Boson
Helium
nn
p
pn
p
p
4He 3He
2 e-, S = 0Electron shell:
S =
Fermion
12
Tc = ???
Quantum liquids
Fermi liquid theory
Phasetransition
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Fermi gas: Ground state
kx
ky
kz
Fermi sea
Fermi surface
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kx
ky
kz
Fermi gas: Excited states (T>0)
Switch on interaction adiabatically (d=3)
Exact k-states ("particles"): infinite life time
Particle
Hole
Fermi sea
Fermi surface
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Landau Fermi liquid
kx
ky
kz
1-1 correspondencebetween k-states
(Quasi-) Particle
(Quasi-) Hole
Prototype: Helium-3• Large effective mass• Strongly enhanced spin susceptibility• Strongly reduced compressibility
= elementary excitation
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Lev Davidovich Landau USSR
b. 1908, d. 1968
The Nobel Prize in Physics 1962"for his pioneering theories for condensed matter,
especially liquid helium"
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kx
ky
kz
Instability of Landau Fermi liquid
+ 2 non-interacting particles
Fermi sea
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kx
ky
k
k
Arbitrarily weak attraction Cooper instability
kz
Universal fermionic property
Cooper pair
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ξ0, k
,k
S=0 (singlet)
0,2,4,... ( )L r
S=1 (triplet)
1,3,5,... ( )L + r
0 ( ) r
( ) - r
L = 0: isotropic wave functionL > 0: anisotropic wave function
Helium-3: Strongly repulsive interaction L > 0 expected
Arbitrarily weak attraction Cooper pair ( , ; , ) k k
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Generalization to macroscopically many Cooper pairs
BCS theory Bardeen, Cooper, Schrieffer (1957)
EF
εc<<EF
"Pair condensate" with macroscopically coherent wave function
"weak coupling theory"
1.13 exp( 1/ 0) )(c c LT ε N V Transition temperature
Energy gap Δ(T)
εc, VL: Magnitude ? Origin ? Tc ?
Thanksgiving 1971: Transition in 3He at Tc = 0.0026 KOsheroff, Richardson, Lee (1972)
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The Nobel Prize in Physics 1996"for their discovery of superfluidity in helium-3"
David M. LeeCornell (USA)
Douglas D. OsheroffStanford (USA)
Robert C. RichardsonCornell (USA)
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NormalFermi liquid
Solid (bcc)disorderedspins
orderedspins
Phase diagram of Helium-3
P-T phase diagram Dense, simple liquidisotropicshort-range interactionsextremely purenuclear spin S=1/2
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Phase diagram of Helium-3
P-T-H phase diagram
“Very low temperatures”: T << Tboiling ~ 3-4 K<< Tbackgr. rad. ~ 3 K
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Theory + experiment: L=1, S=1 in all phases
Superfluid phases of 3He
anisotropy directionsin a 3He Cooper pair
orbital part
spin part d̂
l̂
Attraction due to spin fluctuations Anderson, Brinkman (1973)
LeggettWölfleMermin, …
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… and a mystery!
Osheroff et al. (1972)3m
T
Larmor frequency: L H
NMR experiment on nuclear spins I= 12
TTC,A
?!
L
superfluid normal
22 2 ( )L T
Shift of ωL spin-nonconserving interactions nuclear dipole interaction
710D CTg K
Origin of frequency shift ?! Leggett (1973)
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The superfluid phases of 3He
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B-phase
Balian, Werthamer (1963)Vdovin (1963)
(pseudo-) isotropic state s-wave superconductor
Weak-coupling theory: stable for all T<Tc
0( ) k
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A-phase
Strong-coupling effect
0( ) sin(ˆ ˆ, )ˆk k l Anderson, Morel (1961)
l̂
strong anisotropy
Cooper pair orbital angular momentum
“Axial state” has point nodes
“unconventional” pairing in• heavy fermion/high-Tc superconductors• Sr2RuO4
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3He-A: Spectrum near poles Volovik (1987)
EkEnergy gap
Excitations
l^
Fermi sea = Vacuum
Ek
k
22 2 2 2F 0v sin ( ˆ ˆ, )F k lE k k k
2 chiralities1
1
ˆ ˆ
ˆ ˆ
k l
k le
ˆFlke
Ap k A
ijg= i jp p2
ij 2Fg v ( )i j ij i j
F
l l l lk
Lorentz invariance:Symmetry enhancement at low energies
l̂l̂
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Energy gap
Excitations
l^
Fermi sea = Vacuum
Ek
k
3He-A: Spectrum near poles
22 2 2 2F 0v sin ( ˆ ˆ, )F k lE k k k
1
1
ˆ ˆ
ˆ ˆ
k l
k le
2 chiralities
ijg= i jp p
Volovik (1987)
The Universe in a Helium Droplet,Volovik (2003)
Massless, chiral leptons, e.g., neutrino ( )E cpp
Chiral anomaly of standard model
Fermi point:spectral flow
2ij 2
Fg v ( )i j ij i jF
l l l lk
l̂l̂
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A1-phase
finite magnetic field
Long-range ordered magnetic liquid
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Broken Symmetries, Long Range Order
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Normal 3He 3He-A, 3He-B:2. order phase transition
Broken Symmetries, Long Range Order
T<Tc: higher order, lower symmetry of ground state
0M0M
I. Ferromagnet
Order parameter
T>Tc T<Tc
Average magnetization:Symmetry group: SO(3) U(1) SO(3)
T<TC: SO(3) rotation symmetry in spin space spontaneously broken
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Broken Symmetries, Long Range Order
2. order phase transition
T<Tc: higher order, lower symmetry of ground state
II. Liquid crystal
T>Tc T<Tc
Symmetry group: SO(3) U(1) SO(3)
T<TC: SO(3) rotation symmetry in real space spontaneously broken
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Broken Symmetries, Long Range Order
III. Conventional superconductor
2. order phase transition
T<Tc: higher order, lower symmetry of ground state
. ..
...
.
.
.
T>Tc T<Tc
Pair amplitude ie “Order parameter“c c † †k k 0
Gauge transf. :ic c e † †
k k gauge invariant not gauge invariant
Symmetry group U(1) —
..
..
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Broken Symmetries, Long Range Order
T<Tc: higher order, lower symmetry of ground state
. ..
...
.
.
.
T>Tc T<Tc
T<TC: U(1) “gauge symmetry“ spontaneously broken
2. order phase transition
III. Conventional superconductor
..
..
U(1) gauge symmetry also broken in BEC
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Cooper pairing of Fermions vs. Bose-Einstein condensation
Conventional superconductors
High TC superconductors
Superfluid 3He
0 10000 Å
0 150 Å
0 10 Å
BCS
Tightly packed bosons BEC0 1 Å
Cont
inuo
us
cros
sove
r?
Leggett (1980)New insights from BEC of cold atoms
Cooper pair: “Quasi-boson“
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Cooper pair:
L=1, S=1 in all phases
orbital part
spin part d̂
l̂
Leggett (1975)SO(3)S´SO(3)L´U(1)φ symmetry spontaneously broken
3x3 order parameter matrix Aiμ3x3 order parameter matrix Aiμ
(2 1) (2 12 )L S Characterized by = 18 real numbers
Superfluid,magneticliquid crystal
phaseanisotropy direction for spinanisotropy direction in real spaceQuantum coherence in
Broken symmetries in superfluid 3He
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SO(3)S+L_
Spontaneously broken spin-orbitsymmetry Leggett (1972)
„Unconventional" superfluidity
SO(3)S´SO(3)L´U(1)φ symmetry broken
Broken symmetries in superfluid 3He
Fixed relative orientation
Cooper pairs
Mineev (1980)Bruder, DV (1986)
3He-B
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SO(3)S+L_
Broken symmetries in superfluid 3He Mineev (1980)Bruder, DV (1986)
Relation to high energy physics
chiral invariance
Goldstone excitations (bosons) 3 pions
IsodoubletL
ud R
ud
,
Global symmetry SU(2)L´SU(2)R
SU(2)L+R
condensation (”Cooper pair”)qq
SO(3)S´SO(3)L´U(1)φ symmetry broken
„Unconventional" superfluidity
3He-B
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U(1)Sz
U(1)Lz- φ´
l̂
Broken symmetries in superfluid 3He
Cooper pairs
Fixed absolute orientation
„Unconventional" pairing
Mineev (1980)Bruder, DV (1986)
SO(3)S´SO(3)L´U(1)φ symmetry broken3He-A
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Resolution of the NMR puzzle
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What determines the actual relative orientation of ? ˆ,d l̂
Cooper pairs in 3He-A
Unimportant ?!
Dipole-dipole coupling of 3He nuclei:710D CTg K
Anisotropic spin-orbit interaction of nuclear dipoles:
Fixed absolute orientation
Superfluid 3He - a quantum amplifier
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Long-range order in : tiny, but lifts degeneracy of relative orientation710Dg K
Quantum coherence
NMR frequency increases: 22 2 (( )) D TgH Leggett (1973)
Nuclear dipole interaction macroscopically measurable
locked in all Cooper pairsˆ,d l̂
ˆ,d l̂••
Cooper pairs in 3He-A Fixed absolute orientation
Superfluid 3He - a quantum amplifier
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The Nobel Prize in Physics 2003"for pioneering contributions to the theory of superconductors
and superfluids"
Alexei A. AbrikosovUSA and Russia
Vitaly L. GinzburgRussia
Anthony J. Leggett UK and USA
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Order parameter texturesand topological defects
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Order parameter textures
Orientation of anisotropy directions in 3He-A ?ˆ,d l̂
“Textures“ in liquid crystalsˆ,d l̂
Magnetic field
Walls
d̂
l̂
Topologically stable defects: Classification by homotopy theory
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Order parameter textures and topological defects
D=2: domain walls in ord̂ l̂
Single domain wall
Domain wall lattice
l̂ l̂
l̂
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D=1: Vortices
Order parameter textures and topological defects
e.g., Mermin-Ho vortex(non-singular)
Vortex formation(rotation experiments)
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D=0: Monopoles
Order parameter textures and topological defects
Defect formation by, e.g., • rotation• geometric constraints• rapid crossing through phase transition
“Boojum” in -texture of 3He-A (geometric constraint)
l̂
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Big bang simulationin the low temperature lab
BANG!
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Universality in continuous phase transitions
T=Tc
T>Tc
T<Tc
Phase transition
High symmetry,short-range order
Broken symmetry,long-range order
Spins:para-magnetic
ferromagnetic
Defects: domainwalls
Helium:normalliquid
superfluid
vortices, etc.
nucleation of galaxies?
Universe:Unified forcesand fields
elementaryparticles,fundamentalinteractions
cosmic strings,etc. Kibble (1976)
BANG!
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3.
Estimate of density of defects Zurek (1985)
”Kibble-Zurek mechanism”: How to test?
4.
CT T : Vortex tangle
Defects overlap
1.
2.
Rapid thermal quench through 2. order phase transition Kibble (1976)
Local temperature
Expansion + rapid cooling
Nucleation of independently ordered regions
Defects
Clustering of ordered regions
CT T
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Grenoble: Bäuerle et al. (1996), Helsinki: Ruutu et al. (1996)
Big bang simulation in the low temperature laboratory
Measured vortex tangle density: Quantitative support for Kibble-Zurek mechanism
3He-B
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Present research on superfluid 3He: Quantum Turbulence
Quantum Turbulence = Turbulence in the absence of viscous dissipation (superfluid at T0)
Vinen, Donnelly: Physics Today (April, 2007)Test system: 3He-B
• Why are quantum and classical turbulence so similar?• What provides dissipation in the absence of friction?
Leonardo da Vinci (1452-1519) Flow through grid
Classical Turbulence
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Superfluid Helium-3:
• Anisotropic superfluid
• Large symmetry group broken
- 3 different bulk phases- Cooper pairs with internal structure
- Close connections to particle theory- Zoo of topological defects- Kibble-Zurek mechanism quantitatively verified
Conclusion