Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Superfluid Superfluid 33He in Confined GeometriesHe in Confined GeometriesBroken Symmetry, Excitations and Possible New Broken Symmetry, Excitations and Possible New
PhasesPhases
Superfluid Superfluid 33He in Confined GeometriesHe in Confined GeometriesBroken Symmetry, Excitations and Possible New Broken Symmetry, Excitations and Possible New
PhasesPhases
James A. Sauls
Anton B. Vorontsov
Physics
Lake Michigan
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Superfluid 3He in Confined GeometrySuperfluid 3He in Confined Geometry
❖ Introduction‣ Bulk Phases of 3He - Symmetry‣ Superfluid 3He Near Surfaces - Pair Breaking
❖Kinks, Domain Walls - Confined Fermions❖Chiral Edge States in 3He-A - 2D limit
‣ Edge Currents and Angular Momentum of 3He-A
‣ Robustness: Non-Specular Boundary Conditions
❖Surface Excitation Spectrum - 3He-B‣ Majorana Fermions‣ Andreev Fermions
❖Phase Diagram for 3He Films‣ Translationally Invariant Phases‣ Broken Translational Symmetry‣ Instabilities and Domain Wall Proliferation
❖SuperSolid Phase of 3He Films‣ Order Parameter ⟿Density Wave Amplitude‣ Optical Detection
❖Conclusions & Outlook
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
1. Introduction to Superfluid 1. Introduction to Superfluid 33He He 1. Introduction to Superfluid 1. Introduction to Superfluid 33He He Unconventional BCS Superfluid: S=1 - Spin TripletL=1 - Orbital p-wave
Cooper Pair Amplitude
Inhomogeneous States:- relative momentum (p) - Center-of-Mass (R)
9 complex amplitudes
S=1
L=1
pxpx pypy pzpz
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Bulk Phase Diagram of Superfluid Bulk Phase Diagram of Superfluid 33HeHeBulk Phase Diagram of Superfluid Bulk Phase Diagram of Superfluid 33HeHe
B - phase (``isotropic’’)Balian-Werthamer
A - phase (``axial’’)Anderson-MorelNodal QuasiparticlesChiral Axis: Lz = ℏ
Fully Gapped
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
‣ Balian & Werthamer Balian & Werthamer (1963)(1963)
Weak Nuclear Dipole Weak Nuclear Dipole EnergyEnergy
violation::violation::A. Legge
tt
A. Legge
tt
Nuclear Spin Nuclear Spin DynamicsDynamics
Nuclear Spin Nuclear Spin DynamicsDynamics
Superfluid Superfluid 33He-B He-B
Approximate Approximate particle-holeparticle-hole symmetrysymmetry
violation::violation::
Possible SuperSolid Phase
Possible SuperSolid Phase
A.Vorontsov & JAS
A.Vorontsov & JAS
FS
Translational InvarianceTranslational Invariance
Fully Gapped, TRI Superfluid with Spontaneously generated Spin-Orbit Coupling
GeneratorGenerator
Broken Broken relativerelative spin-orbit spin-orbit symmetrysymmetry
Broken Broken relativerelative spin-orbit spin-orbit symmetrysymmetry
Transverse SoundTransverse Sound
Acoustic Faraday Acoustic Faraday EffectEffect
Transverse SoundTransverse Sound
Acoustic Faraday Acoustic Faraday EffectEffect
G. Moor
es
& JAS
G. Moor
es
& JAS
Y. Lee et al.
Nature 1999
Y. Lee et al.
Nature 1999
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Superfluid Superfluid 33He-P (``Planar phase’’) He-P (``Planar phase’’)
Degenerate with the Degenerate with the Axial StateAxial State
Possible Ground State in Confined Possible Ground State in Confined 33He FilmsHe Films
Strong-Coupling - Strong-Coupling - FluctuationsFluctuations
Un-realized in Bulk Un-realized in Bulk 33HeHe
Nodal Quasiparticles
Non-Chiral Axis
2D TRI ``B-2D TRI ``B-phase’’phase’’
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Superfluid Superfluid 33He-A (``Axial phase’’) He-A (``Axial phase’’)
Broken Broken time-reversal time-reversal symmetrysymmetry
Ground state Orbital Angular Ground state Orbital Angular MomentumMomentum
Broken relative spin-orbit Broken relative spin-orbit symmetrysymmetry
Broken relative gauge-orbit symmetry
‣Anderson & Morel Anderson & Morel (1962)(1962)‣Anderson & Morel Anderson & Morel (1962)(1962)
Ans: Chiral Edge States and Edge Currents
Lz =(N/2)ℏ
(Δ/Ef)p
p = 0,1,2 ?
Chirality: Lz = ℏBroken 2D ParityBroken T-Symmetry
Broken Broken 2D 2D parityparity
Spin-Mass Vortices
Chiral Fermions
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Chiral Superfluids
Spin AFM Orbital FM
✤Chiral Spin-Triplet Superconductivity
UPt3Sr2RuO4 ?
‣A-phase of A-phase of 33HeHe
‣A-phase of A-phase of 33HeHe
Anderson & Morel (PR,1962)Anderson & Morel (PR,1962)
strong spin-orbit coupling
tetragonal
?
hexagonal
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Dirac Fermions
Degenerate Vacuum States:
R. Jackiw and C. Rebbi, Phys. Rev. D 1976
❖Zero Energy Bound State
Domain Wall
Mass
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Quasiparticle States Confined near Boundaries and Quasiparticle States Confined near Boundaries and InterfacesInterfaces
Quasiparticle States Confined near Boundaries and Quasiparticle States Confined near Boundaries and InterfacesInterfaces
Δ eiϕ1Δ eiϕ2
Toplogical defect (kink) - Jackiw,Rebbi (PRD 1976)
⟿ due to interface scatteringC.-R. Hu 1994, Buchholtz et al. 1995
|Δ| - Quantum Well Particle-hole InterferenceA. Andreev 1964
L
⇉⇉
L grows - more states enter from continuumL grows - more states enter from continuum
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Andreev States and Edge Currents in d-wave SCs
Andreev States and Edge Currents in d-wave SCs
Fogelström,Rainer & JAS PRL’97
Walter et al PRL ’98
Covington et al PRL ’97
Andreev Surface States
d-wave near pairbreaking surface
Surface states ⇓
Anomalous edge currents
Large N(0) ⟿ Broken Time-reversal Edge currents Surface d+i s Superconductivity
Paramagnetic Meissner current
Doppler Splitting of the ZEBS
Doppler Splitting of the ZEBS
Sub-dominant pairing: d+is
Tunnel Splitting from dI/dV
Burkhart, Rainer & JAS
pairbreaking
ZEBS
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Zero Modes, Sub-Gap States & Bound States in ...Josephson Point Contacts
Vortex Core Excitations
Sub-Gap States in Superfluid 3He Films
Edge States of dx2-y2 Superconductors
Δe-iφ/2
Δe+iφ/2
Δe+iφ
φ
Specular
Diffuse
[110]
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
NTRI p-wave
Chiral p-wave2D Chiral p-
wave
Specular - no surface bound states - no pairbreaking
Chiral Edge States in Chiral Edge States in 33He-AHe-A Chiral Edge States in Chiral Edge States in 33He-AHe-A
‣Majorana Fermion at ε
l - parallel to the edge ⇒ edge currents Dipole-
Locked
Diffuse - gapless band - surface pairbreaking
Skew Scatterng
‣Chiral Edge State - Weyl Fermion G. E. Volovik M. Stone & R. Roy
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Triplet P-wave CondensatesTriplet P-wave Condensates
Singlet S-wave Condensates
``Scalar BEC’’
``Chiral P-wave molecular BEC’’⟿
Ground State Angular MomentumIntrinsic Angular Momentum Density
Molecular BEC
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Molecular BEC vs. BCS PairingMolecular BEC vs. BCS Pairing
✤Loosely Bound Cooper Pairs:
ξ ≫ a
✤Overlapping Pairs ⟿ Internal Exchange
✤Cancellation of Orbital Currents?
ξ
⟿
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Molecular BEC
FermiSea
✤Momentum Space: Pair Correlations on the Fermi Shell
✤Intrinsic Angular Momentum Density in the BCS limit
# of pair-correlated Fermions# of pair-correlated Fermions
... vs ...BEC limit
A. J. Leggett, RMP 1975, M. Cross JLTP 1975 & G. Volovik & V. A. J. Leggett, RMP 1975, M. Cross JLTP 1975 & G. Volovik & V. Mineev JETP 1976Mineev JETP 1976
vs BCS Condensation
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Angular Momentum Paradox
✤Real Space Formulation in Cylindrical Geometries
✤Integrated Angular Momentum Density in the BCS
~10-6
... vs ...BEC limits
P.W. Anderson and P. Morel 1962 & M. Cross 1975, A. P.W. Anderson and P. Morel 1962 & M. Cross 1975, A. Leggett RMP 1975Leggett RMP 1975
z
independent of (a /ξ)!
✤McClure-Takagi Result: M. McClure, S. Takagi, PRL (1979)M. McClure, S. Takagi, PRL (1979)
For any cylindrically symmetric chiral texture and pair wave function that vanishes on the boundary:
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
McClure-Takagi gives the correct answer:
but ... so ...Currents are on the boundary
G. E. VolovikV. P. MineevM. IshikawaP. MuzikarD. Mermin
T. KitaM. StoneA. GargR. Roy
...
M. Stone & R. Roy, PRB 2006
J. A. S., PRB 2011
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
⟿
2D Chiral A-phase with
Bulk SolutionBulk Solution
Bulk spectrumBulk spectrum Bound State PoleBound State Pole
Propagators for States Near an Edge
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Surface Surface Confinement ...Confinement ...
Surface Surface StatesStates
occupied
unoccupied
Pair of Time-Reversed Edge States
➡ ➡
Chiral Edge StatesChiral Edge States
Surface CurrentSurface Current
≪ L a ≪
Asymmetry in the Occupation
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Local Spectral Local Spectral DensityDensity
in
out
Pair Time-reversed Trajectories Pair Time-reversed Trajectories ⟿⟿ Spectral Current Spectral Current DensityDensity
in
out
α
p’
_p’
Exact CancellationAsymmetry in the Occupation
x = 0.5 ξΔ
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Bound-State Current & Angular Bound-State Current & Angular Momentum Momentum
z
Rx
✤Number of Fermions:
✤Galilean Invariance:
r
Mass Current
⨉ 2 Too Big vs. MT
Continuum States contribute to the
Edge CurrentsM. Stone & R. Roy PRB
2006J.A.S. PRB 2011
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
ξ
CR
Resonance Resonance EffectsEffects
T = 0T = 0
+iΔ
-iΔ
C1
C2
MT !!
Continuum Spectral Continuum Spectral CurrentCurrent
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Finite Finite TemperatureTemperature
ξ
CR
C1
+iΔ
C2
⨯⨯⨯⨯
✤
Matsubara Matsubara RepresentationRepresentation
T. Kita ``conjecture’’T. Kita ``conjecture’’J. Phys. Soc. Jpn. J. Phys. Soc. Jpn. 67 (1998) pp. 216-224 (1998) pp. 216-224
3D Mesoscale (R3D Mesoscale (R≃≃ 2 2ξξ))Numerical BdG Numerical BdG
?
LLzz(T) is ``soft’’ (2D or 3D) due to thermal excitation of (T) is ``soft’’ (2D or 3D) due to thermal excitation of Excited Edge StatesExcited Edge Statesρρs|| s|| (T) is ``soft’’ (3D) due to thermal excitation (T) is ``soft’’ (3D) due to thermal excitation of of Nodal QPsNodal QPs
LLzz(T)(T)
ρρs|| s|| (T)(T)
ρρss⊥⊥ (T)(T)
YYzz(T) = 1- (T) = 1- c c TT22
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
NoNo Chiral Currents Chiral Currents
Robustness of the Chiral Edge Robustness of the Chiral Edge StatesStates
Chiral Edge StatesChiral Edge States
Specular Specular ReflectionReflection
in
out
➡
in
out
pp_
TinyTiny Angular Angular MomentumMomentum
!!
Facetted Facetted SurfaceSurface
Chirality Invisible!Chirality Invisible!
Retro ReflectionRetro Reflection
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
RR1,1, R R2,2, (R (R11 - R - R22) ) ⋙⋙ ξΔΔ
Edge Currents in a Toroidal Edge Currents in a Toroidal GeometryGeometry
Specular EdgeSpecular Edge
Angular MomentumAngular Momentum
Sheet Current
x
Counter-Propagating CurrentsCounter-Propagating Currents
MT ResultMT Result !!
J1
VolumeVolume
J2
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
RR1,1, R R2,2, (R (R11 - R - R22) ) ⋙⋙ ξΔΔ
Non-Extensive Scaling of Non-Extensive Scaling of LLzz
Non-Specular Non-Specular ScatteringScattering
Fraction of Forward Scattering Fraction of Forward Scattering TrajectoriesTrajectories
Sheet Current - Non-Specular Edge
Incomplete Screening of Counter-Propagating Incomplete Screening of Counter-Propagating CurrentsCurrents
LLzz ≉ ≉ VV
J1
J2
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
‣Topology and Non-specular Scattering ⟿ Lz is Non-Extensive: Lz >> (N/2)ℏ or Lz << - (N/2)ℏ ⟿ Direct Evidence of Edge Currents
‣Topology and Non-specular Scattering ⟿ Lz is Non-Extensive: Lz >> (N/2)ℏ or Lz << - (N/2)ℏ ⟿ Direct Evidence of Edge Currents
Conclusions
‣Detailed models of surface scattering <-> Edge Currents ‣Gyroscopic Dynamics of Toroidal Disks of 3He-A‣A.C. rotational dynamics of Edge States and Edge currents
‣Detailed models of surface scattering <-> Edge Currents ‣Gyroscopic Dynamics of Toroidal Disks of 3He-A‣A.C. rotational dynamics of Edge States and Edge currents
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
2. New States of 2. New States of 33He in Confined He in Confined GeometryGeometry
2. New States of 2. New States of 33He in Confined He in Confined GeometryGeometry
Multi-component Order Parameter with Broken Spin- Orbital and Gauge Symmetries. A & B are Topological Superfluids ↳ Low-Energy Topological Surface/Interface States
↳ Confinement: Strongly deformed Order Parameter
➡ D≈10ξ0 Interactions of Surface & Interface States ↳ new superfluid phases in confined geometries
↳ low energy transport & thermodynamics ↳ phase transitions
NTRI p-wave
Chiral p-wave
Stripe Phase
SuperSolid
S=1
L=1
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Superfluid Superfluid 33He-B near a He-B near a wallwall
Superfluid Superfluid 33He-B near a He-B near a wallwall
Specular Trajectories
Specular Diffuse
Pair Breaking & Pair Pair Breaking & Pair EnhancementEnhancement
Pair Breaking & Pair Pair Breaking & Pair EnhancementEnhancement2D Translationally invariant B-planar state2D Translationally invariant B-planar state2D Translationally invariant B-planar state2D Translationally invariant B-planar state
s
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
ϴϴ
specular
‣New Andreev Bound States
Surface Fermionic Spectrum of Fermionic Spectrum of 33He-He-BBSpecular Scattering
Non-Specular Scattering
is conserved is not conserved
‣Majorana Fermion at ε‣Dispersion:
diffuse
‣Spectral Wt.: N(0)≠0 for all
‣Continuum Edge disperses ‣Broad Low-energy Band
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
4. Edge Spectrum in the Film4. Edge Spectrum in the Film
‣ Wall: Angle-resolved FS averaged
‣ QP spectroscopy: ‣ Ballistic Emmitters ‣ Momentum-resolved Detectors
‣Longitudinal and Transverse Acoustic Impedance Spectroscopy
‣Transverse Acoustic Impedance Spectroscopy R. Nomura et al. PRL 2009
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Translationally invariant phases of 3He films
Translationally invariant phases of 3He filmsdeformed bulk B- P- & A-
phasesconfinement in z - invariant in x-y
3He-B-planar
A & P phases are degenerateQuantum Critial
Point Re-entrance or Inhomogeneous
Phase?
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Crystalline Phase of Superfluid 3He
Crystalline Phase of Superfluid 3He
‣ Spontaneously Broken Translation Symmetry in the x-y plane of the film‣ Spontaneously Broken Translation Symmetry in the x-y plane of the film
‣ Dc1 - Domain Walls Proliferation‣ Dc2 - Single-Q Mode
Instability
‣ new OP components
‣2nd order‣2nd order ‣Degenerate Vacua
smallsmall
A. Vorontsov & JAS, PRL 2007 & Review 2012
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Mechanism at Dc1 = Domain Wall Proliferation
Surface - Domain Wall Interaction
Net Energy gain from a domain wall
“perpendicular DW”
Condensaton Energy
loss to surface states
“parallel DW” Condensation Energy loss to Andreev states
Perpendicular domain wall costs less enegy/length than a Parallel domain wall
‣Different ``healing lengths’’ for ⊥ and || components ‣ξ⊥ = √3ξ||
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Spontaneous Currents in d-wave films
Spontaneous Currents in d-wave films
SC - Normal transition in films
Inhomogeneous State: New structure of the order parameter !
A. Vorontsov, PRL ’09
Free Energy gain from paramagnetic edge states undergoing spontaneous TRS
breaking
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Non-trivial “vacuum”
azx ~ 0.6 azz➡Couples q and -q at the instability
TRI Superfluid Planar-Stripe transitionTRI Superfluid Planar-Stripe transitionTRI Superfluid Planar-Stripe transitionTRI Superfluid Planar-Stripe transition
↳ n=0 - Broken symmetry “Vacuum” - Planar↳ n=1 - new state (perturbation - linearized)
Coupling of Δ with Δ* in the presence of non-
trivial “vacuum”
Order Parameter Equation - Mode-Instability - Dc2
TRI
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Azz
Edge
Center
1 2 3‣ Wall: Angle-resolved FS averaged
4. Edge & Domain Wall Fermions in the Film
4. Edge & Domain Wall Fermions in the Film
spectral weight transfer to higher energy E3 ⇒ E1
spectral weight transfer to lower energy C3 ⇒ C1
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
E1 E3
C3C1
Azz
Edge
CentCenter er
13
‣ QP spectroscopy: Ballistic Emmitters and Momentum-resolved Detectors
Andreev Statesnear continuumAndreev Statesnear continuum
Angular resolved DOS in FilmsAngular resolved DOS in Films
Low Energy States
Continuum
spectral weight transfer to higher energy
spectral weight transfer to higher energy
spectral weight transfer to lower energy
Tomasch
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Induced DLRO ~ Density ModulationInduced DLRO ~ Density Modulation
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
OP and Density Wave in the Stripe OP and Density Wave in the Stripe PhasePhase
OP and Density Wave in the Stripe OP and Density Wave in the Stripe PhasePhase
Axx
Azz
domain wall
δn / n0
variations in the film δn/n ~ 10-5variations in the film δn/n ~ 10-5
‣ Possible Detection: Light Scattering by density fluctuations?
OP Domain Wall
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Superfluid film formationSuperfluid film formationSuperfluid film formationSuperfluid film formation‣ Competition: gravity vs. Van der Waals attractions of atoms to the substrate or wall‣ Non-uniform film surface: surface tension & density modulation ‣ Affects surface waves as well (third sound, etc)
density = (n0 m) ~ 81.5 kg/m3
surface tension σ ~ 0.156 mN/m mass of 3He atom ~ 5 .10 -27
kg
vdW constant α ~ 10 .10-9
m4/3
see e.g. Steel, Harrison et al JLTP 95, 1994 “Film flow on a round rim beaker”
hydrostatic equlibrium
chemical potential per particle
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Crystalline PhaseCrystalline Phase Film thickness Film thickness variationsvariations
Crystalline PhaseCrystalline Phase Film thickness Film thickness variationsvariations
average density variations ~10-5 film height driven by density fluctuations and α(n) -dependence
‣ energy scales
‣ VdW / gravity
‣ Surface tension / gravity
‣ Dominates at D ~ 10 ξ0
D ~ 10 ξ0 = 750 nmD ~ 10 ξ0 = 750 nm
‣ Overall change in film thickness
ζ~0.1 Å Possible Capacitive Detection? A. Schechter et al., Nature 396, 554 (1998).
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
SummarySummary
‣ Confinement ⟷ Surface States‣ Majorana & Andreev States‣ Interactions Between Surface States ⇓‣ Broken Translational Symmetry ‣ Density Wave ⟶ ``SuperSolid’’‣ Particle-hole asymmetry:✓ variations of the film height✓ tension dominates at D ~ 10 ξ0
‣Possible Detection:✓ Momentum Resolved QP spectroscopy✓Capacitance detection of height fluctuations✓Optical detection of density fluctuations (?)✓NMR and Mass/Heat Transport
Momentum-Resolved Fermion Spectrum
Crystalline Superfluid Phases
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
Spin Triplet Pairing in UPt3Spin Triplet Pairing in UPt3
Anisotropic Pauli limiting
C. Choi & JAS, PRL (1991)
B. Shivaram et al. PRL (1986)
No pair breaking
pair breaking
C
➡ Spin-Triplet, w/ strong Spin-Orbit Coupling - E2u
=
T [mK]
H [kOe]
return
Superfluid 3He in Confined Geometries AWG - P-wave , RHUL March 2012
J. Kycia et al., PRB (1998)
NFL
C
S. Adenwalla et al. PRL (1990)
ABB. Shivaram et al. PRL
(1986)
➡ Spin-Triplet, E2u, w/ strong Spin-Orbit Coupling
Andrew Huxley et al. Nature (2000).
Realignment of the flux-line lattice in UPt3
T/Tc
➡ E2u orbital symmetry
M. Graf, S.K. Yip & JAS, PRB (1996)
✓Heat Capacity Anomalies ✓Anisotropy Transverse Sound✓Anistropic Thermal Conductivity
Weak Symmetry Breaking - AFM orderD. Hess, et al., J. Phys.: Cond. Mat. 1, 8135
(1989).R. A. Fisher et al., Phys. Rev. Lett. 1989.
Unconventional Pairing in UPt3Unconventional Pairing in UPt3L. Gorkov (1987)
T. Champel & V. Mineev (2001)
Anisotropic Pauli limiting - S=1H-T phase diagram - Tetracritical point - E2u
C. Choi & JAS, PRL (1991)
No pair breaking
pair breaking
➡ E2u orbital symmetry
C
J. A. Sauls, Adv. Phys. 43, 113 (1994).
Cv/T
B. Lussier et al.., Phys. Rev. B 53 (1996)B. Ellman et al.., Phys. Rev. B 54 (1996)
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