superfluid 3 he in confined geometries awg - p-wave, rhul march 2012 superfluid 3 he in confined...

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 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


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


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


‣ 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 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 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


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


Dirac Fermions

Degenerate Vacuum States:

R. Jackiw and C. Rebbi, Phys. Rev. D 1976

❖Zero Energy Bound State

Domain Wall

Mass


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


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


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]


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


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


Molecular BEC vs. BCS PairingMolecular BEC vs. BCS Pairing

✤Loosely Bound Cooper Pairs:

ξ ≫ a

✤Overlapping Pairs ⟿ Internal Exchange

✤Cancellation of Orbital Currents?

ξ

⟿


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


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:


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


⟿

2D Chiral A-phase with

Bulk SolutionBulk Solution

Bulk spectrumBulk spectrum Bound State PoleBound State Pole

Propagators for States Near an Edge


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


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


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


ξ

CR

Resonance Resonance EffectsEffects

T = 0T = 0

+iΔ

-iΔ

C1

C2

MT !!

Continuum Spectral Continuum Spectral CurrentCurrent


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

http://jpsj.ipap.jp/archive/JPSJ-67.html


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


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


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


‣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


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 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


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


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


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?


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


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||


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


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


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


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


Induced DLRO ~ Density ModulationInduced DLRO ~ Density Modulation


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 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


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).


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


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


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)

return

return

superfluid 3 he in confined geometries awg - p-wave, rhul march 2012 superfluid 3 he in confined...

Documents

confined geometries

andreev surface states

continuumandreev states

dwave scssuperfluid

axial phase superfluid

planar phase superfluid

filmsedge states of

relative momentum p