from spontaneous symmetry breaking to topological...
TRANSCRIPT
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Physics Department Colloquium, SUNY Buffalo, NY, March 24, 2016
From Spontaneous Symmetry Breaking to Topological Order
Anomalous Hall Effect in Superfluid 3He
J. A. Sauls
Northwestern University
• Hao Wu • Oleksii Shevtsov
Unconventional Superconductivity
Spin-Fluctuation Mediated Pairing
The Helium Paradigm
Broken T & P Symmetry
Topological Order in 3He
Chiral Edge Currents in 3He
NSF Grant DMR-1508730
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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BCS Pairing from 10−9 K to 10+9 K
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1972 Discovery of Triplet, P-wave, Superfluid 3He Phases1979 Discovery of Heavy Electron Superconductors1982 Exotic Pairing in U-based Heavy Fermions1986 High Tc Superconductivity in Oxides1994 Exotic Pairing in Sr2RuO4
1995 D-wave Pairing Discovered in YBCO2001 Coexistent Ferromagnetism & Superconductivity2008 Superconductivity in Fe-based Materials
1908 Helium is liquified1911 Superconductivity discovered in Hg1933 Diamagnetism - Meissner Effect1935 London Theory 1950 Ginzburg-Landau Theory
1957 BCS Theory 1957 Landau Fermi Liquid Theory 1957 Abrikosov’s Theory of Type II SC1959 Gauge-Invariant Pairing Theory1959 Field Theory formulation of BCS Pairing1959 Pairing in Nuclei and Nuclear Matter1961 Theory of Spin-Triplet Pairing1962 Josephson Effect
1995 Discovery of Bose-Einstein Condensation of Rb1998 Discovery Quantized Vortices in BEC2003 Degeneracy of Cold Fermionic Atoms - 6Li, 40K2007 BEC-BCS Condensation in 6Li, 40K
1992 - 2008 Topological Superfluids and Superconductors
1967 Pulsars discovered - Hewish & Bell1969 Pulsar Glitches observed in Vela 1971 - 1985 - Superfluid Hydrodynamics NS
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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BCS Pairing from 10−9 K to 10+9 K
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1972 Discovery of Triplet, P-wave, Superfluid 3He Phases1979 Discovery of Heavy Electron Superconductors1982 Exotic Pairing in U-based Heavy Fermions1986 High Tc Superconductivity in Oxides1994 Exotic Pairing in Sr2RuO4
1995 D-wave Pairing Discovered in YBCO2001 Coexistent Ferromagnetism & Superconductivity2008 Superconductivity in Fe-based Materials
1908 Helium is liquified1911 Superconductivity discovered in Hg1933 Diamagnetism - Meissner Effect1935 London Theory 1950 Ginzburg-Landau Theory
1957 BCS Theory 1957 Landau Fermi Liquid Theory 1957 Abrikosov’s Theory of Type II SC1959 Gauge-Invariant Pairing Theory1959 Field Theory formulation of BCS Pairing1959 Pairing in Nuclei and Nuclear Matter1961 Theory of Spin-Triplet Pairing1962 Josephson Effect
1995 Discovery of Bose-Einstein Condensation of Rb1998 Discovery Quantized Vortices in BEC2003 Degeneracy of Cold Fermionic Atoms - 6Li, 40K2007 BEC-BCS Condensation in 6Li, 40K
1992 - 2008 Topological Superfluids and Superconductors
1967 Pulsars discovered - Hewish & Bell1969 Pulsar Glitches observed in Vela 1971 - 1985 - Superfluid Hydrodynamics NS
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Broken Symmetry, Phase Transitions and Long-Range Order
2
Solid Nematic Ferromagnet Super-liquid
Translations Space Rotations Spin Rotation Gauge
ρ(q) =
dre−iq·r δn(r) M = γ SQij = Q(T )
ninj −
1
3δij
Ψ = ψ(r)
SO(3)L SO(3)S U(1)NGtrans
N/V eiϑ
I Break one or more spin/space-group symmetries in conjunction with U(1)N
Superconductivity with Unconventional BCS Pairing
I Phases of liquid 3He exhibit all of these broken symmetries!
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Long-Range Order in G2(x1, x2;x3, x4) - BCS Condensation
I Two particle correlations for s = 1/2 Fermions: x1 = (r1, α1) etc.
G2(x1, x2;x3, x4) = 〈ψ(x1)ψ(x2)ψ†(x3)ψ†(x4)〉
I Cooper Instability Long-range Order of Bound Fermion Pairs
b
b
b
b
ψ†(x3)
ψ(x1)
ψ†(x4)
ψ(x2)
|R12 − R34|
G2(x1, x2;x3, x4) −−−−−−−−−−→|R12−R34|→∞
Ψ(x1, x2) Ψ(x3, x4)
Macroscopic 2-particle state
Ψα1α2(r1, r2)→(
Ψ↑↑ Ψ↑↓Ψ↓↑ Ψ↓↓
)
I Total Spin S = 0, 1 and Orbital Angular Momentum L = 0, 1, 2, 3, . . .
I Internal Structure of Cooper Pairs “Unconventional Superconductivity”
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Pairing Symmetry Classes for Isotropic Normal Fermi Systems
I Total Spin, S = 0, 1 and I relative Angular Momentum L = 0, 1, 2, 3, . . .
Name Spin S Orbital LSinglet Pairing 0 evenTriplet Pairing 1 oddS-wave Pairing 0 0D-wave Pairing 0 2P-wave Pairing 1 1F-wave Pairing 1 3
S-wave Pairing (only U(1)N is broken) [“Conventional” Superconductors, NbSe2 ]
D-wave Pairing (U(1)N and SO(3)L are broken) [ YBa2Cu3O7-x ,CeCoIn5,URu2Si2]
P-wave Pairing (U(1)N, SO(3)S and SO(3)L are broken) [ 3He, Sr2RuO4 ,UGe2]
F-wave Pairing (U(1)N, SO(3)S and SO(3)L are broken) [ UPt3 ]
I Superconducting Classes, G. E. Volovik and L. P. Gorkov, JETP 61, 843 (1985)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Superfluidity and Superconductivity in Neutron Star Interiors
Physics at the Falls Workshop:I “Pairing Phenomena from Neutron Stars to Cold Atoms”
2
SuperfluidCore
n ≈ 95%p , e− ≈ 5%
Crust & Superfluid
Solid Crust
10 km
I Baym and Pethick, Ann. Rev. Nucl. Phys. 25, 27 (1975)
Superfluid Neutrons and Superconducting Protons
Crust: 1S0 Neutron Pairs
Core: 3P2 Neutron Pairs & 1S0 Proton Pairs
I JAS et al., Phys. Rev. D 17, 1524 (1978)I Pairing Gaps: R. Tamagaki, Prog. Theor. Phys. 44, 905 (1970)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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The Helium Paradigm: Superfluid Phases of 3He
Symmetry of Normal Liquid 3He : G = SO(3)S × SO(3)L × U(1)N × P× T
Phase Diagram of Bulk 3He
0.0 0.5 1.0 1.5 2.0 2.5T [mK]
0
5
10
15
20
25
30
P [b
ar]
A
B
PCP
3HeSuperfluid
Spin-Triplet, P-wave Order Parameter(Ψ↑↑ Ψ↑↓Ψ↑↓ Ψ↓↓
)=
(−dx + idy dz
dz dx + idy
)
Chiral ABM State ~l = z
Lz = 1, Sz = 0
dz = ∆ (px + ipy)
“Isotropic” BW State
J = 0, Jz = 0dα = pα, α = x, y, z
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Spin Fluctuation Exchange: Ferromagnetic Odd-Parity, Spin-Triplet Pairing for 3He
I A. Layzer and D. Fay, Int. J. Magn. 1, 135 (1971)
Vsf(p,p′) =
p ↑
p′ ↑ −p
′ ↑
−p ↑
=−g/4
1−g χ(p−p′)
−gl = (2l + 1)∫ dΩp
4π
∫ dΩp′4π
Vsf(p,p′)Pl(p · p′)
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
2.5
3.0
q ≈ h/ξsf
Vsf =−g/4
1− g χ(q)
q/pf
−gl is a function of g ≈ 0.75 and ξsf ≈ 5 ~/pf
l = 1 (p-wave) is dominant pairing channelI p-wave basis functions:
pz ∼ cos θp
px + ipy ∼ sin θp e+iφp
px − ipy ∼ sin θp e−iφp
l = 3 (f-wave) is the sub-dominant channel
S = 1 pairing fluctuations in Vsf A-phaseW. Brinkman, J. Serene, and P. Anderson, PRA 10, 2386 (1974)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Are there electronic superconductors with broken symmetry phases analogous to 3He?
UPt3 Tc=0.56 K !S=1 ! ``f-wave’’, Lz = 2
! chiral
Are there electronic superconductors with broken symmetry phases analogous to 3He?
A. Huxley et al. - Nature, 406, 160 (2000)
∆(p) ∼ pz (px + ipy)2
multiple SC phasesJAS, Adv. Phys. 43, 113(1994)
T. Champel & V. Mineev - PRL, 86, 4903 (2001)
UPt3 Tc=0.56 K !S=1 ! ``f-wave’’, Lz = 2
! chiral
Are there electronic superconductors with broken symmetry phases analogous to 3He?
J. Strand et al. - PRL 103, 197002 (2009)
A. Huxley et al. - Nature, 406, 160 (2000)
"
∆(p) ∼ pz (px + ipy)2
multiple SC phasesJAS, Adv. Phys. 43, 113(1994)
T. Champel & V. Mineev - PRL, 86, 4903 (2001)
Josephson Interferometry
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rb. u
nits
)
0.35 0.4 0.45 0.5 0.55 0.6−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Temperature (K)
#$ K (µ
rad)
Tc+TKerr Tc+TKerr
UPt3 Tc=0.56 K !S=1 ! ``f-wave’’, Lz = 2
! chiral
Are there electronic superconductors with broken symmetry phases analogous to 3He?
Kerr rotation
E. Schemm et al. - Aspen 2014
J. Strand et al. - PRL 103, 197002 (2009)
A. Huxley et al. - Nature, 406, 160 (2000)
"
∆(p) ∼ pz (px + ipy)2
multiple SC phasesJAS, Adv. Phys. 43, 113(1994)
T. Champel & V. Mineev - PRL, 86, 4903 (2001)
Josephson Interferometry
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Are there electronic superconductors with broken symmetry phases analogous to 3He?UPt3 Tc=0.56 K !S=1
! ``f-wave’’, Lz = 2
! chiral
Are there electronic superconductors with broken symmetry phases analogous to 3He?
A. Huxley et al. - Nature, 406, 160 (2000)
∆(p) ∼ pz (px + ipy)2
multiple SC phasesJAS, Adv. Phys. 43, 113(1994)
T. Champel & V. Mineev - PRL, 86, 4903 (2001)
UPt3 Tc=0.56 K !S=1 ! ``f-wave’’, Lz = 2
! chiral
Are there electronic superconductors with broken symmetry phases analogous to 3He?
J. Strand et al. - PRL 103, 197002 (2009)
A. Huxley et al. - Nature, 406, 160 (2000)
"
∆(p) ∼ pz (px + ipy)2
multiple SC phasesJAS, Adv. Phys. 43, 113(1994)
T. Champel & V. Mineev - PRL, 86, 4903 (2001)
Josephson Interferometry
!" (a
rb. u
nits
)
0.35 0.4 0.45 0.5 0.55 0.6−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Temperature (K)
#$ K (µ
rad)
Tc+TKerr Tc+TKerr
UPt3 Tc=0.56 K !S=1 ! ``f-wave’’, Lz = 2
! chiral
Are there electronic superconductors with broken symmetry phases analogous to 3He?
Kerr rotation
E. Schemm et al. - Aspen 2014
J. Strand et al. - PRL 103, 197002 (2009)
A. Huxley et al. - Nature, 406, 160 (2000)
"
∆(p) ∼ pz (px + ipy)2
multiple SC phasesJAS, Adv. Phys. 43, 113(1994)
T. Champel & V. Mineev - PRL, 86, 4903 (2001)
Josephson Interferometry
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Polar Kerr Effect Measurements on UPt3
Kerr effect - optical polarization rotation on reflection: θK ≈ 4πn(n2−1)ω
σxy(ω)
I θK 6= 0 broken P and TI S.-K. Yip & JAS, J. Low Temp. Phys. 86, 257 (1992).
E. Schemm et al. Science (2015)Standford (Kapitulnik) - Northwestern (Halperin)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Evidence for Broken P & T with E2u Symmetry in UPt3
Polar Kerr Effect: q||z
E. Schemm et al. Science (2014)
A: ~d = z∆pz(p2x − p2
y) T Symmetric
B: ~d = z∆pz(px ± ipy)2 T BrokenE2u Pairing: JAS, Adv. Phys. (1994)
Polarization rotation onsets: T ≤ Tc2θK is weak field-trainableSingle Chiral Domain
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Emergent Topological Order
Spontaneous Symmetry Breaking Topological Order
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Real-Space vs. Momentum-Space Topology
Topology in Real SpaceΨ(r) = |Ψ(r)| eiϑ(r)
C
Phase Winding
NC =1
2π
∮C
dl· 1
|Ψ| Im[∇Ψ] ∈ 0,±1,±2, . . .
I Massless Fermions confined in theVortex Core
Chiral Symmetry Topology in Momentum SpaceΨ(p) = ∆(px ± ipy) ∼ e±iϕp
Topological Quantum Number: Lz = ±1
N =1
2π
∮dp · 1
|Ψ(p)| Im[∇pΨ(p)] = Lz
I Massless Chiral Fermions
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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The Helium Paradigm: Superfluid Phases of 3He
Symmetry of Normal Liquid 3He : G = SO(3)S × SO(3)L × U(1)N × P× T
Phase Diagram of Bulk 3He
0.0 0.5 1.0 1.5 2.0 2.5T [mK]
0
5
10
15
20
25
30
P [b
ar]
A
B
PCP
3HeSuperfluid
Spin-Triplet, P-wave Order Parameter(Ψ↑↑ Ψ↑↓Ψ↑↓ Ψ↓↓
)=
(−dx + idy dz
dz dx + idy
)
Chiral ABM State ~l = z
Lz = 1, Sz = 0
dz = ∆ (px + ipy)
“Isotropic” BW State
J = 0, Jz = 0dα = pα, α = x, y, z
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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The Helium Paradigm: Superfluid Phases of 3He
Symmetry of Normal Liquid 3He : G = SO(3)S × SO(3)L × U(1)N × P× T
3He Films: A. Vorontsov & JAS PRL 2007
0 10 20D /
!
0
0.2
0.4
0.6
0.8
1
T / T
c
TAB
BA
Stripe Phase
Spin-Triplet, P-wave Order Parameter(Ψ↑↑ Ψ↑↓Ψ↑↓ Ψ↓↓
)=
(−dx + idy dz
dz dx + idy
)
Chiral ABM State ~l = z
Lz = 1, Sz = 0
dz = ∆ (px + ipy)
“Isotropic” BW State
J = 0, Jz = 0dα = pα, α = x, y, z
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Angular Momentum of Chiral P-wave Condensates
Chiral P-wave BEC Molecules or BCS Pairs (N Fermions):
|ΦN 〉 =[∫∫dr1dr2 ϕs1s2(r1 − r2) ψ†s1(r1)ψ†s2(r2)
]N/2| vac 〉
ϕs1s2(r) = f(|r|/ξ) (x+ iy) χs1s2(S = 1,MS = 0)
BEC (ξ < a) vs. BCS (ξ > a)
l || z
Lz = (N/2)~
Lz = (N/2)~ (a/ξ)2 (N/2)~ ? (P.W. Anderson & P. Morel, 1960, A. Leggett, 1975)
I Lz|ΦN 〉 = (N/2)~ |ΦN 〉 independent of (a/ξ)! - McClure-Takagi (PRL, 1979)
BCS Limit: Currents are confined on the Edge
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Angular Momentum of Chiral P-wave Condensates
Chiral P-wave BEC Molecules or BCS Pairs (N Fermions):
|ΦN 〉 =[∫∫dr1dr2 ϕs1s2(r1 − r2) ψ†s1(r1)ψ†s2(r2)
]N/2| vac 〉
ϕs1s2(r) = f(|r|/ξ) (x+ iy) χs1s2(S = 1,MS = 0)
BEC (ξ < a) vs. BCS (ξ > a)
l || z
Lz = (N/2)~
Lz = (N/2)~ (a/ξ)2 (N/2)~ ? (P.W. Anderson & P. Morel, 1960, A. Leggett, 1975)
I Lz|ΦN 〉 = (N/2)~ |ΦN 〉 independent of (a/ξ)! - McClure-Takagi (PRL, 1979)
BCS Limit: Currents are confined on the Edge
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Angular Momentum of Chiral P-wave Condensates
Chiral P-wave BEC Molecules or BCS Pairs (N Fermions):
|ΦN 〉 =[∫∫dr1dr2 ϕs1s2(r1 − r2) ψ†s1(r1)ψ†s2(r2)
]N/2| vac 〉
ϕs1s2(r) = f(|r|/ξ) (x+ iy) χs1s2(S = 1,MS = 0)
BEC (ξ < a) vs. BCS (ξ > a)
l || z
Lz = (N/2)~
Lz = (N/2)~ (a/ξ)2 (N/2)~ ? (P.W. Anderson & P. Morel, 1960, A. Leggett, 1975)
I Lz|ΦN 〉 = (N/2)~ |ΦN 〉 independent of (a/ξ)! - McClure-Takagi (PRL, 1979)
BCS Limit: Currents are confined on the Edge
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Angular Momentum of Chiral P-wave Condensates
Chiral P-wave BEC Molecules or BCS Pairs (N Fermions):
|ΦN 〉 =[∫∫dr1dr2 ϕs1s2(r1 − r2) ψ†s1(r1)ψ†s2(r2)
]N/2| vac 〉
ϕs1s2(r) = f(|r|/ξ) (x+ iy) χs1s2(S = 1,MS = 0)
BEC (ξ < a) vs. BCS (ξ > a)
l || z
Lz = (N/2)~
Lz = (N/2)~ (a/ξ)2 (N/2)~ ? (P.W. Anderson & P. Morel, 1960, A. Leggett, 1975)
I Lz|ΦN 〉 = (N/2)~ |ΦN 〉 independent of (a/ξ)! - McClure-Takagi (PRL, 1979)
BCS Limit: Currents are confined on the Edge
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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2D Chiral A-phase
3He-A confined in a thin cylindrical cavity - h ξ0 and R ξ0.
hp_
pα
x
Rr
ϕx y
∆ + pip 2D Chiral ABM State:~d(p) = ∆ z (px + ipy)/pf ∼ e+iϕp
Fully Gapped: |~d(p)|2 = ∆2
Bogoliubov Equations for Fermionic Excitations: p→ ~i∇
|p|2/2m∗ − µ ∆ (px + ipy)/pf
∆ (px − ipy)/pf −|p|2/2m∗ + µ
uv
= ε
uv
Anderson’s Iso-spin Representation with particle-hole (Nambu) matrices ~τ = (τ1, τ2, τ3)
H =(|p|2/2m− µ
)τ3 + [∆ px τ1 ∓∆ py τ2] /pf = ~m(p) · ~τ
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Topological Invariant for 2D 3He-A and Fermionic Spectrum
Nambu-Bogoliubov Hamiltonian for 2D 3He-A : H = ~m(p) · ~τ ~m = ( cpx , ∓cpy , ξ(p)) with | ~m(p)|2 =
(|p|2/2m− µ
)2+ c2|p|2 > 0 , µ 6= 0
Topological Invariant for 2D 3He-A↔ QED in d = 2+1 [G.E. Volovik, JETP 1988]:
N2D = π∫
d2p(2π)2
m(p) ·(∂m
∂px× ∂m
∂py
)=
±1 ; µ > 0 and ∆ 6= 0
0 ; µ < 0 or ∆ = 0
“Vacuum” (∆ = 0) with N2D = 0
∣∣∣∣∣ 3He-A(∆ 6= 0) withN2D = 1
Zero Energy Fermions ↑ Confined on the Edge
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Topological Invariant for 2D 3He-A and Fermionic Spectrum
Nambu-Bogoliubov Hamiltonian for 2D 3He-A : H = ~m(p) · ~τ ~m = ( cpx , ∓cpy , ξ(p)) with | ~m(p)|2 =
(|p|2/2m− µ
)2+ c2|p|2 > 0 , µ 6= 0
Topological Invariant for 2D 3He-A↔ QED in d = 2+1 [G.E. Volovik, JETP 1988]:
N2D = π∫
d2p(2π)2
m(p) ·(∂m
∂px× ∂m
∂py
)=
±1 ; µ > 0 and ∆ 6= 0
0 ; µ < 0 or ∆ = 0
“Vacuum” (∆ = 0) with N2D = 0
∣∣∣∣∣ 3He-A(∆ 6= 0) withN2D = 1
Zero Energy Fermions ↑ Confined on the Edge
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Edge Fermions in the 2D Chiral Sr2RuO4and 3He-A Films
Edge Fermions: GRedge(p, ε;x) = π∆|px|
ε+iγ−εbs(p||)e−x/ξ∆
Confinement: ξ∆ = ~vf/2∆ ≈ 102 − 103 A ~/pf
εbs = −c p|| withc = ∆/pf vf
Broken P & T Edge Current
p’p’ p’
_
p’_ p_
p_
ap
bp
xx
−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
p||/pf
−1.0
−0.5
0.0
0.5
1.0Chiral Edge State Dispersion
εbs(p||)
Fermi Energy
−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
p||/pf
−1.0
−0.5
0.0
0.5
1.0Chiral Edge State Dispersion
εbs(p||)
Unoccupied
Occupied
Fermi Energy
I M. Stone, R. Roy, PRB 69, 184511 (2004)
I J. A. Sauls, Phys. Rev. B 84, 214509 (2011)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Edge Fermions in the 2D Chiral Sr2RuO4and 3He-A Films
Edge Fermions: GRedge(p, ε;x) = π∆|px|
ε+iγ−εbs(p||)e−x/ξ∆
Confinement: ξ∆ = ~vf/2∆ ≈ 102 − 103 A ~/pf
εbs = −c p|| withc = ∆/pf vf
Broken P & T Edge Current
p’p’ p’
_
p’_ p_
p_
ap
bp
xx
−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
p||/pf
−1.0
−0.5
0.0
0.5
1.0Chiral Edge State Dispersion
εbs(p||)
Fermi Energy
−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
p||/pf
−1.0
−0.5
0.0
0.5
1.0Chiral Edge State Dispersion
εbs(p||)
Unoccupied
Occupied
Fermi Energy
I M. Stone, R. Roy, PRB 69, 184511 (2004)
I J. A. Sauls, Phys. Rev. B 84, 214509 (2011)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Ground-State Angular Momentum of 3He-A in a Toroidal Geometry
3He-A confined in a toroidal cavity
J2J1
R1
R2D
w R1, R2, R1 −R2 ξ0
Sheet Current: J = 14n ~ (n = N/V = 3He density)
Counter-propagating Edge Currents: J1 = −J2 = 14n ~
Angular Momentum:
Lz = 2π h (R21 −R2
2)× 14n ~ = (N/2) ~ McClure-Takagi Result
I J. A. Sauls, Phys. Rev. B 84, 214509 (2011)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Long-Standing Challenge: The Ground-State Angular Momentum of 3He-A
Possible Gyroscopic Experiment to Measure of Lz(T )S. Davis, J. Parpia & J. Saunders (Cornell-RHUL)
TOR SIONFIBER
MAGNETIC AXISFIELD
SUPERF LUIDPERSISTENT
ENT
APPLIED ROTATIONALVELOCITY
Lp
PERSISTENT CURRENTANGULAR MOMENTUM
SUPERFLUID GYROSCOPEv=uxL
J. Clow and J. Reppy, Phys. Rev. A 5, 424–438
DissipationlessChiral Edge
Currents
EquilibriumAngular Momentum
Non-Specular Edge
Specular Edge
Thermal Signature of Chiral Edge States
IPower Law for T . 0.5Tc
Lz = (N/2)~ (1− c (T/∆)2 )
Toroidal Geometry with Engineered Surfaces
I Incomplete Screening
Lz > (N/2)~
Direct Signature of Edge Currents
I J. A. Sauls, Phys. Rev. B 84, 214509 (2011)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Detection of Broken Time-Reversal Symmetry & Edge Currents
Anomalous Hall Effect in 3He-A
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Detection of e− Bubble Edge Currents in 3He-A
3He
v = µ · E
v
Detection of Broken Time-Reversal Symmetry of Cooper pairs in Superfluid 3He-A
Hiroki Ikegami, Yasumasa Tsutsumi, Kimitoshi Kono, Science 341, 59-62 (2013)
RIKEN, Japan
Electron Mobility:
z
xy V
e-5 nm
3He
µ = µ
1 0 00 1 00 0 1
v = µ · E
v
Detection of Broken Time-Reversal Symmetry of Cooper pairs in Superfluid 3He-A
Hiroki Ikegami, Yasumasa Tsutsumi, Kimitoshi Kono, Science 341, 59-62 (2013)
RIKEN, Japan
Electron Mobility:
z
xy V
B-phase Mobility
IsotropicFully Gapped
e-5 nm3He
! ! ! ! !!! ! ! !
ℓ⟶
µ = µ
1 0 00 1 00 0 1
v = µ · E
v
µ =
µ⊥ µxy 0
− µxy µ⊥ 0
0 0 µ
Detection of Broken Time-Reversal Symmetry of Cooper pairs in Superfluid 3He-A
Hiroki Ikegami, Yasumasa Tsutsumi, Kimitoshi Kono, Science 341, 59-62 (2013)
RIKEN, Japan
Electron Mobility:
z
xy V
B-phase Mobility A-phase Mobility
IsotropicFully Gapped
Anisotropic Transverse Force
e-5 nm
r !
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Detection of e− Bubble Edge Currents in 3He-A
3He
v = µ · E
v
Detection of Broken Time-Reversal Symmetry of Cooper pairs in Superfluid 3He-A
Hiroki Ikegami, Yasumasa Tsutsumi, Kimitoshi Kono, Science 341, 59-62 (2013)
RIKEN, Japan
Electron Mobility:
z
xy V
e-5 nm3He
µ = µ
1 0 00 1 00 0 1
v = µ · E
v
Detection of Broken Time-Reversal Symmetry of Cooper pairs in Superfluid 3He-A
Hiroki Ikegami, Yasumasa Tsutsumi, Kimitoshi Kono, Science 341, 59-62 (2013)
RIKEN, Japan
Electron Mobility:
z
xy V
B-phase Mobility
IsotropicFully Gapped
e-5 nm
3He
! ! ! ! !!! ! ! !
ℓ⟶
µ = µ
1 0 00 1 00 0 1
v = µ · E
v
µ =
µ⊥ µxy 0
− µxy µ⊥ 0
0 0 µ
Detection of Broken Time-Reversal Symmetry of Cooper pairs in Superfluid 3He-A
Hiroki Ikegami, Yasumasa Tsutsumi, Kimitoshi Kono, Science 341, 59-62 (2013)
RIKEN, Japan
Electron Mobility:
z
xy V
B-phase Mobility A-phase Mobility
IsotropicFully Gapped
Anisotropic Transverse Force
e-5 nm
r !
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Detection of e− Bubble Edge Currents in 3He-A
3He
v = µ · E
v
Detection of Broken Time-Reversal Symmetry of Cooper pairs in Superfluid 3He-A
Hiroki Ikegami, Yasumasa Tsutsumi, Kimitoshi Kono, Science 341, 59-62 (2013)
RIKEN, Japan
Electron Mobility:
z
xy V
e-5 nm3He
µ = µ
1 0 00 1 00 0 1
v = µ · E
v
Detection of Broken Time-Reversal Symmetry of Cooper pairs in Superfluid 3He-A
Hiroki Ikegami, Yasumasa Tsutsumi, Kimitoshi Kono, Science 341, 59-62 (2013)
RIKEN, Japan
Electron Mobility:
z
xy V
B-phase Mobility
IsotropicFully Gapped
e-5 nm3He
! ! ! ! !!! ! ! !
ℓ⟶
µ = µ
1 0 00 1 00 0 1
v = µ · E
v
µ =
µ⊥ µxy 0
− µxy µ⊥ 0
0 0 µ
Detection of Broken Time-Reversal Symmetry of Cooper pairs in Superfluid 3He-A
Hiroki Ikegami, Yasumasa Tsutsumi, Kimitoshi Kono, Science 341, 59-62 (2013)
RIKEN, Japan
Electron Mobility:
z
xy V
B-phase Mobility A-phase Mobility
IsotropicFully Gapped
Anisotropic Transverse Force
e-5 nm
r !
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Skew Scattering of Quasiparticles by Bubble Edge Currents
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Chiral “Edge” Currents Circulating an Electron Bubble
Jφ(r, φ)
4π3vFNF kBTcat kfr = 30 for kfR = 11.17
I Oleksii Shevtsov & JAS, 2016Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Detection of Broken Time-Reversal Symmetry in 3He-A
electron bubbles
f = 0.06 – 6 HzVin = 0.1 – 1 V
Measurement of the Transverse e- mobility in Superfluid 3He Films
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
left electrode
right electrode
electron bubbles
f = 0.06 – 6 HzVin = 0.1 – 1 V
v =
µ⊥ E + µxy × E
Measurement of the Transverse e- mobility in Superfluid 3He Films
=−z
Transverse Force from Skew Scattering
=+z
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
left electrode
right electrode
∆I = IR − IL
∆Iim
∆Ire
= −z= +z
|IR − IL|IR + IL
≈ 6%
Transverse e- bubble current in 3He-A
Single Domains:
Run 1
Tc
Tc
Run 2
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Zero Transverse e- current in 3He-B (T - symmetric phase)
µµN
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Zero Transverse e- current in 3He-B (T - symmetric phase)
µµN
∆Iim
∆Ire
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Detection of Broken Time-Reversal Symmetry in 3He-A
electron bubbles
f = 0.06 – 6 HzVin = 0.1 – 1 V
Measurement of the Transverse e- mobility in Superfluid 3He Films
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
left electrode
right electrode
electron bubbles
f = 0.06 – 6 HzVin = 0.1 – 1 V
v =
µ⊥ E + µxy × E
Measurement of the Transverse e- mobility in Superfluid 3He Films
=−z
Transverse Force from Skew Scattering
=+z
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
left electrode
right electrode
∆I = IR − IL
∆Iim
∆Ire
= −z= +z
|IR − IL|IR + IL
≈ 6%
Transverse e- bubble current in 3He-A
Single Domains:
Run 1
Tc
Tc
Run 2
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Zero Transverse e- current in 3He-B (T - symmetric phase)
µµN
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Zero Transverse e- current in 3He-B (T - symmetric phase)
µµN
∆Iim
∆Ire
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Detection of Broken Time-Reversal Symmetry in 3He-A
electron bubbles
f = 0.06 – 6 HzVin = 0.1 – 1 V
Measurement of the Transverse e- mobility in Superfluid 3He Films
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
left electrode
right electrode
electron bubbles
f = 0.06 – 6 HzVin = 0.1 – 1 V
v =
µ⊥ E + µxy × E
Measurement of the Transverse e- mobility in Superfluid 3He Films
=−z
Transverse Force from Skew Scattering
=+z
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
left electrode
right electrode
∆I = IR − IL
∆Iim
∆Ire
= −z= +z
|IR − IL|IR + IL
≈ 6%
Transverse e- bubble current in 3He-A
Single Domains:
Run 1
Tc
Tc
Run 2
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Zero Transverse e- current in 3He-B (T - symmetric phase)
µµN
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Zero Transverse e- current in 3He-B (T - symmetric phase)
µµN
∆Iim
∆Ire
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Detection of Broken Time-Reversal Symmetry in 3He-A
electron bubbles
f = 0.06 – 6 HzVin = 0.1 – 1 V
Measurement of the Transverse e- mobility in Superfluid 3He Films
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
left electrode
right electrode
electron bubbles
f = 0.06 – 6 HzVin = 0.1 – 1 V
v =
µ⊥ E + µxy × E
Measurement of the Transverse e- mobility in Superfluid 3He Films
=−z
Transverse Force from Skew Scattering
=+z
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
left electrode
right electrode
∆I = IR − IL
∆Iim
∆Ire
= −z= +z
|IR − IL|IR + IL
≈ 6%
Transverse e- bubble current in 3He-A
Single Domains:
Run 1
Tc
Tc
Run 2
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Zero Transverse e- current in 3He-B (T - symmetric phase)
µµN
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Zero Transverse e- current in 3He-B (T - symmetric phase)
µµN
∆Iim
∆Ire
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Detection of Broken Time-Reversal Symmetry in 3He-A
electron bubbles
f = 0.06 – 6 HzVin = 0.1 – 1 V
Measurement of the Transverse e- mobility in Superfluid 3He Films
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
left electrode
right electrode
electron bubbles
f = 0.06 – 6 HzVin = 0.1 – 1 V
v =
µ⊥ E + µxy × E
Measurement of the Transverse e- mobility in Superfluid 3He Films
=−z
Transverse Force from Skew Scattering
=+z
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
left electrode
right electrode
∆I = IR − IL
∆Iim
∆Ire
= −z= +z
|IR − IL|IR + IL
≈ 6%
Transverse e- bubble current in 3He-A
Single Domains:
Run 1
Tc
Tc
Run 2
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Zero Transverse e- current in 3He-B (T - symmetric phase)
µµN
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)
Zero Transverse e- current in 3He-B (T - symmetric phase)
µµN
∆Iim
∆Ire
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59-62 (2013)Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Anomalous Hall Mobility of Negative Ions (e− Bubbles) in 3He-A
Anomalous Hall Angle: tanα ≈ α =FxyFxx
H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59 (2013)
I Note: ~FL = ec~v × ~B, is negligible: |FL|/|Fxy| ≈ 10−3 − 10−6
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Theory of the Anomalous Hall Mobility of Negative Ions (e− Bubbles) in 3He-A
Force on a Moving e− Bubble from Thermally Excited Quasiparticles
ek’
k
v
dP
dt=∑k,k′
~(k′ − k)fk(1− fk′)Γv(k′,k), ≡ e [µ]−1 v
fk → fk = fk(Ek − ~k · v),
e(µ−1)ij = n3pF
∫ ∞0
dE
(−2
∂f
∂E
)[〈σsymm
ij (E)〉+ 〈σskewij (E)〉 ], i, j ∈ x, y, z,
(1)
I G. Baym, C. Pethick, M. Salomaa, PRL 38, 845 (1977) - e− Mobility in 3He-B
I Oleksii Shevtsov & JAS (2016) - e− Mobility in Chiral Superfluids
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Theory of the Anomalous Hall Mobility of Negative Ions (e− Bubbles) in 3He-A
Differential Cross-Section atE = 1.01∆
Skew Scattering in 3D Chiral 3He-A
Hard Sphere QP-e− Interaction withkfR = 11.16 (fit µN )
l . 11 channels dominate
Chiral Fermions confined near the e−
Broadened by Nodal Fermions
Skew Scattering Resonances
I Oleksii Shevtsov & JAS (2016) - e− Mobility in Chiral Superfluids
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Longitudinal Mobility of Negative Ions (e− Bubbles) in 3He-A
Drag Force - Longitudinal Mobility:I Expert: H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59 (2013) µxx = vx/Ex
0.0 0.2 0.4 0.6 0.8 1.0T/Tc
100
101
102
103
104µxx/µ
N
theory
experiment
0 5 10 15 20
l
-0.5
0.0
0.5
δ l[π
]
3He-e− Interaction: Hard-sphere with kf R = 11.16 fit to normal-state mobility (µN )I Oleksii Shevtsov & JAS (2016) - e− Mobility in Chiral Superfluids
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Anomalous Hall Mobility of Negative Ions (e− Bubbles) in 3He-A
Anomalous Hall Angle: tanα ≈ α =FxyFxx
=µ−1xy
µ−1xx
I Expt: H. Ikegami, Y. Tsutsumi, K. Kono, Science 341, 59 (2013)
I Theory: Oleksii Shevtsov & JAS (2016) - e− Mobility in Chiral Superfluids
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order
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Ongoing & Challenges
Experiment
I Detect Ground State Angular Momentum of 3He-AI Discover and Characterize New Phases of Nano-scale 3HeI Local probes to detect and control Majorana states in 3He-B
Theory
I Develop quantum transport theory for coupled nano-fluidicmechanical resonantors & oscillators
I Develop theory for acoustic, NMR and optical spectroscopy oftopological edge/surface states of 3He
I Develop theory of topological quantum matter beyond themean-field level; strong-coupling, interactions and non-locality
Anomalous Hall Effect in Superfluid 3He J. A. Sauls From Spontaneous Symmetry Breaking to Topological Order