SLIDE 1
Thesis defense Excitations in topological superfluids and - - PowerPoint PPT Presentation
Thesis defense Excitations in topological superfluids and - - PowerPoint PPT Presentation
Thesis defense Excitations in topological superfluids and superconductors Hao Wu Supervisor: James A. Sauls Northwestern University December 3, 2016 DMR-1106315 Introduction superconductivity and superfluidity Heike Onnes: superconductor
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Introduction – superconductivity and superfluidity
Superfluid 4He – frictionless flow, quantized vortices
Bose-Einstein condensation: macroscopically occupied ground state Gauge symmetry breaking
Superfluid 3He
Anisotropic Cooper pair of 3He atoms
I Repulsive interaction non-swave I Spin susceptibility spin-triplet pairing
Attractive interaction – spin exchange Unconventional superfluid
SLIDE 4
Spin-Triplet, P-wave Pairing - Superfluid Phases of 3He Symmetry Group of Normal 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 2.5 T [mK] 5 10 15 20 25 30 P [bar]
A B
PCP
3He
Superfluid
Spin-Triplet, P-wave Order Parameter : ∆αβ (p) =~ d(p)· (i~ σσy)αβ dµ(p) = Aµi pi Chiral ABM State ~ l = ˆ m⇥ ˆ n Aµi = ∆ ˆ dµ ( ˆ m+iˆ n)i Lz = 1 , Sz = 0 “Isotropic” BW State Aµi = ∆δµi J = 0 , Jz = 0
SLIDE 5
Symmetry and topology of superfluid phases Nambu-Bogoliubov Hamiltonian for 3He-A : b H = ✓ ξ(p) c(px ±ipy) c(px ⌥ipy) ξ(p) ◆ = ~ m(p)·b ~ τ I Symmetry: C⇥U(1)N + L I Topological Invariant G.E. Volovik, JETP 67(9), 1804-1811 (1988): N2D = π
Z
d2p (2π)2 ˆ m(p)· ✓ ∂ ˆ m ∂ px ⇥ ∂ ˆ m ∂ py ◆ = ±1 Nambu-Bogoliubov Hamiltonian for 3He-B : b H = ξ(p)b τ3 +cp· ˆ σb τ1 I Symmetry: C⇥T⇥SO(3)L+S I Topological Invariant G.E. Volovik, JETP Lett. 90: 398 (2009): N3D =
Z
d3p 24π2 εi jk Tr n Γ( b H1∂pi b H)⇥( b H1∂pj b H)( b H1∂pk b H)
- = 2,
Γ = CT Bulk-edge correspondence I Nontrivial bulk topology gapless boundary states I Bogoliubov quasiparticle, particle-hole mixing Majorana Fermion
SLIDE 6
Superfluid in confined geometry
Porous medium – aerogels Phase diagram
- J. Pollanen et al., Nature Physics 8, 317-320 (2012)
Polar phase
- V. Dmitriev et al., PRL 115, 165304 (2015)
Slab geometry
- L. Levitin et al., Science 340, 841-844 (2013)
- A. Vorontsov and J. A. Sauls, PRL 98, 045301 (2007)
SLIDE 7
Confinement induced inhomogeneous superfluid phase
Missing Stripe phase
L. Levitin et al., Science 340, 841-844 (2013)
Exist in slab geometry
- A. Vorontsov and J. A. Sauls, PRL 98, 045301 (2007)
Exist with diffusive surface
- K. Aoyama, arXiv:1605.07302 (2016)
Generic phenomena, in cylinders
- K. Aoyama, PRB 89, 140502 (2014)
- J. Wiman and J. A. Sauls, to be published (2016)
Strong-coupling effect
- J. Wiman and J. A. Sauls, JLTP 184, 10541070 (2016)
Chiral stripe phase
- H. Wu and J. A. Sauls, to be published (2016)
I Periodic chiral domain walls (PDW) I Competing phases – Polar, PDW, ABM
SLIDE 8
Chiral superconductivity
Chiral superconductors Complex order parameter: ∆(p) = |∆(p)|eiφ(p) Phase winding:
I
Cp
dφ(p) = n⇥2π Time reversal symmetry breaking Nontrivial bulk topology – b H = ~ m(p)·b ~ τ ∆(p) = ∆(px +ipy) Majorana Fermions Chiral Weyl Fermion Spontaneous charge current Half quantum vortex Phase vortex + d-vector rotation Non-abelian statistics quantum computing
ϕ=0 ϕ=π ϕ=3π/4 ϕ=π/2 ϕ=π/4
- D. Ivanov, PRL 86, 268 (2001)
- J. Jang et al., Science 331, 186-188 (2011)
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Is Sr2RuO4 a chiral superconductor? Sr2RuO4 Tc = 1.15K Evidence for chiral superconductivity I Spin triplet pairing S = 1 NMR I Broken time reversal symmetry I Two dimensional E1u representation, px ±ipy Evidence against chiral superconductivity I Missing edge current I No second transition Kerr rotation
cooled in -43 Oe! warmup in ZF
- J. Xia et al. - PRL 97, 167002 (2006)
Uniaxial strain
- C. Hicks et al., Science 344, 283-285 (2014)
- J. Xia et al., PRL 97, 167002 (2006)
- R. Cava, Chem. Commun. (2005)
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Outline Goal : Search for Majorana Fermions Novel superfluid phases under confinement Indentification of chiral superconductor Content :
1
Introduction of quasiclassical theory (Ch 2)
2
Majorana states and currents on superfluid 3He-B (Ch 3-4)
- H. Wu and J. Sauls, PRB 88, 184506 (2013)
3
Edge spectrum, currents and phase transitions in 3He-A film (Ch 5-6)
- H. Wu and J. Sauls, to be published (2016)
4
Collective modes and power absoptions in chiral superconductor (Ch 7-8)
- H. Wu and J. Sauls, to be published (2016)
5
Conclusion (Ch 9)
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Elements of quasi-classical theory
Quasiclassical Green’s function: b G(R,p;ε) = ˆ g ˆ f ˜ ˆ f ˜ ˆ g ! , ∆ ⌧ Ef , ¯ h/pf ⌧ ξ0 ... gαβ(R,p) =
Z ϖ
ϖ dξ(p)
Z
dreip·r ⌦ TrΨ(R+r/2)Ψ†(Rr/2) ↵ fαβ(R,p) =
Z ϖ
ϖ dξ(p)
Z
dreip·r hTrΨ(R+r/2)Ψ(Rr/2)i Equilibrium Eilenberger’s transport equation h ε ˆ τ3 b ∆(R,p), b G(R,p;ε) i +i¯ hv(p)·∇R b G(R,p;ε) = 0 Self-consistent equation ˆ ∆(R,p) =
Z
dp0V(p,p0)
Z ϖ
ϖ dε ˆ
f(R,p0;ε) Linear response theory b G(R,p;ε,t) = b G(R,p;ε)eq +δ b G(R,p;ε,t), b ∆(R,p;t) = b ∆(R,p)eq +δ b ∆(R,p;t) p = pf ˆ p
SLIDE 12
Superfluid 3He-B in a film with width D
D
x y z
∆
3He-B
Bulk 3He-B I Fully gapped E(p) = q ξ 2(p)+∆2 I Helicity eigenstate Surface 3He-B I Gapless Fermions E(p) = ±cpk I Ising spin state
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Quasi-classical theory for confined 3He-B
B Phase Order parameter ˆ ∆ = ~ d ·(i~ σσy) dx = ∆k(z)px, dy = ∆k(z)py, dz = ∆?(z)pz, Nambu Green’s function b G(p,z;ε) = g+~ g·~ σ ~ f·(i~ σσy) ˜ ~ f·(iσy~ σ) ˜ g+ ˜ ~ g·~ σ tr ! Eilenberger’s equation h b HA , b G i +i¯ hvp ·∇b G = 0 Andreev Hamiltonian b HA = εb τ3 b ∆ Integrate along trajectories p, p
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Green’s function for specular 3He-B surface – Quasiparticle propagator
Quasiparticle Green’s function (Scalar) g = πε p ∆2 ε2 " 1+ ∆2
? cos2 θ
∆2
k sin2 θ ε2 e2 p ∆2ε2z/vz
# Local density of states N (p,z;ε) = 1 π Img(p,z;ε) Quasiparticle Green’s function (Spin vector) ~ g = π∆?∆k sinθ cosθ ∆2
k sin2 θ ε2
e2
p ∆2ε2z/vz~
e2 Local spin density of states ~ S (p,z;ε) = 1 π Im~ g(p,z;ε)
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Green’s function for specular 3He-B surface – Cooper pair propagator Anomalous Green’s function fz = π∆? cosθ p ∆2 ε2 " 1e2
p ∆2ε2 z/vz + iε
p ∆2 ε2 ∆2
k sin2 θ ε2 e2 p ∆2ε2 z/vz
# fk = π∆k sinθ p ∆2 ε2 " 1+ ∆? cos2 θ ∆2
k sin2 θ ε2 e2 p ∆2ε2 z/vz
#
1 2 3 4 5 6 z/ξ∆ 0.0 0.2 0.4 0.6 0.8 1.0 1.2
∆k/∆ ∆?/∆ I ξ∆ = vf /2∆ I Supression of pz orbital I Enhancement of pk orbital I Change in length scale ξ∆
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Local density of states
ε/∆
2 1 1 2
pk / pf
1.0 0.5 0.0 0.5 1.0 2 4 6
N(p||, ε)
Surface bound states Bound state energy εb = ±cpk, c = (∆k/p f ) Spectral weight π∆? cosθ 2 e2∆?z/vf Continuous spectrum |ε| > ∆ lim
z!∞Nc(p,z;ε) = Nbulk
SLIDE 17
Spontaneous spin current
Spin spectral function: ~ S = π∆? cosθ 2 [δ(ε +c pk)δ(ε c pk)]e2∆?z/vf ~ e2 Confining length: ξ∆ = vf /2∆? Spin polarized along~ e2 = z⇥p Spectral Spin current: ~ Jα(p,z;ε) = 2Nf vp ⇥ Sα(p,z;ε)Sα(p0,z;ε) ⇤ Time reversal p ! p0: Sα(p,z;ε) = Sα(p0,z;ε), ~ Jα(p,z;ε) = ~ Jα(p0,z;ε) spin current tensor J (p,z;ε)⇥ @ 1 +1 1 A
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Surface spin current
−1.0 −0.5 0.0 0.5 1.0
x
−1.0 −0.5 0.0 0.5 1.0
Spin current structure on the surface
Flow direction Spin polarization
y
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Spectral spin current density pk = 0.5p f
−2 −1 1 2 1 2 3 4 5 −2 −1 1 2
Fermi Distribution
ε / ∆
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Finite temperature spin current
Sheet current – current per unit area K(T) ⌘
Z
dz
Z
dp
Z
dε J (p,z;ε) ' ✓ 1 27πζ(3) 4 T 3 ∆3 ◆ K(0)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
T/TC
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
∆(T)/∆(0) K(T)/K(0) 1 − 27πζ(3)
4 T 3 ∆3
Ground state current: K(0) = 1 6n2Dv f ¯ h/2 Temperature dependence: power law T 3 No static fields couple to spin current
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Mass current in a channel
D
ϕ1
ϕ2
~ ps = ~ 2 ~ r'
x y z
Constant phase gradient ˆ
∆(p,r) = ˆ ∆0(p,r)eiϕ(r)
~ ps = ¯ h 2∇ϕ( ~ r) – DC superflow Local Gauge transform ε ! ε ps ·v(p)
1
Solve transport equation, gauge transform back: b g0 g
2
Current response: j(z,T) = Nf
Z
dpπT ∑
εn
g(p,z;εn)
3
Landau molecular field: vMF(p,z) =
Z
dp0 As(p,p0)πT ∑
εn
g(p0,z;εn)
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Surface mass current reduction
1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0
pk/pf
1.0 0.5 0.0 0.5 1.0
εb(p||)/∆|| Dispersion of 3He-B Surface Bound State
εb(p||) = (±∆||/pf vs) ⇥ p|| εb(p||) = ±∆||/pf ⇥ p||
No bound state contribution to zero temperature mass current j(z,0) = nps
D ξ∆
Thermal excitation of gapless states larger reduction at surface Low temperature power law dependence: J(T) = nps(1aT 3) I Surface to bulk ratio D I Fermi-liquid effect – pressure P
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Mass flow with Fermi liquid effect
Mass flow at various pressure
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
T [mK]
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ρs/ρ
Total Mass Current at P = 34 bar Bulk Mass Current at P = 34 bar leading order correction ∼ T 3
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Two dimensional superfluid A phase Topological superfluid b H = ~ m·b ~ τ ~ m = (∆px,∆py,ξ(p)) Detection of edge current – transverse mobility
- H. Ikegami et al., Science 341(6141), 59-62 (2013)
- O. Shevtsov and J. Sauls, PRB 94, 064511 (2016)
Edge state: gapless Weyl fermion
1.0 0.5 0.0 0.5 1.0 pk/pf 1.0 0.5 0.0 0.5 1.0 ε(pk)∆
- ccupied
empty
ε(pk) = c pk
continuum
Spontaneous ground state current j(x) = Nf vf ∆ex/ξ∆
J1 J2 D 3He-A
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Confined geometry D ! 10ξ0
Main focus: Hybridization of bound fermions at opposing edges New ground states in confinement superfluid film
Chiral A-Phase order Parameter ∆(p) = ∆?(x)px +i∆k(x)py Specular reflection at both edges px = px, py = py Spatial dependent gap amplitudes ∆?(0) = ∆?(D) = 0
∆ ˆ p ˆ p
1 2 3 4 5 6 7 8 9 10 11 12
x/ξ0
0.0 0.1 0.2 0.3 0.4
∆bulk ∆k(x) ∆?(x)
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Quasiclassical wave function – Andreev’s Equation
Andreev’s Equation b HA |Ψ(r;p)i+i¯ hvp ·∇|Ψ(r;p)i = 0 |Ψi = (u,v)T b HA = εb τ3 b ∆ Boundary condition
Ψ(0,y;p) = Ψ(0,y;p) Ψ(D,y;p) = Ψ(D,y;p)
Bloch wave solution
Ψ(x,p) = eikx ¯ Ψ(x,p)
Bound state dispersion
ε(p,k)2 = ∆2p2
y +∆2p2 x(1cos(2kD))/(cosh(2λD)cos(2kD))
with
λ = p ∆2 ε2 x = 0 x = 2D ∆?px +i∆kpy x = −2D x = 4D ∆?px +i∆kpy 2D
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Bound state under confinement – local spectrum at x = D
1.0 0.5 0.0 0.5 1.0 pk/pf 1.0 0.5 0.0 0.5 1.0 ε/∆ 4 8 12 16 20 24 28 32 36 40
∆?px +i∆kpy x = D x = 0
ε(pk) = c pk
continuum
ε(pk) = cpk
continuum
I Energy band at each pk I Van Hove singularities
−1.0 −0.5 0.0 0.5 1.0
ε/∆
2 4 6 8 10
N(ε)/N0
Van Hove Singularity normal incidence py = px(45o)
ε2 = ∆2p2
y +∆2p2 x/cosh(λD)2
ε2 = ∆2p2
y
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Continuum spectrum under confinement – band structure at x = D
1.0 0.5 0.0 0.5 1.0 pk/pf 4 2 2 4 ε/∆ 4 8 12 16 20 24 28 32 36 40 Band gap in continuum Band gap: largest pk = 0 smallest pk = pf Van Hove sigularities
SLIDE 29
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T[Tc] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 J[n¯
h 4 ]
D = ∞ D = 15ξ0 D = 12ξ0 D = 11ξ0 D = 10.4ξ0 D = 10ξ0 D = 9.75ξ0 D = 9.6ξ0
Edge currents
∆?px +i∆kpy x = D x = 0
D/2
J(T) =
Z D/2
dx jy(x;T)
SLIDE 30
Edge currents
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T[Tc] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 J[n¯
h 4 ]
D = ∞ D = 15ξ0 D = 12ξ0 D = 11ξ0 D = 10.4ξ0 D = 10ξ0 D = 9.75ξ0 D = 9.6ξ0
SLIDE 31
Phase Diagram I – Polar to A phase transition
6 8 10 12 14 16 18 20 D/ξ0 0.0 0.2 0.4 0.6 0.8 1.0 T/Tc
Polar Chiral
D = 9.75ξ0
SLIDE 32
Linear instability analysis Instability in polar phase ∆(p,x,y) = i∆kpy + A31(x,y)px +A32(x,y)py
I Pertubative term ∆1(p,r) in orbital basis
Eigenvalue equations for fourier component ∆1(p,Q)
greens function:
ωm = q ∆2
k +ε2 m
f1(p,Qx,Qy) = ∆1(p,Qx,Qy)πωm/ ⇣ ω2
m +
- vp ·Q
2 /4 ⌘ self-consistency: ∆1(p,Qx,Qy) =
Z
dp0V(p,p0)T
εm<ωc
∑
εm
f1(p0,Qx,Qy;εm)
The smallest D for nonvanishing ∆1 Qy(T) ⇠ π/15ξ0 – single mode instability
SLIDE 33
Phase diagram II – polar to periodic domain wall(PDW) phase transition
6 8 10 12 14 16 18 20
D/ξ0
0.0 0.2 0.4 0.6 0.8 1.0
T/Tc
Polar Chiral PDW
Dc2 Dc1
SLIDE 34
Phase diagram II – polar to periodic domain wall(PDW) phase transition
6 8 10 12 14 16 18 20
D/ξ0
0.0 0.2 0.4 0.6 0.8 1.0
T/Tc
(10.5, 0.7) Polar Chiral PDW
Dc2 Dc1
SLIDE 35
Self-consistent PDW order parameter T = 0.7Tc,D = 10.5ξ0, ∆ = ∆?px +∆kpy Base state pure polar ∆0 = i∆py Small (104) random fluctuations |∆?|
0.4 0.8 1.2 ⇥104
|∆k|
0.02 0.10 0.20 30 15 15 30
x[ξ0]
5 10
y[ξ0] cos(φ?)
1.0 0.5 0.0 0.5 1.0
cos(φk)
3 3 ⇥104
Self-Consistent calculation ∆?(r),∆k(r) ∆(r,p) = ∆?(r)px + ∆k(r)py h d HA , b G(r,p;εm) i + i¯ hvp · ∇b G(r,p;εm) = 0
∆(r,p) = T ∑
m Z
dp0 v(p,p0)f(r,p0;εm)
SLIDE 36
Self-consistent PDW order parameter T = 0.7Tc,D = 10.5ξ0, ∆ = ∆?px +∆kpy
|∆?|
0.02 0.04 0.06 0.08
|∆k|
0.269 0.271 0.273 30 15 15 30
x[ξ0]
5 10
y[ξ0] cos(φ?)
1.0 0.5 0.0 0.5 1.0
cos(φk)
0.15 0.05 0.05 0.15
SLIDE 37
Self-consistent PDW order parameter T = 0.7Tc, ∆ = ∆?px +∆kpy D = 10.5ξ0
−0.08 0.00 0.08
∆1,R 2πTc
−0.05 0.00 0.05
∆2,R
−0.0003 0.0000 0.0003
∆1,I
center edge
−30 −20 −10 10 20 30
y/ξ0
0.267 0.270 0.273
∆2,I
px + ipy px + ipy −px + ipy
D = 11.5ξ0
−0.1 0.0 0.1
∆1,R 2πTc
−0.10 −0.05 0.00
∆2,R
center edge
−0.001 0.000 0.001
∆1,I
−30 −20 −10 10 20 30
y/ξ0
0.26 0.27
∆2,I
px + ipy −px + ipy
SLIDE 38
Dc1 comparing free energy
Competing phases:
I Homogeneous A-phase I Chiral domain wall
Free energy functional – Luttinger-Ward ∆Ω[b G,b ∆] = 1 2
Z 1
0 dλSp0 n
b ∆(b Gλ b G)
- +∆Φ[b
G]
Self-consistent order parameter b ∆(D,T) auxiliary propagator b Gλ Integration over space, momentum, energy and λ
Find the critical D(T) at which ∆ΩDW < ∆ΩA
6 8 10 12 14 16 18 20 D/ξ0 0.0 0.2 0.4 0.6 0.8 1.0 T/Tc Polar Chiral PDW
Dc2 Dc1
SLIDE 39
Difference in free energy density D = 15ξ0,T = 0.5Tc y/ξ0
−20 −15 −10 −5 5 10 15 20
x/ξ0
2 4 6 8 10 12 14 −1.0 −0.5 0.0 0.5 1.0
∆ΩDW − ∆ΩA
×10−3
SLIDE 40
Phase diagram III – PDW to chiral phase transition, comparing free energy
6 8 10 12 14 16 18 20 D/ξ0 0.0 0.2 0.4 0.6 0.8 1.0 T/Tc
Polar Chiral PDW
Dc2 Dc1
SLIDE 41
Phase piagram III
6 8 10 12 14 16 18 20 D/ξ0 0.0 0.2 0.4 0.6 0.8 1.0 T/Tc
Polar Chiral PDW
Dc2 Dc1 Broken symmetry I time reversal I translational
SLIDE 42
Phase diagram III
6 8 10 12 14 16 18 20 D/ξ0 0.0 0.2 0.4 0.6 0.8 1.0 T/Tc
Polar Chiral PDW
time reversal ⇥ half lattice translation Dc2 Dc1
SLIDE 43
Phase diagram III
6 8 10 12 14 16 18 20 D/ξ0 0.0 0.2 0.4 0.6 0.8 1.0 T/Tc
Polar Chiral PDW
D PDW D 3He-A
Dc2 Dc1
SLIDE 44
Superconducting Bosonic modes – eigenmodes for order parameter fluctuation
Conventional superconductor ∆ = |∆|eiφ I Higgs mode – amplitude fluctuation ω = 2|∆|
- R. Sooryakumar and M. Klein, PRL 45, 660 (1980)
I Bogoliubov mode – phase fluctuation ω = 0 Unconventional pairing, e.g. 3He ˆ ∆ = ~ d(p)·(i~ σσy)
VOLUME 45, NUMBER 4
PHYSICAL REVIEW LETTERS
28 JUx,v 1980 two 20-MHz fundamental
transducers separated
by 0.318 cm." This combination allowed meas-
urements at 10, 20, 30, 50, and 60 MHz. Both
cells were in good thermal contact with a lantha-
num-diluted cerium-magnesium-nitrate ther mom-
eter whose susceptibility
was monitored
by a SQUID detection
system. The thermometer
was
calibrated against 7, as a function
- f pressure
with the Pt Helsinki scale." The detection
scheme used in the experiment is described
in detail else-
- where. " Briefly,
we used a phase-sensitive
sys-
tem capable of resolving both the in- and out-of- phase components
- f the received
sound signal,
relative to a stable reference signal operating
in a phase-locked loop fixed at the drive frequen-
- cy. The gain of the system was independent
- f
phase to within 3-4P& The sound attenuation
and
phase velocity shift,
bc, were obtained
from the changes
- f the signal amplitude
and phase,
re-
spectively.
As an overall check on all of our measurements, we have selectively compared the data obtained using the phase-sensitive
sys-
tem to that obtained from a non-phase-sensitive detector and a boxcar integrator. The attenua- tion data showed
no dependence upon the detec-
tion scheme used,
as long as the variation
in c was less than -1%. With larger changes in c, the
phase-sensitive
method becomes less reliable
because of changes
in the resonant frequency
- f
the transducers, and for these cases, the data were obtained by our second method. In Fig. 1, we show the attenuation
- f 60-MHz
zero sound versus reduced temperature, at a pressure
- f 5.3 bars.
The peak location of the main attenuation feature near T, cannot be deter- mined because of loss of the signal in this region, but we have marked the expected position
- f the
collective-mode peak as predicted
by Eq. (1), as well as the pair-breaking
cutoff temperature, using the Ginzburg-Landau expression for the en- ergy gap:
- 2 ~g
Z (T) =mt
Tc
C
where 6C/C„ is the specific heat jump at T„ which was taken from the recent measurements by the Alvesalo et al." The main attenuation peak is observed to split into two parts, in agree- ment with earlier observations by Paulson and W'heatley. The new feature, marked
y, is completely un-
expected and, to our knowledge, not previously
- bserved.
This extraordinarily narrow attenua- tion peak was observed at 30, 50, and 60 MHz,
l5-
.
/12/5h .. 2b, . E
l0-
5-
I-
CI
20jd&
I
0.6
I I
0.8 0.9
~ ~ ~ ~
l.o
- FIG. 1. Relative
attenuation
- f 60-MHz sound at 5.3
bars.
Arrows
show the predicted
location of the collec- tive-mode
and pair-breaking attenuation
peaks.
The height of the p peak is not known because of loss of the signal for relative attenuation &16 cm '.
h(u =Ass(T «),
(3) where
A is a constant and 6~(T) is given by Eq.
(2). In Table I, we give the values
- f the y peak
location relative to T„as well as the values"
- f AC/C„used
to calculate
A from the data. The
depending
- n pressure.
At each pressure,
there
is a characteristic
frequency above which this fea- ture can be resolved. This presumably
results from the combined effect of the variation
with frequency
- f the location in temperature
- f the
y feature and the broadening
near T, of the main attenuation peak, whose width depends
- n the
ratio h~/r,
where
7 is the quasiparticle
lifetime
at the Fermi surface (-10-' sec).' Thus, at too
low frequencies
the expected position in tempera-
ture of the y feature falls within
the region of high attenuation, where it cannot be resolved. As the pressure
is lowered,
the height
- f this peak
(for a constant frequency) increases
and its width in temperature
decreases.
This decrease of
width is expected if the broadening
is due to quasi- particle
lifetime effects. Using the phase-sensi- tive detection scheme,
we also observe a sound-
velocity change at the y peak. In Fig. 2, we show both the sound attenuation and the velocity shift in the vicinity
- f this feature.
The velocity change
is superimposed
- n a smooth variation
- f c which
is observed'
at temperatures
below the main attenuation
peak.
In the absence of any prediction
for this fea- ture,
we have tried to characterize
it in the most
- bvious
way, which is to assume 263
Sound attenuation, J = 2+
- R. Giannetta et al., PRL 45, 262 (1980)
Acoustic Faraday effect, J = 2+
- Y. Lee et. al., Nature 400, 431-433 (1999)
SLIDE 45
Fermi surfaces and pairing models for Sr2RuO4 ARPES Fermi Surface
ARPES
- A. Damascelli et al., PRL 85, 5194 (2000)
0.0 0.5 1.0
φ/π
0.0 0.5 1.0 1.5
|∆(p)|/∆
κ = 0.00 κ = 0.20 κ = 0.40 κ = 0.60 κ = 0.80 κ = 1.00
S = 1, p-wave Cooper Pairs (E1u) ⇡ 2D Fermi surface (γ-band)
- T. M. Rice and M. Sigrist, J. Phys. Cond. Mat. 7(47) (1995)
~ d(p) = ˆ d
- Ax ˆ
px +Ay ˆ py
- Anisotropic E1u Cooper Pairs:
- Q. H. Wang et. al., EPL 104(1) (2013)
- T. Scaffidi et al., PRB 89, 220510 (2014)
ˆ px ! Yx(p) = ˆ px I(p) ˆ py ! Yy(p) = ˆ py I(p) I( ˆ px, ˆ py) invariant under D4h I(p) = 1+κ
- |2 ˆ
px ˆ py|1
- 1+4κ(1κ)/π 2κ(13κ/4)
1 2
0 ε 1 I Multi-component order + Anisotropy Spectroscopy of Pairing Symmetry
SLIDE 46
Linear responses in chiral superconductors Order parameter (Nambu): b ∆(p) = ✓ ˆ d ·(i~ σσy)∆(p) ˆ d ·(iσy~ σ)˜ ∆(p) ◆ Chiral basis: Y± = Yx ±iYy, equilibrium state ∆(p) ⇠ ∆Y+ Coupling to vector field: b vext = vp ·A(~ q,ω) b τ3 Dynamic gap in chiral basis: d(p) ! ∆(p,~ q,ω) = ∆(p)+d(p,~ q,ω) ∆(p,~ q,ω) = ∆Y+ + D(~ q,ω)Y+ + E(~ q,ω)Y ˜ ∆(p,~ q,ω) = ∆Y + D⇤(~ q,ω)Y + E⇤(~ q,ω)Y+ Current response: d± ⌘ d ± ˜ d δ~ j = Nf
Z
dpf v f (pf ) ⇢ 1+ η2 ω2 η2 (1λ)
- vext +η ¯
λ
- ∆Rd i∆Id+
I Fermionic quasiparticle contribution I Bosonic mode contribution
SLIDE 47
Dynamical equation for order parameter fluctuation Self-consistent equation for order parameter fluctuations: d(p,~ q,ω) =
Z
dp0V(p,p0)
Z
dε 4πi δfK(p0,~ q;ε,ω) Coupled equations for: d(p) = DY+ + EY , ˜ d(p) = D⇤Y + E⇤Y+
d(p) = 1 2
Z
dp0V(p,p0) ⇢ 1 2 η ¯ λ∆(p0)vext + γ(p0)+ 1 2(ω2 η02 2|∆(p0)|2)¯ λ
- d(p0)¯
λ∆2(p0) ˜ d(p0)
- ˜
d(p) = 1 2
Z
dp0V(p,p0) ⇢ +1 2 η ¯ λ∆⇤(p0)vext + γ(p0)+ 1 2(ω2 η02 2|∆(p0)|2)¯ λ
- ˜
d(p0)¯ λ∆⇤(p0)2d(p0)
- Energy integrals:
η(p) = v(p)·~ q, γ(p0) ' 2/V Tsuneto function: λ(p,ω) ⌘ |∆|2 ¯ λ = |∆(p)|2
Z ∞
|∆(p)|
dε q ε2 |∆(p)|2 tanh ✓ βε 2 ◆ 1 ε2 (ω/2)2
I S.K. Yip and JAS, J. Low Temp. Phys. 86 (1992)
Normal Modes: D± = D±D⇤ E± = i(E ±E⇤)
SLIDE 48
Eigen Mode Frequency and Lifetime
Anderson-Bogoliubov (Goldstone) Mode ω2D = 0 Anderson-Higgs Mode (ˆ Y+) ✓ ω2 4λ10∆(T)2 λ00 ◆ D+ = 0 λmn =
I dφ
2π λ(p)Im(p)cosn(2φ)
SLIDE 49
Eigen Mode Frequency and Lifetime
Anderson-Bogoliubov (Goldstone) Mode ω2D = 0 Anderson-Higgs Mode (ˆ Y+) ✓ ω2 4λ10∆(T)2 λ00 ◆ D+ = 0 Anderson-Higgs Mode (ˆ Y) ✓ ω2 4∆(T)2λ11 λ00 ◆ E+ = 0 ✓ ω2 4∆(T)2(λ10 λ11) λ00 ◆ E = 0 λmn =
I dφ
2π λ(p)Im(p)cosn(2φ)
SLIDE 50
Eigenmode Frequencies and Lifetimes
0.0 0.2 0.4 0.6 0.8 κ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
E+ E− Γ+ Γ− 2∆max 2∆min
SLIDE 51
Mode coupling to a transverse EM field
x y
A Λ
EM field parallel to the surface Transverse field in suprconductor
- q
- A
- q
- A
Polarization [100] direction – gap minimum h (ω +iΓ)2 ω2
E+
i E+ = ie c Λ100 (ω)qv2
f ∆A(q,ω)
Polarization [110] direction – gap maximum h (ω +iΓ)2 ω2
E-
i E = e c Λ110 (ω)qv2
f ∆A(q,ω)
SLIDE 52
Superconducting power absorption
Current response δ~ j(q,ω) = K(q,ω)~ A(q,ω), K(q,ω) = Kex(q,ω)+Kmode(q,ω) Joule’s law P(ω) =
Z ∞
0 dxRe
h ~ E⇤(x,ω)·~ j(x,ω) i Power absorption P
S(ω) = 2ω|B0(ω)|2
c
Z dq
2π ImK(q,ω)
- q2 + 4π
c K(q,ω)
- 2
P . J. Hirschfeld et al., PRB 40, 10 (1989)
SLIDE 53
Power absorption for isotropic κ = 0 chiral superconductor
0.0 0.5 1.0 1.5 2.0 2.5
ω[2∆]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
P[P0]
×10−2
PS Pex I P
S = P ex +P mode ,
P
0 = |B0(ω)|2vf
π I Quasiparticle absorption ω = 2∆ I Collective Mode ω = p 2∆ I Mode dispersion ω2(q) = ω2
E +
v2
f q2
2 narrow peak width
SLIDE 54
Power absorption for anisotropic κ 6= 0 chiral superconductor Polarization [100]
0.0 0.5 1.0 1.5 2.0 2.5
[2∆]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
P[P0]
×10−2
2∆min 2∆max
- A
- q
0.0 0.5 1.0 1.5 2.0 2.5 [2∆] 0.0 0.2 0.4 0.6 0.8 P[P0] ×10−2
= ME+
2∆min 2∆max
- A
- q
Polarization [110]
0.0 0.5 1.0 1.5 2.0 2.5
[2∆]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
P[P0]
×10−2
2∆min 2∆max
- q
- A
0.0 0.5 1.0 1.5 2.0 2.5 [2∆] 0.0 0.4 0.8 1.2 1.6 2.0 P[P0] ×10−3
= ME−
2∆min 2∆max
- q
- A
SLIDE 55
Impurity effect – order parameter and quasiparticle spectrum
Impurity Model – s-wave: nimp, u0 lifetime Γ = ¯ h τ , cross section ¯ s I ¯ s ! 0 weak scattering Born limit I ¯ s ! 1 strong scattering unitary limit
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T/Tc0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Γ = 0.00 Γ = 0.05 Γ = 0.10 Γ = 0.15 Γ = 0.20 Γ = 0.25
0.0 0.1 0.2 0.3 0.4 0.5 ε/2πTc0 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 Γ = 0.02 Born Γ = 0.02 Unitary Γ = 0.1 Born Γ = 0.1 Unitary clean
Isotropic gap κ = 0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 ε/2πTc0 0.0 0.6 1.2 1.8 2.4 3.0 Γ = 0.02 Born Γ = 0.02 Unitary Γ = 0.1 Born Γ = 0.1 Unitary clean
Anisotropic gap κ = 1
SLIDE 56
Impurity effect – collective mode spectrum
Γ = 0.02, ¯ s = 0
0.0 0.2 0.4 0.6 0.8 κ 0.0 0.2 0.4 0.6 E+ E− Γ+ Γ− 2∆max 2∆min
Γ = 0.1, ¯ s = 0
0.0 0.2 0.4 0.6 0.8 κ 0.0 0.2 0.4 0.6 E+ E− Γ+ Γ− 2∆max 2∆min
Γ = 0.02, ¯ s = 0
0.0 0.2 0.4 0.6 0.8 κ 0.0 0.2 0.4 0.6 E+ E− Γ+ Γ− 2∆max 2∆min
Γ = 0.1, ¯ s = 0
0.0 0.2 0.4 0.6 0.8 κ 0.0 0.2 0.4 0.6 E+ E− Γ+ Γ− 2∆max 2∆min
SLIDE 57
Impurity effect – power absorption
0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0
P[P0]
×10−1
PS Pex
0.0 0.5 1.0 1.5 2.0
ω[2∆]
Isotropic case κ = 0 Γ = 0.02, Left ¯ s = 0, Right ¯ s = 0.9 Born: absorption peak ω = p 2∆ Unitary: large absorption ω ⇠ ∆
0.0 0.5 1.0 1.5 2.0
ω[2∆]
1 2 3 4
P[P0]
×10−2 2∆min 2∆max
PS Pex
Polarization [100]
0.0 0.5 1.0 1.5 2.0
ω[2∆]
1 2 3 4
P[P0]
×10−2 2∆min 2∆max
PS Pex
Polarization [110]
SLIDE 58
Summary
Topological superfluid 3He-B Surface Majorana spectrum E(p) = ±cpk Helical spin current Mass current and temperature dependence j(T) = nps(1aT 3) Confined chiral superfluid 3He-A film Fermionic quasiparticle bands under lateral confinement Prediction of new ground state – periodic chiral domain walls Chiral superconductor with anisotropic gap Bosonic modes with opposite chirality to ground state Anisotropic transverse wave power absorption Impurity effects on mode spectrum and power absorption
SLIDE 59
Acknowledgement
Supervisor: Prof. James A. Sauls NU Condensed matter group: ... Domestic and international collaborators:
- Prof. Takeshi Mizushima, Prof. Suk Bum Chung, Prof. Erhai Zhao, Prof.
Anton Vorontsov, Prof. Bill Halperin, Prof. John Saunders, Prof. Jeevak Parpia, Prof. John Davis
SLIDE 60
Majorana and Bogoliubov eigenspinor
1
Define projection operator b P± = 1 2 h b 1± b G(p,z;ε) i b P+(): Project a spinor into particle(hole) sector
2
Act on a particular state i.e. Ψ = b P+ 1 T
3
Projected state at energy εb Ψ = u(θ,z) 2π h Ψ(+)δ(ε cpk)+Ψ()δ(ε +cpk) i
SLIDE 61
Majorana and Bogoliubov eigenspinor
4
Spinor with bound state energy εb = ±cpk Ψ(+) = Φ+ eiφΦ, Ψ() = Φ+ +eiφΦ Amplitude u(θ,z) = π2∆? cosθ 2 e2∆?z/vf
5
Sz eigenstates Φ± respectively Φ+ =
- 1
i T , Φ =
- i
1 T
6
Ψ(±): Eigenstates of spin operator ⌥~ e2 ·~ S
SLIDE 62
Majorana and Bogoliubov eigenspinor
4
Spinor with bound state energy εb = ±cpk Ψ(+) = Φ+ eiφΦ, Ψ() = Φ+ +eiφΦ Amplitude u(θ,z) = π2∆? cosθ 2 e2∆?z/vf
5
Sz eigenstates Φ± respectively Φ+ =
- 1
i T , Φ =
- i
1 T
6
Ψ(±): Eigenstates of spin operator ⌥~ e2 ·~ S Same spinor as obtained from solving Andreev Equation (K.Nagai et al 2008 , T.Mizushima 2012)
SLIDE 63
Selfconsistent spectrum
2 4 6 8
ε and ∆ are in units of Tc N in units of N0 ∆px + i∆py
2 4 6 8 −6 −5 −4 −3 −2 −1 1 2 3 4 2 4 6 8
py = 0 (normal) py = px (45o) px = 0 (grazing)
1 2
∆ = ∆
1 2
∆ = 0.5∆
6 12 1 2
Self-consistent gap
SLIDE 64
Confined chiral domain wall
∆?px + i∆kpy
−∆?px + i∆kpy
Lower energy per unit length
- Y. Tsutsumi, JLTP
, 2014
PDW domain wall
SLIDE 65
Confined chiral domain wall
x = D x = 0 ∆?px + i∆kpy
−∆?px + i∆kpy
Lower energy per unit length
- Y. Tsutsumi, JLTP
, 2014
PDW domain wall Apparent violation of current conservation
SLIDE 66
Confined chiral domain wall
x = D x = 0 ∆?px + i∆kpy
−∆?px + i∆kpy
Lower energy per unit length
- Y. Tsutsumi, JLTP
, 2014
PDW domain wall Apparent violation of current conservation
∆(r,p) = ∆+(r)y+(p)+∆(r)y(p) with y±(p) = px ±ipy
SLIDE 67
Local current density
−30 −20 −10 10 20 30
y/ξ0
5 10 15 20
x/ξ0 j(x, y)
SLIDE 68
Local current density
−0.2 0.0 0.2 0.4 0.6
jx [Nf(2πTc)m3vf]
x = 10ξ0 x = 5.5ξ0 x = 1.9ξ0 x = 0.0ξ0
−30 −20 −10 10 20 30
y/ξ0
−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6
jy
SLIDE 69
Normal state power absorption
1 2 3 4 5 6 ω[vf/Λ] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 PN[P0] ×10−2
Normal state power absorption
2 4 6 ω[vf/Λ] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 PN[P0] ×10−2 Λ/vfτ = 0.01 Λ/vfτ = 0.03 Λ/vfτ = 0.05 2 4 6 ω[vf/Λ] 0.0 0.2 0.4 0.6 0.8 1.0 ×10−1 Λ/vfτ = 0.1 Λ/vfτ = 0.5 Λ/vfτ = 0.9