Superfluid 3He and Unconventional Superconductivity
Introduction The Fermi-liquid normal state of 3He The pairing interaction in 3HeSuperfluid phases of 3HeUnconventional superconductors
I. Eremin, Max-Planck Institut für Physik komplexer Systeme, Dresden, Germany
Institut für Mathematische/Theoretische Physik, TU-Braunschweig, Germany
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Introduction: Two superfluids Introduction: Two superfluids ⇒⇒ 44He and He and 33HeHe
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Why helium is the only example of quantum fluid?
2/122⎟⎟⎠
⎞⎜⎜⎝
⎛=
TmkBdB
hπλ and typical interatomic distance d
0.270.4
0.30.07
d(nm)(nm)dBλ
Ne
He
He is essentially a quantum liquid !!
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Introduction: Two superfluids Introduction: Two superfluids ⇒⇒ 44He and He and 33HeHe
4He 3He
S=0 particles - bosons
Exp: Kapitza, 1938Nobel Prize 1978
Exp: Osheroff, Richardson, Lee, 1972Nobel Prize 1996
S=1/2 particles - fermions
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Introduction: Two superfluids Introduction: Two superfluids ⇒⇒ 44He and He and 33HeHe
Normal state
( )∑∑≠=
−+∇−=ji
jiNi
i Vm
H rr21
2ˆ
,1
22h
( )jiV rr −
van der Waals like forces
( ) 6RCaeRV bR −= −
( ) ( )NNnnNNn EH σσσσ ,,...,,,,...,,ˆ1111 rrrr Ψ=Ψ
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Introduction: Normal state of Introduction: Normal state of 33HeHe
For 3He the many-particle wave function has to be odd under permutation of any pair spatial and spin coordinates
1) Let us put V=0 and T=0
( )
( ) ( ) ( )( ) ( ) ( )
( ) ( )NNNN
NN
NN
NNn Nσϕσϕ
σϕσϕσϕσϕσϕσϕ
σσ
rr
rrrrrr
rr
.........
...
...
!1,,...,,
11
2222112
1221111
11 =Ψ
kF ( ) 3/123 nkF π=
KF 9.4=ε FcT ε0001.0~
degenerate Fermi gas
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FermiFermi--liquid concept of the normal state ofliquid concept of the normal state of 33HeHe
2) V(r) is strong and cannot be neglected
( )∑∑≠=
−+∇−=ji
jiNi
i Vm
H rr22
ˆ,1
22 λλ h
Turn on the interaction adiabatically
10 =⇒= λλ
)1()()0(nnn Ψ⇒Ψ⇒Ψ λ
FermiFermi--liquid concept of the normal state ofliquid concept of the normal state of 33HeHe
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a) momentum distribution
)()( λσσ
λσ ΨΨ= +
kkk ccn⎩⎨⎧
><
=F
F
kkkk
n01
σk
0=λ
1=λ
1) kF is defined at the discontinuity
2) For spherically symmetric systems position of kF remains unchanged
Luttinger theorem !
b) single-particle excited states)0(
0)0( Ψ=Ψ +
σσ kk c( )
e
F
mkk
2
222)0( −=h
kε
( )*
222)1(
2mkk F−
≈= hλεkKmm Fe 1,3* ≈= ε
FermiFermi--liquid concept of the normal state ofliquid concept of the normal state of 33HeHe
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c) two-particle excitations
)0(0''
)0('', Ψ=Ψ ++
σσσσ kkkk cc
)0('
)0(kk εε +=energy
due to interaction
( )'',' σσεε kkkk f++
( )'', σσ kkf Effective interaction between quasiparticles
( )'',21
'',''0 σσδδδε
σσσσσ
σ
kkkk
kkkk
k fnnnEE ∑∑ ++=
Total energy (Landau)
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FermiFermi--liquid concept of the normal state ofliquid concept of the normal state of 33HeHe
Distinction of quasiparticles due to the spin orientation:
( )αβαβαβ δδδ σskk .ˆ21 += nn
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
1001
,0
0,
0110
zyx ii
σσσ
( ) ( )( )∑∑ ⋅+++='',
'21'0 ˆˆ',',21
σσ
δδδεkk
kkkkkk
k sskkkk ffnnnEE
( ) ( ) ( )
( ) ( ) ( )∑
∑=
=
lllF
lllF
PZfN
PFfN
ϑε
ϑε
cos',
cos',
2
1
kk
kkexpansion
FermiFermi--liquid concept of the normal state ofliquid concept of the normal state of 33HeHe
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The success of the Fermi-liquid theory in 3He is that one needs to know only few parameters: m*, F0 , F1 , Z0 , Z1
emm 3* = F0 , F1 large and positive Z0 = -3, Z1 is small
The spin susceptibility
( )4/1 0
0
Z+=
χχ
the system is close to the ferromagnetism Leggett (1975)
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The pairing interaction in liquid The pairing interaction in liquid 33He He
Candidate 1: van der Waals potential
( ) ( ) ( ) ( ) ( )∑∫ +−− ==l
llli PVrdVeV θcos', 2
123' rkk rkk
gives too small Tc !
Candidate 2: Spin fluctuations
( ) ( ) '04
10 ˆˆ
11', kk sskk ⋅
+≈
ZZ
NV
Fε
30 −=Z 41ˆˆ ' +=⋅ kk ss
43ˆˆ ' −=⋅ kk ss
attractive for ferromagnetic fluctuations
repulsive for antiferrmomagnetic fluctuations
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Superfluid phases Superfluid phases 33He He
BCS-like model
( ) ( )∑∑ −+−
++ +−=',
'',
',ˆkk
kkkkk
kkk kk δγβααβγδσ
σσμε ccccVccH
Mean-field approximation
( )( ) ...',ˆ',
''''int ++≈∑ −+−
+−
+−
+
kkkkkkkkkkkk δγβαδγβααβγδ ccccccccVH
Analog of BCS gap:
( ) ( ) ( )( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛ΔΔΔΔ
⇒⎟⎟⎠
⎞⎜⎜⎝
⎛=
↓↓↓↑
↑↓↑↑
↓↓−↑↓−
↓↑−↑↑−
kkkk
kkkkk
kkkk
cccccccc
F
consists of triplet and singlet parts
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Superfluid phases Superfluid phases 33He He
to devide the triplet and singlet component αββα kkkk −− −= cccc
( ) ( )( ) ( ) ( )( ) yIi σσkd
kkkk
k ⋅+Δ=⎟⎟⎠
⎞⎜⎜⎝
⎛ΔΔΔΔ
↓↓↓↑
↑↓↑↑
( ) ( )( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛++Δ−+Δ+−
=⎟⎟⎠
⎞⎜⎜⎝
⎛ΔΔΔΔ
↓↓↓↑
↑↓↑↑
kkkkkkkk
kkkk
yxz
zyx
iddddidd
scalar (singlet) component has even parity ( ) ( )kk −Δ=Δ
Vector (triplet) component has odd parity ( ) ( )kk −−= dd
Gap equation can have solutions either even or odd
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Superfluid phases Superfluid phases 33He He
Strong experimental evidence that the triplet pairing occur in 3He !
restrict to the small k-values
( ) ( )∑= kkk ϕθηνν ,lmlmYd
for l=1 – p-wave pairing
zyx ,,=ν
( ) ( )∑= kkk ϕθηνν ,ii pd zyx ppp ,,
In total d(k) can depend on nine parameters
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
zzzyzx
yzyyyx
xzxyxx
i
ηηηηηηηηη
ην
ν refers to the d-orientation (spin projection)
i refers to the spatial coordinates
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Superfluid phases Superfluid phases 33He He
Two important phases of 3He
1) Anderson-Brinkmann-Morel (ABM) state
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
00000001 i
iνη
( ) )0,0,( yx ippd +=k2) Balian-Werthamer (BW) state
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
100010001
iνη ( ) ),,( yyx pppd =k
Superfluid phases Superfluid phases 33He He
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( ) )0,0,( yx ippd +=k ( ) ),,( yyx pppd =k
1) Anderson-Brinkmann-Morel (ABM) state
2) Balian-Werthamer (BW) state
constant gap on the whole Fermi surfacegap vanishes at k = (0, 0, ± kF )
Superconducting gap can have zeros!
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Superfluid phases Superfluid phases 33He He
Excitation energy:
( ) ( ) ( ) ( ) ( )kdkdkdk k*22 ×±+−= μεE
in ABM and BW states ( ) ( ) 0* =× kdkd
( ) ( ) ( ) 22 kdk k +−= μεE Similar to the usual BCS state
Unconventional Superconductors Unconventional Superconductors
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In 3He l=1 state means a Cooper-pair wave function which is zero if r1 = r2
kk Δ=Δ R̂Conventional superconductor
Unconventional superconductor kk Δ≠Δ R̂
For at least one symmetry operation R̂
Inversion symmetry:
( ) ( )kk −= εε
( ) ( )kk −−= dd
All triplet superconductors are unconventional
System with strong Coulomb repulsion!
Unconventional Superconductors Unconventional Superconductors
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Unconventional Superconductors: Group Theory Application Unconventional Superconductors: Group Theory Application
• The microscopic mechanism in most unconventional superconductors is not yet firmly known
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Unconventional Superconductors: Group Theory Application Unconventional Superconductors: Group Theory Application
• Classification according to the group theory of the gap function
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Unconventional Superconductors: Group Theory Application Unconventional Superconductors: Group Theory Application
( ) ( )kkm
m
mfΓ
ΓΓ∑=Δ η
( ) ( )kk ννν ηm
m
mfd Γ
ΓΓ∑=
( )km
fΓ basis functions of the irreducible representation
(i) The energy eigenstate can be labelled by the irreducible representation of the corresponding point group symmetry G of the crystal
(ii) The degeneracy of the each energy state is determined by the dimension of the irreducible representation Γm
HighHigh--TTcc cuprates cuprates ⇒⇒ Group Theory ApplicationGroup Theory Application
• Crystal structure of the copper oxides: [Bednorz and
Mueller 1986]4
232 CuOSrLa ++− xx
1.5Sr2RuO4
39La1.85Sr0.15CuO4
92YBa2Cu3O7
138HgBa2Ca2Cu3O8+δ
Tc (K)Material
Singlet superconductor S=0, the point group symmetry is D4h
• Layered structure: importance of the reduced dimensionality
• Tc depends on the number of the CuO2 layers per unit cell
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HighHigh--TTcc cuprates cuprates ⇒⇒ Group Theory ApplicationGroup Theory Application
B2g1
B1g1
A2g1
A1g1
d mΓ ( )km
fΓ
yx kk coscos + s-wave
dx2-y2-wave
dxy-wave
( )yxyx kkkk coscossinsin +
yx kk coscos −
yx kk sinsin
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HighHigh--TTcc cuprates cuprates ⇒⇒ Group Theory ApplicationGroup Theory Application
for d-wave components superconducting gap changes sign in the first BZ
superconducting gap has points of nodes
+++
+
−−
−
−
Which of this states is realized?
( )yx kk coscos2
+Δ ( )yx kk coscos
2−
Δ ( )yx kk sinsinΔ
HighHigh--TTcc cuprates cuprates ⇒⇒ Symmetry of the order parameterSymmetry of the order parameter
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1) Linear density of states below Δmax
HighHigh--TTcc cuprates cuprates ⇒⇒ Symmetry of the order parameterSymmetry of the order parameter
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dx2-y2-wave state is experimentally observed
Alternative mechanisms of the CooperAlternative mechanisms of the Cooper--pairingpairing
1) Pairing from purely repulsive interaction (Kohn, Luttinger, 1965)
( )rkrrV F2cos)( 3−∝Friedel oscillations
pairing in high-angular momentum l > 0
( ){ }42exp~/ lTT Fc − too low !!
2) Pairing due to spin fluctuations (Berk and Schrieffer)
( )tSItHB
,),( rrr
h
r
μ−=
Induced spin polarization
( ) ( ) ( )','','',' 3 tHttrdtdtS B rrrr ∫ −−=rr
χμ
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Alternative mechanisms of the CooperAlternative mechanisms of the Cooper--pairingpairing
Spin density-spin density interaction
( ) ( ){ } ( ) ( )',',','',''2
332
2
tStSttttrrddIHsf rrrrrrrr
h ∫ −−−−−−= χχ
Effective pairing interaction
( ) ( )kkqq −=== 'Re4
3 20 χIV S
( ) ( )kkqq −=−== 'Re4
21 χIV S
Theory I (AFI):(R.B. Laughlin, S. Sachdev, P. Lee, P.W. Anderson, T.M. Rice ... )
Theory II (FLmetallic):(J.-R. Schrieffer, D. Scalapino, A. Chubukov, D. Pines...)
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HighHigh--TTcc cuprates cuprates ⇒⇒ possible mepossible mechanism of the Cooperchanism of the Cooper--pairingpairing
)( 42x
3x2 CuOSrLa ++
−
Possible mechanismPossible mechanism of of superconductivity for superconductivity for highhigh--TTcc cuprates cuprates ⇒⇒ spin spin fluctuation mediated fluctuation mediated CooperCooper--pairingpairing
( )sf
Qeffeff
iUUV
ωωξ
χω
−−+=)(
2AF
21,
Qqq
• d-wave symmetry and high-Tc are possible !
∑ )(Δ−
−=)(Δ'
')'(2
)'(
k
kk
kkk
EVeff
( )yx kk coscos2
0 −Δ
=)(Δ k
•Repulsive nature of the interaction
• (Monthoux, Scalapino, Millis, Monien, Pines, PRB (1989, 1991, 1994))
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Triplet superconductivityTriplet superconductivity in Srin Sr22RuORuO44
•• SrSr22RuORuO44 isostructural to La2CuO4 ⇒ basic element RuO2-planes• superconductor at Tc= 1.5K (Y. Maeno et al., Nature (1995))
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Pairing due to spin fluctuations?
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Spin singlet versus spin tripletSpin singlet versus spin triplet pairingpairing
( ) ( )kkqq −=== 'Re4
3 20 χIV S
( ) ( )kkqq −=−== 'Re4
21 χIV S
Condensation energy ( )FS
cond NE 2
21
kk Δ−= ε
( )yx kk coscos2
0 −Δ ( )yxz kkd sinsin ±
( )kεN
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Conclusions and OutlookConclusions and Outlook
(i) Unconventional superconductors give rise to a new phenomena
(ii) The theory of unconventional superconductivity is not finished
(iii) New materials are discovered in accelerating pace
HighHigh--TTcc cuprates cuprates ⇒⇒ feedback on the spin susceptibilityfeedback on the spin susceptibility
at T=0 )(1),(Im 0 QkkQkk
QkkQkkQ ++
++ −−⎟⎟⎠
⎞⎜⎜⎝
⎛ ΔΔ+−∝Ω ∑ EE
EEkωδ
εεχ
for d-wave symmetry
( ) kQk Δ−=+−+Δ=Δ + )cos()cos(0 ππ yx kk
),(Im 0 ΩQχ shows a jump at |Δk|+ |Δk+Q| = 70meV
• effect of the coherence factors
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• via Kramers-Kronig relation real part shows a log singularity
Qq =
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HighHigh--TTcc cuprates cuprates ⇒⇒ feedback on the spin susceptibilityfeedback on the spin susceptibility
( ) ( )202
0
0
),(Im),(Re1),(Im),(Im
Ω+Ω−Ω
=Ωqq
qqχχ
χχUU 0
1
H. Fong et al., PRB 61, 14773 (2000)
So-called resonance peak forms!