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Theory of the pairbreaking superconductor-metal transition
in nanowires Talk online: sachdev.physics.harvard.edu
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Theory of the pairbreaking superconductor-metal transition
in nanowiresAdrian Del Maestro, Bernd Rosenow, Nayana Shah, and Subir Sachdev, Physical Review B 77, 180501 (2008).
Adrian Del Maestro, Bernd Rosenow, Markus Mueller, and Subir Sachdev, Physical Review Letters 101, 035701 (2008).
Adrian Del Maestro, Bernd Rosenow, and Subir Sachdev, Annals of Physics 324, 523 (2009).
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Adrian Del MaestroUniversity of British Columbia
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T
Superconductor
α
Metal
Tc
αc
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Pair-breaking in nanowires
Evidence for magnetic moments on the wire’s surface
A. Rogachev et al., PRL (2006)
B. Spivak et. al, PRB (2001)R! ξ
V > 0
V < 0Inhomogeneity in the pairing interaction
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T
Superconductor
α
Metal
Tc
αc
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Field theory for pair-breakingsuperconductor-metal transition
Outline
Dirty wires and the strong disorder renormalization group
Field theory for pair-breaking superconductor-metal transition
Evidence for an infinite randomness fixed point
Finite temperature transport (weak disorder)
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Computation of fluctuation conductivity in metal at low temperatures
T
Superconductor
α
Metal
Tc
αc
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Computation of fluctuation conductivity in metal at low temperatures
T
Superconductor
α
Metal
Tc
αc
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Computation of fluctuation conductivity in metal at low temperatures
T
Superconductor
α
Metal
Tc
αc
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Computation of fluctuation conductivity in metal at low temperatures
T
Superconductor
α
Metal
Tc
αc
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Theory for quantum-critical region, and beyond
T
Superconductor
α
Metal
Tc
αc
Quantum critical
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Theory for quantum-critical region, and beyond
T
Superconductor
α
Metal
Tc
αc
Quantum critical
In one dimension, theory reduces to the Langer-Ambegaokar-McCumber-Halperin theory (Model A dynamics), near mean-field Tc
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Role of charge conservation in quantum critical theory
Dynamics of quantum theory (and model A) does not conserve total charge.
Analogous to the Fermi-liquid/spin-density-wave transition (Hertz theory), where dynamics of critical
theory does not conserve total spin.
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Role of charge conservation in quantum critical theory
Dynamics of quantum theory (and model A) does not conserve total charge.
Analogous to the Fermi-liquid/spin-density-wave transition (Hertz theory), where dynamics of critical
theory does not conserve total spin.
Cooper pairs (SDW) fluctuations decay into fermionic excitations at a finite rate, before any appreciable phase precession due to changes in
chemical potential (magnetic field).
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α
T
αc
MetalAL, MT, DoS fluctuations
Phase fluctuations realize the Mooij-Schön mode
Quantum critical
LAMH
Fluctuating superconductor
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Field theory for pair-breaking superconductor-metal transition
Outline
Dirty wires and the strong disorder renormalization group
Evidence for an infinite randomness fixed point
Finite temperature transport (weak disorder)Finite temperature transport (weak disorder)
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T
Superconductor
α
Metal
Tc
αc
Quantum critical
Theory for quantum-critical region, and beyond in d=1
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T
Superconductor
α
Metal
Tc
αc
Quantum critical
Theory for quantum-critical region, and beyond in d=1
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Accuracy of large-N expansion
In the metallic phase, exactagreement is found between the large-N and microscopic calculations for the electrical and thermal conductivity
δ
M
QC
T
SC
Lopatin, Shah and Vinokur (2005,2007)
σ =π
3e2
h
√DT 2
R5/2T < δ2
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0 1 2 3 4
Temperature0.00
0.04
0.08
0.12
Conductivity
δ = 0.10δ = 0.20δ = 0.50
T 2T−1/2
σ/√
DNon-monotonic temp. dependance
δ > 0
A crossover is observed as the temperature is reduced for
Bollinger, Rogachev and Bezryadin (2005)
T
SC M
QC
δ
(σ−
σm
etal)/√
D
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The Wiedemann-Franz law
The large-N results can be used to evaluate both the electrical and thermal conductivity
QC M
σκ/T T 2
T 2
T−1/2
T−1/2
T
SC M
QC
δ
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0.00 0.05 0.10 0.15 0.20
Λ−1T 1/2
−0.15
−0.10
−0.05
0.00
0.05
NδW
δW | Λ√T→∞ =
0.038
N
δW =(
kB
e
)2 (0.2820 +
0.0376N
)
Corrections to the WF law
All couplings between bosons and fermions scale to universal values and we are left with a universal correction to the WF law
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Field theory for pair-breaking superconductor-metal transition
Outline
Dirty wires and the strong disorder renormalization group
Evidence for an infinite randomness fixed point
Finite temperature transport (weak disorder)
Dirty wires and the strong disorderrenormalization group
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S =∫
dx
∫dτ
[D(x)|∂xΨ(x, τ)|2 + α(x)|Ψ(x, τ)|2 +
u(x)2
|Ψ(x, τ)|4]
+∫
dω
2πγ(x)|ω||Ψ(x,ω)|2
Spatially dependent random couplings
Hoyos, Kotabage and Vojta, PRL 2007
Real space RG predicts the flow to a strong randomness fixed point for z = 2
QC
SC N
T
!IRFP αRTFIM
Fisher’s
T = 0
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strong field
h1 h2 h3 h4 h5
J4J1 J = J2J3/h3
strong bond
h1 h2 h3 h4 h5
J4
J1 J2 J3 J4
h = h2h3/J2
J1 J3
J1 J2 J3 J4
H = −∑
i
Jiσzi σz
i+1 −∑
i
hi σxi
Strong disorder renormalization group
D. Fisher, PRL, (1992); PRB (1995)
First developed by Ma, Dasgupta and Hu, applied to the RTFIM by D.S. Fisher
Clusters are createdor decimated as the energy is reduced
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Manifestations of strong disorder
Under renormalization, probability distributions for observables become extremely broad
Dynamics are highly anisotropic in space and time: activated scaling
Averages become dominated by rare events(spurious disorder configurations)
ln ξτ ∼ ξψ
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ln(1/Ω) ∼ Lψ µ ∼ ln(1/Ω)φ
ξ ∼ |δ|−ν
Exact predictions from the RTFIM
For a finite size system
D. Fisher, PRL (1992); PRB (1995)
critical exponents
ψ = 1/2
ν = 2
φ = (1 +√
5)/2
In the disordered phase
C(x) ∼exp
[−(x/ξ)− (27π2/4)1/3(x/ξ)1/3
]
(x/ξ)5/6
H = −∑
i
Jiσzi σz
i+1 −∑
i
hi σxi
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S =∫
dτ
L−1∑
j=1
Dj |Ψj(τ)−Ψj+1(τ)|2 +L∑
j=1
[αj |Ψj(τ)|2 +
uj
2|Ψj(τ)|4
]
+ cl|Ψ1(τ)|2 + cr|ΨL(τ)|2
+∫
dω
2π
L∑
j=1
γj |ω||Ψj(ω)|2
P (Dj) P (αj)
Discretize to a chain of L sites
Measure all quantities with respect to andset uj = 1
γ2
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SN =∫
dω
2π
L∑
i,j=1
Ψ∗i (ω) [ |ω|δij + Mij ]Ψj(ω)
Take the large-N limit
Enforce a large-N constraint by solving the self-consistent saddle point equations numerically
Y. Tu and P. Weichman, PRL (1994); J. Hartman and Weichman, PRL (1995)
rj = αj + 〈|Ψj(τ)|2〉SN
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Numerical investigation of observables
Measure observables for L = 16, 32, 64, 128 averaged over 3000 realizations of disorder
δ ∼ α− αc
Tune the transition by shifting the mean of the distribution,
αj
Equal time correlators are easily determined from the self-consistency condition
C(x) = 〈Ψ∗x(τ)Ψ0(τ)〉SN
Numerical solution by solve-join-patch method
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C(x) ∼exp
[−(x/ξ)− (27π2/4)1/3(x/ξ)1/3
]
(x/ξ)5/6
Equal time correlation functions
Fisher’s asymptotic scaling form
5 10 15 20 25 30x
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
102
C(x
)
αc = −0.93(3)ν = 1.9(2)
α0.00-0.25-0.50-0.65-0.75
−2.0 −1.5 −1.0 −0.5 0.0
ln δ
0
1
2
3
lnξ
αc ! −0.93ν = 1.9(2)
ξ ∼ δ−νL = 64
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P (x) =1β
e
»( x−µ
β )−e(x−µ
β )–
Energy gap statistics
Juhász, Lin and Iglói (2006)
−20 −15 −10 −5
ln Ω10−3
10−2
10−1
P(l
nΩ
)
L = 128
δ = 0.18Gumbel
The minimum excitation energy is due to a rare event, an extremal value
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ψ = 0.53(6)
Activated scaling
| lnΩ| ∼ Lψ ∼ δ−νψ
0.5 1.0 1.5 2.0 2.5
δ
0
2
4
6
8
10
12
14
16
|lnΩ|
ψ = 0.53(6)
−2.0 −1.5 −1.0 −0.5 0.0
ln δ
0.0
0.5
1.0
1.5
2.0
2.5
3.0
ln|ln
Ω|
Divergence cannot be fit
with any power law!
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Im χ(ω) =1L
∑
x
Im 〈Ψ∗x(iω)Ψ0(iω)〉SN
∣∣∣iω→ω+iε
Im χloc(ω) = Im 〈Ψ∗0(iω)Ψ0(iω)〉SN
∣∣∣iω→ω+iε
Dynamic susceptibilites
We have direct access to real dynamical quantities
real frequency
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Im χloc =1L
L∑
i=1
ω
ω2 + λ2i
Local order parameter suceptibility
10−510−410−310−210−1 100 101 102 103 104 105
ω10−3
10−2
10−1
100
101
102
103
104
105
χlo
cδ
0.93
0.68
0.43
0.28
0.18
0 2 4 6 8 10
δνψ| ln ω|
100
101
102
103
ωλ
1δλ
3χ
loc
δ ∼ 0
δ ∼ 1
∼ ω−1
∼ ω ∼ ω−1
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Im χ =1L2
L∑
i,j,k=1
ωVikVjk
ω2 + λ2j
Average order parameter suceptibility
10−510−410−310−210−1 100 101 102 103 104 105
ω10−5
10−4
10−3
10−2
10−1
100
101
102
103
104
χδ
0.93
0.68
0.43
0.28
0.18
0 2 4 6 8 10
δνψ| ln ω|
0.0
0.5
1.0
1.5
2.0
2.5
ωλ
1δλ
2χ
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Susceptibility scaling forms
The scaling forms for the disorder averaged order parameter susceptibilities follow from an analysis of a single cluster
Im χ ∼ δ1/ψ−φνψ(1+δ/ψ)
ω1−δ/ψΦ
(δνψ| lnω|
)
Im χloc ∼δ1/ψ−φνδ
ω1−δ/ψΦloc
(δνψ| lnω|
)
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Susceptibility scaling collapse
10−510−410−310−210−1 100 101 102 103 104 105
ω10−5
10−4
10−3
10−2
10−1
100
101
102
103
104
χ
δ0.93
0.68
0.43
0.28
0.18
0 2 4 6 8 10
δνψ| ln ω|
0.0
0.5
1.0
1.5
2.0
2.5
ωλ
1δλ
2χ
10−510−410−310−210−1 100 101 102 103 104 105
ω10−3
10−2
10−1
100
101
102
103
104
105
χlo
c
δ0.93
0.68
0.43
0.28
0.18
0 2 4 6 8 10
δνψ| ln ω|
100
101
102
103
ωλ
1δλ
3χ
loc
average susceptibility
local susceptibility
Observe data collapse consistent with expectations from scaling
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Susceptibility ratioToo much parameter freedom in data collapse to confidently extract the tunneling exponent φ
The ratio of the average to local susceptibility is related to the average cluster moment
R(ω) =Imχ(ω)
Imχloc(ω)∼ δνψ(1−φ)| lnω|
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φ = 1.6(2)
R(ω) =Imχ(ω)
Imχloc(ω)∼ δνψ(1−φ)| lnω|
Activated dynamical scaling
6 8 10 12 14
| ln ω|
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
R
φ = 1.6(2)
δ0.430.280.18
−2.0 −1.5 −1.0 −0.5
ln δ
−6
−5
−4
−3
lnR
| lnω|16.114.513.812.2
dynamic confirmation of infinite randomness
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Putting it all together
ν ψ φ
RTFIM 2 1/2 (1+√5)/2
SMT 1.9(2) 0.53(6) 1.6(2)
Numerical confirmation of the strong disorder RG calculations of Hoyos,
Kotabage and Vojta
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ConclusionsField theory for quantum critical transport near the superconductor-metal transition. Crosses over to phase fluctuation theory (in the superconductor) and conventional pairing fluctuation theory (in the metal) at low temperatures.
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α
T
αc
MetalAL, MT, DoS fluctuations
Phase fluctuations realize the Mooij-Schön mode
Quantum critical
LAMH
Fluctuating superconductor
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ConclusionsField theory for quantum critical transport near the superconductor-metal transition. Crosses over to phase fluctuation theory (in the superconductor) and conventional pairing fluctuation theory (in the metal) at low temperatures.
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ConclusionsField theory for quantum critical transport near the superconductor-metal transition. Crosses over to phase fluctuation theory (in the superconductor) and conventional pairing fluctuation theory (in the metal) at low temperatures.
New Wiedemann-Franz ratio in the quantum critical region.
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Conclusions
First dynamical confirmation of activated scaling from numerical simulations in disordered systems.
Field theory for quantum critical transport near the superconductor-metal transition. Crosses over to phase fluctuation theory (in the superconductor) and conventional pairing fluctuation theory (in the metal) at low temperatures.
New Wiedemann-Franz ratio in the quantum critical region.
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Conclusions
SMT has an infinite randomness fixed point in the RTFIM universality class
First dynamical confirmation of activated scaling from numerical simulations in disordered systems.
Field theory for quantum critical transport near the superconductor-metal transition. Crosses over to phase fluctuation theory (in the superconductor) and conventional pairing fluctuation theory (in the metal) at low temperatures.
New Wiedemann-Franz ratio in the quantum critical region.
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Conclusions
SMT has an infinite randomness fixed point in the RTFIM universality class
First dynamical confirmation of activated scaling from numerical simulations in disordered systems.
Scratching the surface of transport calculations near a strong disorder fixed point
Field theory for quantum critical transport near the superconductor-metal transition. Crosses over to phase fluctuation theory (in the superconductor) and conventional pairing fluctuation theory (in the metal) at low temperatures.
New Wiedemann-Franz ratio in the quantum critical region.