theory of the pairbreaking superconductor-metal transition

Theory of the pairbreaking superconductor-metal transition

in nanowires Talk online: sachdev.physics.harvard.edu

http://sachdev.physics.harvard.edu

http://sachdev.physics.harvard.edu

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).

Adrian Del MaestroUniversity of British Columbia

T

Superconductor

α

Metal

Tc

αc

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

http://www.nanogallery.info


T

Superconductor

α

Metal

Tc

αc

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)

Computation of fluctuation conductivity in metal at low temperatures

T

Superconductor

α

Metal

Tc

αc

Theory for quantum-critical region, and beyond

T

Superconductor

α

Metal

Tc

αc

Quantum critical

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

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.

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).

α

T

αc

MetalAL, MT, DoS fluctuations

Phase fluctuations realize the Mooij-Schön mode

Quantum critical

LAMH

Fluctuating superconductor


Outline



Finite temperature transport (weak disorder)Finite temperature transport (weak disorder)

T

Superconductor

α

Metal

Tc

αc

Quantum critical

Theory for quantum-critical region, and beyond in d=1

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

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

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

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


Outline



Finite temperature transport (weak disorder)

Dirty wires and the strong disorderrenormalization group

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

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



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

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

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

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

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

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

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

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!

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

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

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

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|

)

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

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|

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

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

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.

α

T

αc

MetalAL, MT, DoS fluctuations

Phase fluctuations realize the Mooij-Schön mode

Quantum critical

LAMH

Fluctuating superconductor


New Wiedemann-Franz ratio in the quantum critical region.

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.


Conclusions

SMT has an infinite randomness fixed point in the RTFIM universality class




Conclusions

SMT has an infinite randomness fixed point in the RTFIM universality class


Scratching the surface of transport calculations near a strong disorder fixed point



theory of the pairbreaking superconductor-metal transition

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