Infinite randomness and the superconductor-metal transition - PiTP

Infinite randomness at the
superconductor-metal quantum
phase transition
Phys. Rev. Lett. (2008)
Adrian Del Maestro

Collaborators
Subir Sachdev
Harvard
Bernd Rosenow
MPI Stuttgart
Markus Mueller
University of Geneva
Nayana Shah
UIUC

Diminutive superconductivity
A macroscopic quantum wave function can
be drastically altered by finite size effects
R
Interesting and nontrivial
new behavior
when
R ∼ ξ
ξ
k −1
F

Disorder in quantum systems
Disorder has long
range correlations in
the imaginary time
direction
ln τ ∼ ξ ψ

What role does disorder play
near the superconductor-metal
transition in one dimension

Ultra-Narrow Wires

Fabrication of ultra-narrow wires
Novel fabrication techniques have allowed
for the study of wires with d < 10 nm
www.nanogallery.info
A. Bezryadin et al., Nature (2000)

Transport in ultra-narrow wires
Can tune a phase
transition between a metal
and superconductor via
wire diameter
thin
www.nanogallery.info
thick
A. Bezryadin, et al., PRL (2003)

Pair-breaking in nanowires
V>0
V

Pair-breaking in nanowires
V>0
V

Pair-breaking in nanowires
V>0
V 0
(
Tc
T c0
)
= ψ
( 1
2)
− ψ
( 1
2 + α
2πk B T c
)
A. Abrikosov and G. Gor’kov, JETP (1961)
T/Tc0
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
¯hα/k B T c0

Ramazashvili and Coleman; Herbut; Dalidovich and Phillips; Spivak, etc.
Dissipative Cooperon theory
Diffusive electrons with
Ohmic dissipation coming
from decay of repulsive
Cooper pairs into gapless
fermions
S α =
∫ L
0
dx
+ 1
β
∫ β
0
∑
ω n
z =2
[
dτ ˜D|∂x Ψ(x, τ)| 2 + α|Ψ(x, τ)| 2 + u |Ψ(x, τ)|4]
2
dx γ|ω n ||Ψ(x, ω n )| 2
∫ L
0

Ramazashvili and Coleman; Herbut; Dalidovich and Phillips; Spivak, etc.
Dissipative Cooperon theory
Diffusive electrons with
Ohmic dissipation coming
from decay of repulsive
Cooper pairs into gapless
fermions
S α =
∫ L
0
dx
+ 1
β
∫ β
0
∑
ω n
z =2
[
dτ ˜D|∂x Ψ(x, τ)| 2 + α|Ψ(x, τ)| 2 + u |Ψ(x, τ)|4]
2
dx γ|ω n ||Ψ(x, ω n )| 2
∫ L
0
particle-hole continua

Superconductor-metal transition
T
Phys. Rev. B, (2008)
Annals of Physics (2008)
quantum
critical
LAMH
superconductor
metal
α
c
α

Superconductor-metal transition
T
Phys. Rev. B, (2008)
Annals of Physics (2008)
quantum
critical
LAMH
T dis
superconductor
metal
α
c
α

Infinite Randomness

Spatially dependent random couplings
S =
∫
+
dx
∫
∫ dω
2π
dτ
[
D(x)|∂ x Ψ(x, τ)| 2 + α(x)|Ψ(x, τ)| 2 + u(x)
2 |Ψ(x, τ)|4 ]
γ(x)|ω||Ψ(x, ω)|2
T = 0
Hoyos, Kotabage and Vojta, PRL 2007
Real space RG predicts the
T
flow to a strong randomness
QC
fixed point for z = 2
SC
IRFP
N
α
!

Spatially dependent random couplings
S =
∫
+
dx
∫
∫ dω
2π
dτ
[
D(x)|∂ x Ψ(x, τ)| 2 + α(x)|Ψ(x, τ)| 2 + u(x)
2 |Ψ(x, τ)|4 ]
γ(x)|ω||Ψ(x, ω)|2
T = 0
Hoyos, Kotabage and Vojta, PRL 2007
Real space RG predicts the
flow to a strong randomness
fixed point for z = 2
T
Fisher’s
QC
RTFIM
SC
N
IRFP
α
!

Manifestations of strong disorder
Under renormalization, probability distributions
for observables become extremely broad
Dynamics are highly anisotropic in space and
time: activated scaling
ln ξ τ ∼ ξ ψ
Averages become dominated by rare events
(spurious disorder configurations)

Exact predictions from the RTFIM
D. Fisher, PRL (1992);
PRB (1995)
For a finite size system
ln(1/Ω) ∼ L ψ µ ∼ ln(1/Ω) φ
In the disordered phase
ξ ∼ |δ| −ν
C(x) ∼ exp [ −(x/ξ) − (27π 2 /4) 1/3 (x/ξ) 1/3]
(x/ξ) 5/6 H = − ∑ i
J i σ z i σ z i+1 − ∑ i
h i σ x i

Exact predictions from the RTFIM
D. Fisher, PRL (1992);
PRB (1995)
For a finite size system
ln(1/Ω) ∼ L ψ µ ∼ ln(1/Ω) φ
In the disordered phase
ξ ∼ |δ| −ν
C(x) ∼ exp [ −(x/ξ) − (27π 2 /4) 1/3 (x/ξ) 1/3]
(x/ξ) 5/6 H = − ∑ i
J i σi z σi+1 z − ∑ h i σi
x
i
critical exponents
ψ =1/2
ν =2
φ = (1 + √ 5)/2

Discretize to a lattice of L sites
S =
∫
dτ
{ L−1 ∑
j=1
D j |Ψ j (τ) − Ψ j+1 (τ)| 2 +
+ c l |Ψ 1 (τ)| 2 + c r |Ψ L (τ)| 2 }
+
L∑
j=1
[
α j |Ψ j (τ)| 2 + u j
2 |Ψ j(τ)| 4]
∫ dω
2π
L∑
j=1
γ j |ω||Ψ j (ω)| 2
P (D j ) P (α j )
Measure all quantities with respect to
γ 2
and
set u j =1

Take the large-N limit
Enforce a large-N constraint by solving the selfconsistent
saddle point equations numerically
S N =
∫ dω
2π
L∑
i,j=1
Ψ ∗ i (ω)[|ω|δ ij + M ij ] Ψ j (ω)
r j = α j + 〈|Ψ j (τ)| 2 〉 SN
Y. Tu and P. Weichman, PRL (1994); J. Hartman and Weichman, PRL (1995)

Solve-Join-Patch method
L = 64

Solve-Join-Patch method
L = 64
solve solve solve solve

Solve-Join-Patch method
L = 64
solve solve solve solve
join
join

Solve-Join-Patch method
L = 64
solve solve solve solve
join
join
patch
patch

Solve-Join-Patch method
L = 64
solve solve solve solve
join
join
patch
patch
solve
solve

Numerical investigation of observables
Measure observables for L = 16, 32, 64, 128 averaged
over 3000 realizations of disorder
Tune the transition by shifting the mean of the
distribution,
δ ∼ α − α c
α j
Equal time correlators are easily determined from the
self-consistency condition
C(x) =〈Ψ ∗ x(τ)Ψ 0 (τ)〉 SN

Equal time correlation functions
Fisher’s asymptotic
scaling form
C(x) ∼ exp [ −(x/ξ) − (27π 2 /4) 1/3 (x/ξ) 1/3]
(x/ξ) 5/6
10 2
10 1
10 0
10 −1
L = 64
α c = −0.93(3)
ν =1.9(2)
ln ξ
3
2
1
ξ ∼ δ −ν
C(x)
10 −2
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
α
0.00
-0.25
-0.50
-0.65
-0.75
0
−2.0 −1.5 −1.0 −0.5 0.0
ln δ
5 10 15 20 25 30
x

Equal time correlation functions
Fisher’s asymptotic
scaling form
C(x) ∼ exp [ −(x/ξ) − (27π 2 /4) 1/3 (x/ξ) 1/3]
(x/ξ) 5/6
10 2
10 1
10 0
10 −1
L = 64
α c = −0.93(3)
ν =1.9(2)
ln ξ
3
2
1
ξ ∼ δ −ν
C(x)
10 −2
10 −3
10 −4
10 −5
10 −6
10 −7
10 −8
α
0.00
-0.25
-0.50
-0.65
-0.75
0
−2.0 −1.5 −1.0 −0.5 0.0
ln δ
5 10 15 20 25 30
x
α c ≃−0.93
ν =1.9(2)

Energy gap statistics
δ ∼ α − α c
10 −1
L
128
64
32
16
P (ln Ω)
10 −2
10 −3
10 −4
δ =0.18
−25 −20 −15 −10 −5 0
ln Ω
δ =0.93
−8 −6 −4 −2 0
ln Ω
P. Young and H. Rieger, PRB (1996)

Energy gap statistics
δ ∼ α − α c
10 −1
L
128
64
32
16
P (ln Ω)
10 −2
10 −3
10 −4
δ =0.18
−25 −20 −15 −10 −5 0
ln Ω
δ =0.93
−8 −6 −4 −2 0
ln Ω
distribution broadens with increasing
system size near criticality
P. Young and H. Rieger, PRB (1996)

Extreme value distribution
The minimum
excitation energy
10 −1
L = 128
is due to a rare
event, an extremal
value
P (ln Ω)
10 −2
»
P (x) = 1 β e ( x−µ
β )−e ( x−µ ) –
β
10 −3
δ =0.18
Gumbel
−20 −15 −10 −5
ln Ω
Juhász, Lin and Iglói (2006)

Activated scaling
|ln Ω|
16
14
12
10
8
6
4
2
0
ψ =0.53(6)
ln |ln Ω|
3.0
2.5
2.0
1.5
1.0
0.5
0.0
−2.0 −1.5 −1.0 −0.5 0.0
ln δ
0.5 1.0 1.5 2.0 2.5
δ
| ln Ω| ∼ L ψ ∼ δ −νψ

Activated scaling
|ln Ω|
16
14
12
10
8
6
4
2
0
ψ =0.53(6)
ln |ln Ω|
3.0
2.5
2.0
1.5
1.0
0.5
0.0
−2.0 −1.5 −1.0 −0.5 0.0
ln δ
0.5 1.0 1.5 2.0 2.5
δ
| ln Ω| ∼ L ψ ∼ δ −νψ
ψ =0.53(6)

Dynamic susceptibilites
We have direct access to real dynamical quantities
Im χ(ω) = 1 L
∑
∣
Im 〈Ψ ∗ ∣∣iω→ω+iɛ
x(iω)Ψ 0 (iω)〉 SN
x
Im χ loc (ω) = Im 〈Ψ ∗ 0 (iω)Ψ 0(iω)〉 SN
∣
∣∣iω→ω+iɛ

Dynamic susceptibilites
We have direct access to real dynamical quantities
Im χ(ω) = 1 L
∑
∣
Im 〈Ψ ∗ ∣∣iω→ω+iɛ
x(iω)Ψ 0 (iω)〉 SN
x
Im χ loc (ω) = Im 〈Ψ ∗ 0 (iω)Ψ 0(iω)〉 SN
∣
∣∣iω→ω+iɛ
real frequency

Dynamic susceptibilites
We have direct access to real dynamical quantities
Im χ(ω) = 1 L
∑
∣
Im 〈Ψ ∗ ∣∣iω→ω+iɛ
x(iω)Ψ 0 (iω)〉 SN
x
Im χ loc (ω) = Im 〈Ψ ∗ 0 (iω)Ψ 0(iω)〉 SN
∣
∣∣iω→ω+iɛ
real frequency
The ratio of the average to local susceptibility is
related to the average cluster moment

Activated Dynamical Scaling
Can derive a
scaling form for the
R
0.14
0.12
0.10
0.08
0.06
ln ˜R
−3
dynamic confirmation
−4
of infinite randomness
−5
−6
| ln ω|
16.1
14.5
13.8
12.2
−2.0 −1.5 −1.0 −0.5
ln δ
cluster moment
R(ω) =
Imχ(ω)
Imχ loc (ω)
∼ δ νψ(1−φ) | ln ω|
0.04
0.02
0.00
φ =1.6(2)
6 8 10 12 14
| ln ω|
δ
0.43
0.28
0.18

Activated Dynamical Scaling
Can derive a
scaling form for the
0.14
−3
cluster moment
R
0.12
0.10
0.08
0.06
ln ˜R
−4
−5
−6
| ln ω|
16.1
14.5
13.8
12.2
−2.0 −1.5 −1.0 −0.5
ln δ
R(ω) =
Imχ(ω)
Imχ loc (ω)
∼ δ νψ(1−φ) | ln ω|
0.04
0.02
0.00
φ =1.6(2)
6 8 10 12 14
| ln ω|
δ
0.43
0.28
0.18
φ =1.6(2)

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!

Origin of correspondence
The physics of infinite randomness comes from
the slow dynamics and large contribution of
rare regions
imaginary time
space

Effective classical theories
RTFIM
SMT
1d
O(N)
1d
Z2
|ω| → 1/τ 2
T. Vojta and J. Schmalian (2005); T. Vojta (2006)

Effective classical theories
RTFIM
SMT
1d
O(N)
1d
Z2
|ω| → 1/τ 2
rare regions
are marginal!
T. Vojta and J. Schmalian (2005); T. Vojta (2006)

Conclusions
First dynamical confirmation of activated scaling from
numerical simulations
SMT has an infinite randomness fixed point in the
RTFIM universality class
Scratching the surface of transport calculations near a
strong disorder fixed point