Vortex pinning by a columnar defect in planar superconductors with point disorder

Anatoli Polkovnikov

Yariv Kafri, David Nelson

Department of Physics, Harvard University.

Plan of the talk

1. Vortex physics in 1+1 dimension. Mapping to a Luttinger liquid.

2. Effects of point disorder. Vortex glass phase. Response to a columnar pin.

3. Unzipping of a single vortex from a columnar pin and a twin plane with point disorder.

4. Unzipping a single vortex from a two-dimensional Luttinger liquid. Revealing the Luttinger liquid parameter.

A single vortex line in a planar superconductor

x

Free energy 2

1

0

( ) ( ( ), )2

Ldx

F x d V xd

L

Partition function:[ ( )]

( ) expF x

Z DxT

Identify with the imaginary time of a quantum particle, ., L T

Many vortex lines in a planar superconductor

x

2 2

44 110

0

( , ) 2 2

Lc cu u

F u x dx dx

L

u is the coarse-grained phonon displacement field

( ) ( ( ))j jx a j u

a

2 2

44 110

0

( , )2 2

Lc cu u

F u x dxdx

1/ 4 1/ 4

11 44 44 11, x x c c c c

2 20 2 x

Fdxd u u

T g

11 44

Tg

c c

Luttinger liquid parameter

noninteracting regime

1 dilute limit of interacting vortices

0 dense array of vortex lines

g

2 2

1( , ) (0,0)

gn x n

x

Point disorder:

1

1 1 2 2 0 1 2 1 2

2 V( , ) cos 2 ( , ) ( , )

V( , )V( , ) ( ) ( )

Fdxd x u x x

T

x x x x

2

01 1 2 2 , 1 2 1 22

h ( , ) ( , ) h ( , ) ( , )

( , ) ( , ) ( ) ( )

x x

Fdxd x u x x u x

T

h x h x x xg

Random phase,[0,2]

Flow equations near g=1

2

0

2 (1 )

dg

dld

gdld

dl

J. Cardy and S. Ostlund, 1982

g1

/ 22 2

1,

1( , ) (0,0) ~ ,

g

n x nx

002 ( 1) ln 1

1g g

g

Vortex liquid phase

2 2 2 2

1,

( , ) (0,0) ~ exp (1 ) ln .

g

n x n g x

vortex glass phase J. Cardy and S. Ostlund, 1982, M.P.A. Fisher 1989.

g

Correlation functions

1

Add a columnar pin

Contribution to free energy

0 cos 2 (0, )pinF V d u Kane-Fisher problem with no disorder:

1dV

g Vdl

1g High-temperature weakly interacting (liquid) phase

Both columnar defect and point disorder are irrelevant. Thermal fluctuations dominate pinning and disorder.

1g Low-temperature strongly interacting (glassy) phase

Columnar pin and point disorder become relevant.

g

g=1

V

gg=1

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20g=1.1

Dis

orde

r (

)

Pinning Potential (V)

()

0.0 0.5 1.0 1.5 2.00.00

0.05

0.10

0.15

0.20g=0.9

Dis

orde

r (

)

Pinning Potential (V)

()

Flow diagram.

Columnar pin is always irrelevant !!!

Friedel oscillations around a columnar pin (linear response in V)

01

1,

cos 2( ) ~ ,

g

n xn x

x

002 ( 1) ln 1

1g g

g

2

0

(1 ) ln

1,

cos 2( ) ~ .

g x

g

n xn x

x

Slowest asymptotic decay at the vortex glass transition (g=1).

Free fermion limit, g=1

Partition function:

0Tr exp ( , )

L

fZ T d dx H x

The ground state of the N-particle system is the Slatter determinant of N-highest eigenstates of the evolution operator:

0exp ( , )

L

fS T d dx H x

Find eigenstates numerically for a given realization of disorder by discretizing space and time.

2

1

( ) ( )N

jj

n x x

0 20 40 60 80 100 120 140 160 180 200

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20Single realization of disorder

Vor

tex

line

dens

ity (

n)

Position (x) at some given fixed time

201 sites, filling factor 0.1

0 20 40 60 80 100 120 140 160 180 200

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Ensemble average over 65546 disorder realizations

Vor

tex

line

dens

ity (

n)

Position (x)

Free fermion limit, g=1

Extract exponent (average over 65536 realizations of point disorder):

0.0 5.0x10-4 1.0x10-3 1.5x10-3 2.0x10-3

1.00

1.05

1.10

1.15

Free fermions with point disorder

Exp

onen

t ()

Disorder (0)

00/ 2 1

cos 2( ) , 2

n xn x

x

RG result:

Response to a weak transverse field.

Np is the number of vortices prevented from tilting by a columnar pin (pinning number)

0

( ,0) p

h u x dxN n

h

Np

h

Traffic jam scenario

No disorder

2

3 2

2 3 2

1 ( 1.5),

1

1

p g

gp

g g

VN hL

h

N V L hL

0

1 ,

ln 11 1

1

p

p

g

hN hL

h g

N Ln hL

I. Affleck, W. Hofstetter, D.R. Nelson, U. Schollwöck (2004)

With point disorder

2

3

2 3

1 (2 3),

1

1

p

p

g

VN hL

h

N V L hL

1

2(1 )

20

1,

1 1max ,exp

4(1 )

g

p

g

gN

g

In an infinite sample g=1 corresponds to the strongest divergence of Np with either L or 1/h.

0.6 0.8 1.0 1.2 1.40

200

400

600

800

1000

1200

1400

1600

g

(a)

L=200 L=400 L=1000 L=2000 L=4000

Np

g

0 1000 2000 3000 4000

0.74

0.76

0.78

0.80

0.82

0.84

0.86

0.88

0.90(b)

g*

L

Unzipping of a single vortex line

0

L

f L

dxF x f f d

d

MFM Tip

fxL

f plays a role of a local transverse magnetic field acting on a vortex

Unzipping transition at the critical force: f=fc

What are the critical properties of this transition?

No disorder, unzipping from a columnar pin

1F is the free energy of the unbound piece

0F is the free energy of the localized piece

2

11

( ) const2

fF

0 0( )F E 1 02cf E

c

Tx

f f

N. Hatano, D. Nelson (1997)

1 0exp ( ) ( )Z d F F

Add point disorder

1 0exp ( ) ( )Z d F F

1 1 1

0 0 0

( ) ( ) ( )

( ) ( ) ( )

F F F

F F F

0,1( )F A

bulk

defect

Relation to anomalous diffusion

2 1 x F

2 1 D. Huse, C. Henley (1985)

Fragmented Columnar pin: =1/2

1/ 20 ( )F A

Disordered twin plane =1/3, =2/3

1/30 ( )F A

2D: =1/31/3

1( )F A

3D: 0.220.22

1( )F A

2/3x

Dominant disorder in the bulk

x

Dominant disorder on the defect

cF f f A

11

10

c

dF

d f f

Disordered columnar pin (=1/2):

2, cx f f D. Lubensky and D. Nelson (2000).

Replica calculation:

( )

/ 20

e

2 1 e

cf f y

ydy y

2

1,

, 2

cc

c

c

f ff f

f ff f

x

1 0F F

F

Replica and numerical calculations for a disordered columnar pin:

0.160 0.165 0.170 0.175 0.1800

200

400

600

800

Ave

rage

dis

plac

emen

t

Force (f)

Numerics, L=8000 Replica Calculation

Replica derivation gives exact result!

Unzipping from a twin plane ( =1/3):

1.5

1,

c

xf f

11

1

cf f

Agrees with exact numerical simulations.

General case. Bulk randomness

Effective disorder on the defect due to finite extent of the localized state.

( )F A

1D 1/ 2

2D 1/ 3

3D 0.22

Asymptotically the main contribution comes from disorder generated on the defect!!!

0 5000 100001.2

1.3

1.4

1.5

1.6

1.7

Exp

onen

t

Lx

Unzipping from a columnar pin in 2D with bulk disorder

-0.005 0.000 0.005 0.010

0.2

0.3

0.4

0.5

x/L

x

Lx=800

Lx=1600

1( )c xf f L

1

x x c

c

x L G L f ff f

Finite size scaling

Extract exponent =1/(1-) from numerics

Anticipate =1.5 from bulk part (=1/3), =2 from columnar pin part (=1/2).

Effectively have unzipping from a disordered pin

0.0 0.2 0.4 0.6 0.80.00

0.02

0.04

0.06

0.08

0.10

f c

Point disorder ()

Critical force versus point disorder in 1+1d

As expected, there is no unbinding transition in 1+1d due to point disorder

Pulling a vortex from a twin plane with an array of flux lines

S

S`

Create a dislocation (magnetic monopole) in the twin plane

Method of images: energy of a dislocation distance from the boundary is equal to the energy of a dislocation pair of opposite signs.

†( ,0) ln ( ) (0)dF T a a

Schulz, Halperin, Henley (1982)

Compute boson-boson correlation function using Luttinger liquid formalism.

1† 2( ) (0) ( ) const ln

2g

d

Ta a F

g

I. Affleck, W. Hofstetter, D.R. Nelson, U. Schollwöck (2004)

11

4

1

4

exp

exp

gc

gc c

d f fx

d f f

1 4 1,

4 4c

T gg x

f f g

8 1

41 1,

8 4

g

g

c

Tg x

f f

1 1 4,

8 1 8

gg x a

g

Discontinuous unbinding transition for g<1/8

Conclusions

1. Columnar pin is always irrelevant in the presence of point disorder.

2. The columnar pin is least irrelevant at the vortex glass transition (g=1).

3. The number of vortices prevented from tilting by a columnar pin in a weak transverse magnetic field has a maximum at g1.

4. Point disorder changes critical properties of an unzipping transition of a single vortex line from an extended defect.

5. Unbinding transition properties from a twin plane in the presence of many flux lines drastically depends on the Luttinger parameter g.

Finite size scaling

x

Lx

x x cx L G L f f

Clean case: =1

absorbing boundary conditions

0.08 0.09 0.10 0.11 0.12 0.130.0

0.2

0.4

0.6

0.8

1.0

Lx=800

Lx=1600

Lx=3200

x/L x

force (f)

-0.04 -0.02 0.00 0.02 0.04 0.060.1

0.2

0.3

0.4

0.5

0.6

x/L x

Lx=800

Lx=1600

1600

c x

c

f f L

f