Recent progress in glassy physics, Paris 27.-30.9.2005

H. Rieger, G. Schehr

Glassy properrties of

lines, manifolds and elastic media

in random environments

Physics Department, Universität des Saarlandes – Saarbrücken, Germany

Strong disorder: Vrand >> ; Vint short ranged, hard core

ibondi

ineH )(

1,0in ]1,0[ie

Magnetic flux lines in a disordered superconductor

Ground state of N-line problem: Minimum Cost Flow problem

N

i

H

ijjiirand

i zrzrVzzrVdz

drdzH

1 0 )(int

2

)]()([]),([2

Continuum model for N interacting elastic lines in a random potential

Lines in 2d - Roughness

222ii xxw Roughness

For H roughness saturates wws(L)ws(L) ~ ln(L) means „super-rough“

L

H

Lines in 3d - Roughness

Line density: =0.2 =N/L2

222ii xxw

For L : w~H1/2 Random walk behavior

Saturation roughness (H): w~L

( elastic media!)

)/(~ 2LHwLw

FSS:L

H

Crossover from single line collective behavior (low density limit):

2d: Scaling with H/ln(L)1/

3d = 0.005

= 0.4

[Petäjä, Lee, HR, Alava: JSTAT P10010 (2004)]

Splayed columnar defects (in 2d)

Single Line Roughness: =3/4

Rougness of a line in a multi-line system: =1/2 (Random Walk)

[Lidmar, Nelson, Gorokhov `01]

[Petäjä, Alava, Rieger `05]

Universality classes: Interacting lines = elastic medium

...},2,1,0{;)( ijjiij

i ndndnH

SOS-model on a disordered substrate

randomd [1,0[T>Tg: Rough phase, <(ni-ni+r)2> ~ ln r Tg=2/T<Tg: Super-rough phase, <(ni-ni+r)2> ~ ln2r

[2,0[)(;))]()(cos())([( 22 rrrrrdH

randomr [2,0[)(

Sine-Gordon model with random phase shifts

N

i

H

ijjiirand

i zrzrVzzrVdz

drdzH

1 0 )(int

2

)]()([]),([2

N interacting elastic lines in a random environment

The SOS model on a random substrate

1,0,,)( 2

)( iiiijij i ddnhhhH

Ground state (T=0):

In 1d: hi- hi+r performs random walk

C(r) = [(hi- hi+r)2]~r

Height profile Flow configuration

[HR, Blasum PRB 55, R7394 (1997)]

In 2d: Ground state superrough,

C(r) ~ log2(r)

stays superrough at temp. 0<T<Tg

)ln(~ r

Dynamics (T>0) : Autocorrelation function

C(t,tw) ~ F(t/tw) ~ (t/tw)- for t/tw>>1

C(t,tw) = [<ni(tw)ni(t)>- <ni(tw)><ni(t)>]av

T>Tg: (T) = 1T<Tg: (T) = 2/z 0, proportional to T

Dynamics (T>0): Spatial correlation functionC(r,t) = [<ni(t)ni+r(t)>- <ni(t)><ni+r(t)>]av

C(r,t) ~ F(r/t1/z) ~ -ln(r/t1/z) for r/t1/z << 1

L(t)=rC(r,t) ~ t1/z

Coarsening (T>0)?

Idea: T>0 non-equilibrium dynamics is coarsening in the overlap with the ground state.

Check: Compute ground state.

At each time calculate the difference: mi(t) = ni(t)-ni0

Identify connected clusters (domains) of site with identical mi(t)

Result: Not quite coarsening, but interesting ...

Overlap w. Ground state (T<Tg)

Overlap w. Ground state (TTg)

Ground state overlap analysis (1)

),(),( )1()1()( trintinrOv

L(t) = linear size of Ground state domains

„Droplet“ size distributionS=Size of connected clusters of sites with ni(t)=ni

0+m with a common m0Initial state is ground state ni

0

Pt(S) ~ S- F(S/L2(t))

~1,85 L(t)~t1/z

[G. Schehr, H.R., (04)]

Disorder chaos in the SOS model

Compare GS hia for disorder configuration di

a

with GS hib for disorder configuration di

b= dia+i

([i2]- [i]2)1/2 = <<1

In 1d: [(hi+ra- hi+r

b)2] ~ 2r when hia = hi

b

i.e. the GS looks different beyond length scale 1/

But: Displacement-correlation function:

Cab(r) = [(hia- hi+r

a) (hib- hi+r

b)] ~ r

Increases with r in the same way as C(r)!

No chaos in 1d.

Disorder chaos (T=0) in the 2d Ising spin glass

[HR et al, JPA 29, 3939 (1996)]

Disorder chaos in the SOS model – 2d

Scaling of Cab(r) = [(hia- hi+r

a) (hib- hi+r

b)]:

Cab(r) = log2(r) f(r/L) with L~-1/ „Overlap Length“

Analytical predictions for asymptotics r:

Hwa & Fisher [PRL 72, 2466 (1994)]: Cab(r) ~ log(r) (RG)

Le Doussal [cond-mat/0505679]: Cab(r) ~ log2(r) / r with =0.19 in 2d (FRG)

Exact GS calculations:q2 C12(q) ~ log(1/q) C12(r) ~ log2(r)

q2 C12(q) ~ const. f. q0 C12(r) ~ log(r)

Numerical reslts support RG picture of Hwa & Fisher. [Schehr, HR `05]

Conclusions

• Superrough ground state in 2d (also for T<Tg)• Weak collective effects in 3d (random walk roughness)• Splay disorder: Random walk roughness

• Dynamics: Autocorrelations C(t,tw) ~ (t/tw)-2/z(T) for t/tw>>1• Spatial correlations C(r,t) ~ -ln(r/t1/z(T) ) for r/ t1/z<<1

• Droplet excitations above ground state P(S) ~ S-F(S/t2/z(T) )

• Weak disorder chaos: Cab(r) ~ log(r)

N hard core interacting lines in 2d, 2d elastic medium with point disorder,SOS model on disordered substrate

Title

Title

Entanglement transition of elastic lines

H =64 H =96 H=128

Conventional 2d percolation transition

=4/3=2,055

df=1,896

[Petäjä, Alava, HR: EPL 66, 778 (04)]

Computation of the ground state

Finding the ground state of the SOS model on a disordered substrate is a minimum cost flow problem

(polynomial algorithm exist)

see Blasum & Rieger,PRB 55, 7394 (1997)

G. Schehr, J.-D. Noh, F. Pfeiffer, R. Schorr

Universität des Saarlandes

V. Petäjä, M. Alava

Helsinki University of Technology

A. Hartmann

Universität Göttingen

J. Kisker, U. Blasum

Universität zu Köln

Collaborators

Further reading:

H. Rieger:Ground state properties of frustrated systems,Advances in computer simulations, Lexture Notes in Physics 501(ed. J. Kertesz, I. Kondor), Springer Verlag, 1998

M. Alava, P. Duxbury, C. Moukarzel and H. Rieger:Combinatorial optimzation and disordered systems,Phase Transitions and Critical phenomena, Vol. 18(ed. C. Domb, J.L. Lebowitz), Academic Press, 2000.

Book:A. Hartmann and H. Rieger,Optimization Algorithms in Physics, (Wiley, Berlin, 2002)New Optimization Algorithms in Physics, (Wiley, Berlin, 2004)