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)