Recent progress in glassy physics, Paris H. Rieger, G. Schehr Glassy properrties of lines, manifolds and elastic media in random environments Physics Department, Universität des Saarlandes – Saarbrücken, Germany

Strong disorder: V rand >>  ; V int short ranged, hard core Magnetic flux lines in a disordered superconductor Ground state of N-line problem: Minimum Cost Flow problem Continuum model for N interacting elastic lines in a random potential

Lines in 2d - Roughness Roughness For H  roughness saturates w  w s (L) w s (L) ~ ln(L) means „super-rough“ L H

Lines in 3d - Roughness Line density:  =0.2  = N/L 2 For L  : w~H 1/2  Random walk behavior Saturation roughness (H  ): w~L (  elastic media!) FSS: L H

Crossover from single line collective behavior (low density limit): 2d : Scaling with H/ln(L) 1/  3d  =  = 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  SOS-model on a disordered substrate T>T g : Rough phase, ~ ln r T g =2/  T ~ ln 2 r  Sine-Gordon model with random phase shifts N interacting elastic lines in a random environment

The SOS model on a random substrate Ground state (T=0): In 1d: h i - h i+r performs random walk C(r) = [(h i - h i+r ) 2 ]~r Height profile  Flow configuration [HR, Blasum PRB 55, R7394 (1997)] In 2d: Ground state superrough, C(r) ~ log 2 (r) stays superrough at temp. 0<T<T g

Dynamics (T>0) : Autocorrelation function C(t,t w ) ~ F(t/t w ) ~ (t/t w ) - for t/t w >>1 C(t,t w ) = [ - ] av T>T g : (T) = 1 T<T g : (T) = 2/z  0, proportional to T

Dynamics (T>0): Spatial correlation function C(r,t) = [ - ] av C(r,t) ~ F(r/t 1/z ) ~ -ln(r/t 1/z ) for r/t 1/z << 1 L(t)=  r C(r,t) ~ t 1/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: m i (t) = n i (t)-n i 0 Identify connected clusters (domains) of site with identical m i (t) Result: Not quite coarsening, but interesting...

Overlap w. Ground state (T<T g )

Overlap w. Ground state (T  T g )

Ground state overlap analysis (1) L(t) = linear size of Ground state domains

„Droplet“ size distribution S=Size of connected clusters of sites with n i (t)=n i 0 +m with a common m  0 Initial state is ground state n i 0 P t (S) ~ S -  F(S/L 2 (t))  ~1,85 L(t)~t 1/z [G. Schehr, H.R., (04)]

Disorder chaos in the SOS model Compare GS h i a for disorder configuration d i a with GS h i b for disorder configuration d i b = d i a +  i ([  i 2 ]- [  i ] 2 ) 1/2 =  <<1 In 1d: [(h i+r a - h i+r b ) 2 ] ~  2  r when h i a = h i b i.e. the GS looks different beyond length scale 1/  But: Displacement-correlation function: C ab (r) = [(h i a - h i+r a ) (h i b - h i+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 C ab (r) = [(h i a - h i+r a ) (h i b - h i+r b )]: C ab (r) = log 2 (r) f(r/L  ) with L  ~  -1/  „Overlap Length“ Analytical predictions for asymptotics r  : Hwa & Fisher [PRL 72, 2466 (1994)]: C ab (r) ~ log(r) (RG) Le Doussal [cond-mat/ ]: C ab (r) ~ log 2 (r) / r  with  =0.19 in 2d (FRG) Exact GS calculations: q 2  C 12 (q) ~ log(1/q)  C 12 (r) ~ log 2 (r) q 2  C 12 (q) ~ const. f. q  0  C 12 (r) ~ log(r)  Numerical reslts support RG picture of Hwa & Fisher. [Schehr, HR `05]

Conclusions Superrough ground state in 2d (also for T<T g ) Weak collective effects in 3d (random walk roughness) Splay disorder: Random walk roughness Dynamics: Autocorrelations C(t,t w ) ~ (t/t w ) -2/z(T) for t/t w >>1 Spatial correlations C(r,t) ~ -ln(r/t 1/z(T) ) for r/ t 1/z<<1 Droplet excitations above ground state P(S) ~ S -  F(S/t 2/z(T) ) Weak disorder chaos: C ab (r) ~ log(r) N hard core interacting lines in 2d, 2d elastic medium with point disorder, SOS model on disordered substrate

Title

Entanglement transition of elastic lines H =64H =96 H=128 Conventional 2d percolation transition  =4/3  =2,055 d f =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, Book: A.Hartmann and H. Rieger, Optimization Algorithms in Physics, (Wiley, Berlin, 2002) New Optimization Algorithms in Physics, (Wiley, Berlin, 2004)