Geometry of Domain Walls in disordered 2d systems C. Schwarz1, A. Karrenbauer2, G. Schehr3, H. Rieger1 Saarland University 2 Ecole Polytechnique Lausanne 3 Universit Paris-Sud 1 Physics of Algorithms, Santa Fe 31.8.-3.9.2009 Applications of POLYNOMIAL combinatorial optimization methods in Stat-Phys. (T=0)

o o o o o o o o o Flux lines with hard core interactions Vortex glass with strong screening Interfaces, elastic manifolds, periodic media Disordered Solid-on-Solid model Wetting phenomena in random systems Random field Ising systems (any dim.)

Spin glasses (2d polynomial, d>2 NP complete) Random bond Potts model at Tc in the limit q ... c.f.: A. K. Hartmann, H.R., Optimization Algorithms in Physics (Wiley-VCH, 2001); New optimization algorithms in Physics (Wiley-VCH, 2004) Paradigmatic example of a domain wall: Interfaces in random bond Ising ferromagnets H J ij Si S j i Find for given random bonds Jij the ground state

configuration {Si} Si= +1 with fixed +/- boundary conditions Find interface (cut) with minimum energy J ij 0, Si 1 Si= -1 The SOS model on a random substrate {n} = height variables

(integer) H (ij ) (hi h j ) 2 , hi ni d i , d i 0,1 Ground state (T=0): In 1d: hi- hi+r performs random walk C(r) = [(hi- hi+r)2]~r In 2d: Ground state superrough, C(r) ~ log2(r) Stays superrough at temperatures 0

{x}, the height differences, is an integer flow in the dual lattice Height profile Flow configuration Minimize with the constraint (mass balance on each node of the dual lattice) Minimum cost flow problem Domain walls in the disordered SOS model Fixed boundaries

Grey tone DW scaling in the disordered SOS model Length fractal dimension df = 1.25 0.01 Energy L E ~ log L

Energy scaling of excitations Droplets for instance in spin glasses (ground state {Si0}): Connected regions C of lateral size ld with Si=Si0 for iC with OPTIMAL excess energy over E0. Droplets of ARBITRARY size in 2d spin glasses [N. Kawashima, 2000] For SOS model c.f. Middleton 2001. Droplets of FIXED size in the SOS model Droplets: Connected regions C of lateral size L/4 < l < 3L/4 with hi=hi0+1 for iC with OPTIMAL energy (= excess energy over E0). Efficient computation: Mapping on a minimum s-t-cut.

Example configurations (excluded white square enforces size) Results: Scaling of droplet energy Average energy of droplets of lateral size ~L/2 saturates at FINITE value for L Probability distribution of excitations energies: L-independent for L. n.b.: Droplet boundaries have fractal dimension df=1.25, too!

Geometry of DWs in disordered 2d models DWs are fractal curves in the plane for spin glasses, disordered SOS model, etc (not for random ferromagnets) Do they follow Schramm-Loewner-Evolution (SLE)? Yes for spin glasses (Amoruso, Hartmann, Hastings, Moore, Middleton, Bernard, LeDoussal) Schramm-Loewner Evolution (1) The random curve can be grown through a continuous exploration process Paramterize this growth process by time t: t D

gt H gt-1 at When the tip t moves, at moves on the real axis At any t the domain D/ can be mapped onty the standard domain H, such that the image of t lies entirely on the real axis Loewners equation:

Schramm-Loewner evolution: If Proposition 1 and 2 hold (see next slide) than at is a Brownian motion: determines different universality classes! Schramm-Loewner Evolution (2) Define measure on random curves in domain D from point r1 to r2 r2 Property 1: Markovian 2 1 D

r1 Property 2: Conformal invariance r2 r2 D r1 D

r1 Examples for SLE = 2: Loop erased random walks = 8/3: Self-avoiding walks = 3: cluster boundaries in the Ising model = 4: BCSOS model of roughening transition, 4-state Potts model, double dimer models, level lines in gaussian random field, etc. = 6: cluster boundaries in percolation = 8: boundaries of uniform spanning trees Properties of SLE 1) Fractal dimension of : df = 1+/8 for 8, df=2 for 8

2) Left passage probability: (prob. that z in D is to the left of ) g(z) z Schramms formula: DW in the disordered SOS model: SLE? Let D be a circle, a=(0,0), b=(0,L) Fix boundaries as shown Cumulative deviation of left passage probability from Schramms formula Minimum at =4!

Local deviation of left passage probability From P=4 Other domains (conformal inv.): D = square Cum. Deviation: Minimum at =4! Local dev. D = half circle Dev. From P=4 larger than 0.02, 0.03,

0.035 Deviation from P=4() DWs in the disordered SOS model are not described by chordal SLE Remember: df = 1.25 0.01 Schramms formula with =4 fits well left passage prob. IF the DWs are described by SLE=4: df = 1+/8 df = 1.5 But: Indication for conformal invariance! Conclusions / Open Problems Droplets for l have finite average energy,

and l-independent energy distribution Domain walls have fractal dimension df=1.25 Left passage probability obeys Schramms formula with =4 [8(df-1)] in different geometries conformal invariance? DWs not described by (chordal) SLE why (not Markovian?) Contour lines have df=1.5 Middleton et al.): Do they obey SLE=4? What about SLE and other disordered 2d systems?

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+ra) (hib- hi+rb)]: 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, Schehr & HR `05: q2 C12(q) ~ log(1/q) C12(r) ~ log2(r) q2 C12(q) ~ const. f. q0 C12(r) ~ log(r) Numerical results support RG picture of Hwa & Fisher.