A Tour of Spin Glasses and Their Geometry

Transcription

1 A Tour of Spin Glasses and Their Geometry P. Le Doussal, D. Bernard, LPTENS A. Alan Middleton, Syracuse University Support from NSF, ANR IPAM: Random Shapes - Workshop I 26 March, 2007

2 Goals Real experiment very slow dynamics memory Edwards-Anderson spin glass L? geometric objects? Numerical experiments evidence for SLE as a challenge

3 An alloy with randomly placed magnetic ions Competition between ferromagnetic and anti-ferromagnetic interactions. Measure response to magnetic field: 1. Slowly cool, then slowly heat. 2. Cool slowly, pause, continue cooling, then slowly heat. = Cu = spin up Mn

4 A more complicated experiment [Miyashita, Vincent] * Slow relaxation * Rejuvenation * Memory

5 Edwards-Anderson Hamiltonian - Statics Energy for Ising spins s i = ±1 on a graph with sites i: H = J ij s i s j, ij where the J ij are i.i.d. Gaussian with zero mean. Temperature T is believed to be irrelevant : At small enough T, changing T modifies the effective J ij, but has no other effect on the statics study minima of H.

6 Given J ij, two ground states (min H) α = {s 0 i }

7 Given J ij, two ground states (min H) β = { s 0 i }

8 Two intertwined questions: Only two states as L? What is the effect of boundary conditions?

9 Ground state description, numerical solution Single ground state (in finite sample) for F ij bonds dual to edges ij : F ij = J ij s i s j. If s i s i for i B, then F ij F ij on B and H = 2 B F ij Find ground state numerically on cylinder or open square: Use Barahona s mapping of planar ISG to a matching problem (1982). The ground state Fij 0 weight γ F ij. has no loops or contractible paths of negative total

10 Thermodynamic limit: does it exist? Numerically: Solve GS for open system of size L. Grow to size L & solve for GS. Compare the F ij in box of size w. Probability of change in window appears to scale as ( w ) f(l d df /L), L L w L with f(x) const as x.

11 Domain Walls in a cylinder Changing J ij J ij on a column (periodic anti-periodic BCs), changes constraints on F ij. Introduces a domain wall γ, the shortest path in Fij 0 from the bottom to the top. E DW = 2 γ F ij 0. γ α β Y X L rows of W spins; X = W, Y = 3 2 L.

12 Domain Wall Wide sample, domain wall in 2D ISG: Display of bonds changed by BC comparison, periodic->antiperiodic in horizontal direction Study or with good statistics (N > 10 4, 1% errors). Domain walls can be unconstrained or have a fixed end.

13 Energies and fractals Domain wall renormalization group (Bray, Moore, McMillan) and droplet theory (Fisher, Huse) give scaling hypotheses, E DW L θ, θ = 0.28(1) γ L d f, df = 1.28(1) where estimates are from simulations.

14 Tests of conformal invariance and domain Markov (SLE) For comparison: Loop-erased random walk (LERW): execute a random walk on a lattice, erasing loops as they form. [Minimal spanning tree (MST): d f = 1.217(3), not SLE.] Numerically, d f for LERW, independent of whether the boundary conditions are reflecting (not SLE) or absorbing (SLE, d f = 5 4 ).

15 Conformal Invariance If SLE, d f = 1 + κ 8 ; prediction of winding number on cylinder of circumference 2π: θ 2 = κl W = 32, > 1000 samples each Fit with κ = 2.32(2) θ x L Inferred κ cyl = 2.32 ± 0.02 is consistent with d f = 1.28 ± 0.01 (for both unconstrained & fixed BCs).

16 Probability of DW passing to right of a point R φ Y X Near start, like upper half plane, so use Schramm: P κ (φ) = Γ( κ) 4 πγ( 8 κ 2κ ) cot(φ) ( 1 2F 1 2, 4 κ, 3 2 ; cot2 (φ) )

17 Probability of DW passing to the right of a point Probability - Probability(kappa=2) Compare forward / back paths with Fixed Bottom, Open Top BCs, smoothed angles 128x1024, from localized end, r=32, N= x1024, from localized end, r=64, N= x1024, from floating end, r=32, N= x1024, from floating end, r=64, N=5254 kappa=2.16 kappa=2.24 kappa=2.32 kappa= Wed Jul 19 16:45: Angle / Pi Plots show residuals P κ P 2, where P 2 (φ) = φ sin(2φ)/2 π.

18 Dipolar maps Wide sample, domain wall in 2D ISG: Sequence of dipolar transformations. t = (100 steps) t = (200 steps) t = (300 steps) t = (500 steps) Sequence of maps g ti (z) for strip height π/2 to infer ξ(t i ), approximates dg t (z) dt = 2 tanh[g t (z) ξ t ] ; g 0(z) = z

19 Dipolar maps Is ξ 2 (t) = κt, d f = 1 + κ? YES for unconstrained, NO for fixed end. 8 4 [ξ 2 (2t)-ξ 2 (t)]/t 3 2 L=256, from constrained end L=128, from constrained end L=128, floating BCs L=64, floating BCs Is ξ Gaussian? Yes. t Is ξ correlated? Doesn t look like it.

20 But ξ(t) is not the best test MST with L = 256, N = 40000: ξ 2 / t (cf. expected κ 1.74) t

21 Markov Property It may be hard to see failure by studying ξ(t). Another approach is to directly study P [γ 2 (b, c) γ 1 (a, c); a, b, c, D]? = P [γ 2 (b, c) c; D\γ 1, a] [Note, approximate study by Hartmann, Amoruso] Must fail microscopically - Hastings.

22 Sample, sample, sample, compute P Say samples of 6 6 spins. On original strip D, compute DWs; for given start γ 1, compute fraction P (γ 2, γ 1 ) that end with γ 2. γ 2 Given γ 1, generate J ij and find DW in D\γ 1 : out of those that start at c, find fraction P (γ 2, γ 1 ) that end in γ 2. c γ 1

23 Compare P and P, ISG - lexicographic order for γ 2 2DISG, FF BCs; γ 1 = RLR, L = P P P, P γ 2

24 Compare P and P, ISG 2DISG, FF BCs; γ 1 = RLR, L = P P P, P 5e-05 0 γ 2, #1200->#1300

25 Test Markov, LERW with absorbing boundary conditions L = 6; γ 1 = RLR, P = P (γ 2 γ 1 ; D), P (γ 2 c; D γ 1 ) P P P P P, P 0.01 P, P γ 2 0 γ 2, #400 to #500

26 Test Markov, LERW, reflecting boundary conditions L = 6; γ 1 = RLR. P, P γ 2, #1 to #100 P P

27 Alternative display p(p /P<r) A-LERW, γ 1 =6, γ 2 = A-LERW, γ 1 =6, γ 2 =4 R-LERW, γ 1 =6, γ 2 =1 R-LERW, γ 1 =6, γ 2 =4 2D ISG, γ 1 =6, γ 2 =1 2D ISG, γ 1 =6, γ 2 = r Cumulative probability p = γ 1,γ 2,P /P <r P D(γ 1, γ 2 )

28 Highlights & Sequels 2DISG: κ eff = 2.30(5) consistent with d f = 1.28(1) for unconstrained paths, but not for paths with fixed root. Markov property generally holds for L > 4. Applied same analysis path for LERW, MST, to study the utility of numerical checks. Other curves? (MST, etc.) Conjecture κ? What are good tests of domain Markov? Understanding of domain Markov? [Hierarchy, reversibility of min path] Modifications to SLE for boundary effects?

29 Correlation Functions 10 L=W=400, FF L=W=720, FF exp(log(x)*(-0.5))*52 exp(log(x)*(-0.5))* L=W=400, FF, correlations of xi(t_i+s)xi(t_i) 1/x/x*6 exp(-x*0.26)* (a) N(t) 1 C_d(n) t s

30 Endpoint distribution on a strip Compare with formula by Bauer & Bernard, plot location of endpoint relative to start: Residual from kappa = LERW, L= W=1024, L=256, LF BC W=1024, L=128, FF BC W=512, L=64, FF BC kappa = kappa = 2.32 kappa = 2.40 kappa = 2.85 kappa = Delta-x / L