Multifractality in random-bond Potts models

Transcription

1 Multifractality in random-bond Potts models Ch. Chatelain Groupe de Physique Statistique, Université Nancy 1 April 1st, 2008

2 Outline 1 Self-averaging 2 Rare events 3 The example of the Ising chain 4 Multifractality 5 Comparison with perturbative results 6 Replica Symmetry Breaking 7 Universality of multifractal spectrum 8 Prob. distrib. and multifractal spectrum

3 Self-averaging Physical observables φ should be averaged over thermal fluctuations AND disorder realizations. Using Bertrand s notations φ = D[K ij ]P[K ij ] φ [Kij ] = D[K ij ]P[K ij ] D[σ i ]φ[σ i ] e βh[k ij,σ i ] Z[K ij ]

4 Self-averaging (2) φ Kij is a random variable (fluctuates from sample to sample) with probability distribution : ( ) (φ) = D[K ij ]P[K ij ]δ φ φ [Kij ] and average φ = φ (φ)dφ Naive assumption: when the system size tends to, all disorder realizations become equivalent, i.e. (φ) 1 2πσ 2 e (φ φ )2 /2σ 2 δ(φ φ )

5 Self-averaging (3) Sample-to-sample fluctuations should vanish R φ = φ 2 φ 2 according to the central limit theorem. φ 2 0 Wiseman and Domany (1995) observed that it is not true: R m and R χ tend to finite values in the thermodynamic limit! m and χ are examples of non self-averaging quantities.

6 Self-averaging (4) lim R <Gσσ (u)> + L q

7 Self-averaging (5) Aharony, Harris and Wiseman (1998) showed that Stable random fixed point: (non self-averaging) R φ (L) R φ ( ) + A L (α/ν)rand. R φ ( ) is a universal value. Unstable random fixed point, i.e. α < 0: (weak self-averaging) R φ (L) L α/ν Out of the critical point (self-averaging) R φ (L) L d

8 Rare events The average is dominated by rare events. Average and typical values are different <G σσ (20)> <G σσ (60)> N G

9 Rare events (2) The distribution (φ) displays a long tail: 0.03 u= 30 u= 60 u= 80 P(<G σσ (u)>) <G σσ (u)> Impossibility of numerical simulations?

10 The Ising chain βh = i K i σ i σ i+1, (σ i { 1; +1}) Spin-spin correlation function for a given disorder realization j 1 G σσ (i, j) = σ i σ j = tanhk k k=i

11 The Ising chain (2) If the K k are uncorrelated independent random variables, the central limit theorem applies to the sum leading to ( i j 1) j 1 ln G σσ (i, j) = lntanhk k (ln G σσ (i, j) = lnc) k=i 1 2πσ 2 e (lnc ln C)2 /2σ 2

12 The Ising chain (3) Typical and average values G σσ (i, j) = Gσσ typical (i, j) = e j i ln tanhk = e ln Gσσ(i,j) e x (x)dx = e j i ln tanhk G typical σσ (i, j) are different (Derrida, 1981). Rare events: large number of strong couplings.

13 The Ising chain (3) One can define two different correlation lengths lng typical σσ (r) r ξ typical, with different critical behavior ln G σσ(r) r ξ avg ξ typical T T c νtypical, ξ avg T T c νavg The stability of the random fixed point imposes ν avg 2/d but ν typical can violate this constraint.

14 Multifractality G σσ (u) <G σσ (u)> e ln <G σσ (u)> Cumulant d ordre 4 Maximum de P(<G σσ (u)>) u

15 Multifractality (2) For a self-averaging quantity, for instance a Gaussian distributed random variable, the cumulants satisfy the recursion relation (Wick Theorem) (φ φ) n = 1 2πσ 2 + = (n 1)σ 2 (φ φ) n 2 dφ (φ φ) n e (φ φ)2 /2σ 2 The scaling dimension x φ (n) of (φ φ) n1/n is thus x φ (n) = x φ (2) = x φ

16 Perturbative results Replica trick 1 [ F = ln Z = lim Z p 0 p p 1 ] For disorder m( r) coupled to the energy density ǫ( r) Z p = D[φ 1 ( r)]... D[φ 1 ( r)]... D[φ p ( r)] e P p α=1(h 0 [φ α]+ P r m( r)ǫ α( r)) D[φ p ( r)] e P α H 0[φ α] e P r [m P α ǫα( r)+ 1 2 (m2 m 2 ) P α,β ǫα( r)ǫ β( r)+...]

17 Perturbative results (2) Renormalization group predictions for the decay exponents of the moments G σσ (u) n1/n u 2xb σ(n) of the spin-spin correlation function of conformal minimal models (Ludwig, Dotsenko, Lewis) x b σ (n) = xb,pure σ n 1 (» 11 y H ln 2 + n π «) y 2 H + O(y 3 H ) where y H = α Pure /ν Pure for disorder coupled to the energy density.

18 Perturbative results (3) 0.15 q=2.25 x σ b(n) er ordre 2 nd ordre Matrice de transfert n

19 Perturbative results (4) 0.15 q=2.5 x σ b(n) er ordre 2 nd ordre Matrice de transfert n

20 Perturbative results (5) 0.15 q=2.75 x σ b(n) er ordre 2 nd ordre Matrice de transfert n

21 Perturbative results (6) 0.15 q=3.5 x σ b(n) er ordre 2 nd ordre Matrice de transfert n

22 Multifractality at boundaries n=0 n=1 n=2 n=3 x σ 1(n) /L

23 Replica Symmetry Breaking Even for uncorrelated disorder, i.e. m( r)m( r ) = m 2 δ( r r ) it appears a coupling between different replicas. All interactions are symmetric under permutation between replicas. But the associated random fixed point may be unstable (random field XY model, φ 4 ) and the system flows under renormalization toward a new fixed point where this symmetry is broken! p g αβ ǫ α ( r)ǫ β ( r) α,β=1

24 Replica Symmetry Breaking (2) For minimal models, renormalization group calculations leads to Replica symmetry x b σ(2) = x b,pur σ 1 16 y H Broken replica symmetry x b σ(2) = x b,pur σ 1 16 y H ( 4 ln2 11 ) yh O(y3 H ) ( 4 ln2 5 ) yh O(y3 H )

25 Replica Symmetry Breaking (3) 0.14 Systeme pur Symetrie des repliques Brisure de symetrie des repliques Matrice de transfert x σ b(2) q

26 Universality Universality of the multifractal spectrum 0.15 q=3 x σ b(n) er ordre 2 nd ordre Distribution binaire Distribution continue n

27 Prob. distrib. and multifractal spectrum The multifractal spectrum is entirely determined by the proba. distribution. G σσ (u) n = 1 0 dg u (G)G n A n u 2X(n) where X(n) = nx b σ(n). Let y = lng and u (y)dy = u (G)dG + By Mellin-Fourier transform 0 u (y) 1 2iπ dy e ny u (y) A n u 2X(n) δ+i δ i ny 2X(n)ln u+ln An dn e

28 Prob. distrib. and multifractal spectrum (2) Let α = y/2 ln u. In the saddle point approximation u (y) 1 2iπ δ+i δ i dn e 2ln u[x(n) nα] 2ln u H(α) e where H(α) is the Legendre transform of the exponent X(n) ( ) X(n) H(α) = X(n ) αn, α = n n

29 Prob. distrib. and multifractal spectrum (3) H( α) H(α) = X(n) αn X(n) 0 α x σ b (n) α 0 α

30 Prob. distrib. and multifractal spectrum (4) er ordre 2 nd ordre Matrice de transfert H(α) α

31 Conclusions Self-averaging because of rare events Multifractality Multifractal spectrum determined by the probability distribution