Effect of Anisotropy on the Continuous Abelian Sandpile Model

Nahid Azimi-Tafreshi Shar if Uni versit y of Tech nol ogy

16th Spring Theoretical Physics Conference May 20-21, 2009 Forword

Historically, the Abelian Sandpile Model (ASM) has been proposed by Bak, Tang and Wiesen fe ld (1987) as a siilmple example of a 1‐ slowly driven, 2‐dissipative, 3‐non‐equilibrium system exhibiting self‐organized criticality (SOC).

Idea was: many critical behaviours (power laws) exist in nature, but unlikely to result from fine‐tuning it is the dynamics that drives the system to a critical state, even if the system is prepared in a non‐critical state.

1 Outlines:

1‐ Abelian Sandpile Model(()ASM) definition of the model‐ dynamics

2‐ Anisotropic Continuous Abelian Sandpile Model directed zigzag pattern

3‐ Random Directed Continuous Sandpile Model

2 The model

• The model is defined on a two dimensional lattice with N sites.

• A random variable h i is assigned to every site (ish i # grains)

• Site i is stable if 1 ≤ hi ≤ 4 4 2 3 1 3 2 4 1 2 1 2 2 1 1 4 3 4 2 3 2 2 2 1 2 4 2 1 3 2 3 3 4 3 2 1 1 3 4 3 4 4 4 3 2 4 3 2 1 2 3 2 3 3 4 4 3 1 1 2 3 2 3 2 4 3 3 4 2 4 3 3 1 3 2 4 2 1 4 4 3

a configurat ion is stable if all sites are stable

# stable configs = 4 N

3 ASM ‐ Dynamics

The sandpile model is a stochastic in discrete 2 + 1.

Dynamics takes configuration Ct into Ct+1 in two steps:

1. on random site i , drop one grain: hi → hi + 1 2. relaxation: all unstable sites topple (avalanche)

h → h − 4 If h > 4 , then ⎧ i i i ⎨ hj = hj − Δij ∀j ⎩ h n .n → h n .n + 1,

Toppling matrix Δ is the Laplacian operator : ∑ Δ ij G jk = δ ik j ⎧4 for i = j Δ ij = ⎨ ⎩− 1 for 〈i, j〉

System spanning avalanches will happen, and induce correlations of heights over long dis tances critica l stttate

4 Recurrent set

Configurations are either

• transient: they are not in the repeated image of the dynamics, and occur only a

finite number of times ⇒ Pt (C ) = 0 for large enough t.

• recurrent: they are in the repeated image of the dynamics and asymptotically occur with non‐zero probability.

Number of recurrent configurations ?

R = det Δ (~( 3.21N << 4N )

5 Field Theory Description

# recurrent configs. partition function of the model θ θ θ Z = det Δ = d d exp( Δ θ ) ∫ i j i ij j

θ and θ are scalar, anticomm. fields with canonical dimension 0.

In continuous limit this partition function defines field theory with the action:

S = d 2 z ∂θ ∂θ 0 ∫

S 0 is the action of c=‐2 .

height variables in ASM scaling fields in c=‐2

6 Anisotropic models The original sandpile model is isotropic such that after a site topples, the same value of sand is distributed uniformely between the neighbor sites of that site.

anisotropy in ASM t (directed model) isotropic model x

directed model reflects beter the dynamics of a real sandpile. (under gravity, particles would only fall down and not up )

directed model solveable in any dimension.

universality class of directed sandpile model is different from the ordinary ASM .

7 Directed Continuous Sandpile Model modified version of ASM Continuous Abelian Sandpile Model (CASM)

height variable is a real number in [0,4)

There is a (many to one) mapping from configurations in CASM to configurations in ASM, which preserves the dynamics. anisotropy in redistribution of sands in CASM: we are able to control the amount of directedness.

Toppling matrix: ⎧2 i = j ⎪ 1−ε 1+ ε Δ ij = ⎨− (1 ± ε ) i = j m 1 ⎪ ⎩0 otherwise ε a positive real parameter less than 1 that controls the amount anisotropy

8 Numerical simulations

Log(Ps) size of avalanche (s): the total number of toppling events.

Log(s) Power law distribution of avalanches : P(s) ∝ s−τ s

The extrapolated value of τ for ε values: ε 0 0.1 0.4 1 1.38 ± 0.08 τ s 1.25± 0.08 1.31± 0.08 1.41± 0.08

directedness changes the universality class. fully directed model (ε = 1) dddirected percolation

9 Zigzag Pattern A B

B A ′ ′ ⎧4 i = i′, j = j′ ⎧4 i = i , j = j ⎪ ⎪ B ⎪− (1± ε ) i = i′ ±1 A ⎪− (1± ε ) i = i′ ±1 Δij,i′j′ = ⎨ Δij,i′j′ = ⎨ − (1 ε ) j = j′ ±1 ⎪− (1m ε ) j = j′ ±1 ⎪ m ⎪0 otherwise ⎩⎪0 otherwise ⎩

ε a positive real parameter less than 1 that controls the amount anisotropy

A sites to direc t the avalhlanche ttdowards up‐lftleft corner and the B sites to direc t the avalanche to down‐right corner, thus on average the sands do not move in any specific direction.

ε =1 one‐dimensional sandpile model

Interesting for us: effect of this type of anisotropy on the critical behaviors 10 Free Energy Function The free energy is obtained by enumerating the corresponding spanning trees on the lattice. LliLaplacian matrix Q (ST) Toppling matrix Δ (ASM)

(‐1,0) (0,1) y

(‐1,‐1) (0,0) (1,1)

x (0,‐1) (1,0)

π π 2 2π 1 dθ1 dθ 2 θ f = ln det F ( 1,θ2 ) 8π 2 ∫0 2π ∫0 2

r r ir .Θ Θ=(θ ,θ ) F (θ1 ,θ 2 ) = 4Ι − ∑ a(r ) e ; 1 2 r a(n, n′)are the 2×2 cell adjacency matrices describing the connectivity between sites of unit cells n , n′ 11 Free Energy Function

(‐1,0) (0,1) adjacency matrices :

⎛ 0 1+ ⎞ T ⎛ 0 0⎞ a(0,0) = ⎜ ε ⎟, a(0,1) = a (0,−1) = ⎜ ⎟ (‐1,‐1) (0,0) (1,1) ⎝1+ ε 0 ⎠ ⎝1+ ε 0⎠

(0,‐1) (1,0) T T ⎛ 0 1− ε ⎞ a(−1,0) = a(−1,−1) = a (1,0) = a (1,1) = ⎜ ⎟ ⎝ 0 0 ⎠

The free energy function: ε π θ 2 2π θ 1 2 2 2 f = dθ1 dθ 2 ln[12 − 4ε − 4(1− ε ) cosθ1 − 4(1+ ε ) cosθ 2 8π 2 ∫0 ∫0 ε 2 2 θ − 2(1− ) cos( 1 + 2 ) − 2(1− ) cos( 1 −θ 2 )]

The model is equivalent to a free fermion 8 vertex model

for all values of ε the free energy ftifunction is analtillytical and the modldel shows no . 12 Numerical simulations Every avalanche can be represented as a sequence of waves of the topplings such that each site at a wave topples only once. (w) log Ps wave toppling distributions:

( w) w −τ s Ps (s) ∝ s

log s

The extrapolated value of τ for three ε values:

ε 0.1 0.4 0.8 ( w ) τ s 1.00 ± 0.01 0.99 ± 0.01 1.01± 0.01

The wave exponents obtained here are independent of the ε values and (w) consistent with the exact value of τ s =1 for the isotropic model.

13 Random Directed Continuous Sandpile Model

there exists a preferred direction for the transportation of 1−ε 1+ ε sands rotational symmetry is broken in this model

we consider a statistical distribution for ε such that it can take both positive values negative values in different sites:

Guassian distribution : ε (z1)ε (z2 ) = g0δ (z1 − z2 )

The question: if such a modification takes the system to a new universality class or not?

method pertubative group technique. the action of perturbed theory: ε ϕ S = S + d 2 z (z, z) ( (z, z) +ϕ (z, z)) 0 ∫ ϕ ϕ = −2θ ∂θ , = −2θ ∂θ perturbing field operators

14 Random Directed Continuous Sandpile Model

replica Method Taking the partition function of n identical copies of the system Z n −1 f = ln Z = εlim n→0 n ϕ N n − S 0 ,a − ∫ ( z ,z )( a ( z ,z )+ϕ a ( z ,z )) Z = ∏ e z a =1 The average of Z n is made with a Gaussian distribution for ε ( z, z ): 1 − ε 2 ( z,z ) n ϕ n 2 g 0 Z = ∫∏ϕdε (z, z) Z e z,z ϕ The effective action: ϕ

n n ϕ S = S + g ( (z, z) (z, z) + (z, z) (z, z) + 2 (z, z)ϕ (z, z)) ∑ 0,a 0 ∫∑ a b a b a b a=1 z a≠b

15 Random Directed Continuous Sandpile Model The renormalization of the coupling constants of each of the perturbing operators will ϕϕ be determined by a perturbative computation. β ϕϕ ϕϕ limit n→0 π β ‐ functions: ϕϕ ϕϕ ∂g 2 3 = a =16 g g −16π α (9g g +7g ϕϕ ) gϕϕ ∂a ∂gϕϕ 2 2 3 2 β = a = 24π (g ϕϕ + g ϕϕ ) −32π α (5g ϕϕ +6g ϕϕ g ) gϕϕ ∂a ϕϕ

gϕϕ

ϕϕ g = gϕϕ = 0

new fixed point α L { 3 ; = 4π ln gϕ = ϕ 20α a

RG flow of the model with quenched randomness g ϕϕ 16 Conclusions

¾ We defined the directed continuous sandpile model and showed that the critical exponents are sensitive to the directed anisotropy.

¾ We sttdidudied the critica l bhbehav ior of the continuous sandildpilemodldel with one kind patterned anisotropy in the toppling matrix. The anisotropic model is in the same universality class of the continuous sandpile model.

¾ The effect of quenched randomness on the critical behavior of continuous directed sandpile model was investigated. We showed that the perturbing fields are relevant and take the system to the new fixed point.

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