Geometric phase transitions: percolation

Transcription

1 Chapter 2 Geometric phase transitions: percolation 2013 by Alessandro Codello 2.1 Percolation Percolation models are the simplest systems that exhibit continuous phase transitions and universality. A percolating model is defined onalattice, where a given site or bond is active with probability p and inactive with probability 1 p. Forsmallvaluesofp the clusters of active sites will remain of finite size as the size of the system L grows to infinity. But when a certain critical probability p c is reached suddenly a cluster of infinite size is generated and the system is said to percolating. For values p>p c the probability that an active states belongs to the giant cluster is non-zero andreaches one for p =1. As we will see this is a first example of a continuous phase transition and a clear setting where to introduce the concept ofuniversality and Renormalization Group (RG) techniques. percolation We will consider a site percolation model on a square lattice in continuous two dimensions. Figure 2.1 shows site percolation on a L =100lattice for phase different values of the occupation probability p. Onecanclearlyseethatas transition p is increased, clusters of larger and larger size get formed. The is a critical value p c at which the first percolating, or giant, cluster is formed and the system undergoes a continuous phase transition between a non percolating 18

2 CHAPTER 2. GEOMETRIC PHASE TRANSITIONS: PERCOLATION19 Figure 2.1: Site percolation on a two dimensional square lattice. From topleft the pictures show a L =100lattice for increasing values of the occupation probability p = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9. Differentclustersare colored with different colors in order to make them visible. One clearly sees that as the probability is increased clusters of larger and larger sizes are formed. The p = 0.6 picture shows the first percolating cluster that spans all the lattice, for larger values of p the probability P (p) that a site is on the (unique) percolating cluster grows and becomes one for p =1.

3 CHAPTER 2. GEOMETRIC PHASE TRANSITIONS: PERCOLATION20 P p p Figure 2.2: The probability P (p) that an occupied site belongs to the percolating cluster as a function of p measured on a L =1000lattice. One observes the characteristic trait of a continuous phase transition happening around the critical value p c and a percolating phase. Numerically one observes p c 0.59 for site percolation on a 2d square lattice. For p>p c the probability P (p) that an occupied site belongs to the per- order colating cluster is non-zero and growing; it becomes one for p =1. Since parameter P (p) =0for p p c, P (p) plays the role of the order parameter of the continuous phase transition. We can evaluate numerically the probability that an occupied site belongs to the percolating cluster. We find the curve shown in Figure 2.2. One can observe that for p just larger than the critical value, the order parameter P (p) behaves as a power law: P (p) (p p c ) β p p + c, (2.1) where β is the order parameter critical exponent. Another quantity of interest is the correlation length ξ(p) defined, in this correlation length context, as the characteristic size of the larger clusters at occupation probability p. Observing Figure 2.1 one notices that the correlation length grows as p is increased from zero and diverges at p c. For p>p c the correlation length is defined as the characteristic size of the larger clusters (percolating cluster excluded) and thus is a monotonically decreasing function of p. Note

4 CHAPTER 2. GEOMETRIC PHASE TRANSITIONS: PERCOLATION21 Figure 2.3: The percolating cluster is a random fractal of infinite size. that we have ξ(0) = ξ(1) = 0. AnobservationofFigure2.1showsthatthese clusters can be identified with the holes of the percolating cluster. Around the critical probability the correlation length behaves as apowerlaw: ξ(p) p p c ν p p c, (2.2) where ν is the correlation length critical exponent. At criticality the percolating cluster is a random fractal of infinitesize(in fractal the limit L ). That the structure is statistically self-similar can be seen dimension observing Figure 2.3. It s interesting now to calculate the fractal dimension d f of the percolating cluster. This can be estimated from the relation: N(L) L d f, (2.3) where N(L) is the number of sites in a L L window in the percolating cluster. Since the fractal is infinite we consider larger and larger windows: d f = lim L log N(L) log L/L 0. (2.4)

5 CHAPTER 2. GEOMETRIC PHASE TRANSITIONS: PERCOLATION22 log N l Log l Figure 2.4: Log log plot of the number of sites in the percolating cluster N(L) in a window of linear size L. Thelinearfitgivesd f =1.91. Note that this is the inverse procedure than that of (1.7), i.e. L 1 l. Here L 0 = a is the microscopic lattice spacing. As shown in Figure 2.4 one finds d f 1.91, (2.5) which is not far from the known exact value d exact f = One can obtain a scaling relation relating the three critical exponentin- (hyper)scaling troduced so far. Since relation P (p) = N(L), (2.6) L d for p p + c we have: N(L) =P (p)l d (p p c ) β L d. We now choose L = ξ, comparewith(2.3)anduse(2.2)toobtain: p p c νd f ξ d f N(ξ) (p p c ) β ξ d p p c β νd,

6 CHAPTER 2. GEOMETRIC PHASE TRANSITIONS: PERCOLATION23 C S R Figure 2.5: RG transformation for the percolation problem. which gives: d d f = β ν. (2.7) Equation (2.7) is the first example of an (hyper)scaling relation between critical exponents. It is valid near criticality, in the so-called, scaling region. 2.2 RG analysis of the percolation phase transition We have seen in the previous chapter that a (deterministic) fractal can be characterized as the fixed point of a similarity transformation and that this ensures that the fractal is self similar. Can we characterize a random fractal in a similar way, in particular the self similarity of the critical percolating system? The answer to this question is yes, and the generalized transformation we are looking for are the renormalization group transformations. We need a transformation that takes a configuration of the system and gen- rg erates a new configuration probed at a different scale 1.Sincewearedealing trans- with random variables we need a procedure that replaces a collection of random variables at the smaller scale with a random variable at the larger scale: tions forma- this is the coarse graining 2 procedure. We indicate this transformation 1 This can be a smaller scale or a larger scale. For the moment we will focus on increasing the scale. 2 The rescaling of a phenomenon into units or cells of size close totheuncertaintyof measurement.

7 CHAPTER 2. GEOMETRIC PHASE TRANSITIONS: PERCOLATION24 as C. Since the coarse graining procedure enlarges the system byafactor λ 1 > 1 we need to compensate this transformation with a scale transformation with scaling factor λ, or more generally with a similarity S. Thus: renormalization group transformation are the composition of acoarse grainingtransformationwithasimilarity: R = S C. (2.8) In the percolation problem we can define the RG transformation C using a majority rule in the following way: R and R where the similarity is just a scale transformation of scale factor λ = 1 2.Ifwe apply the RG transformation to a configuration generated with aprobability p we obtain a configuration generated with probability Rp. Inparticular,the probability for a state to be occupied can be read of from the first picture above: Rp = p 4 +4(1 p)p 3 +2(1 p) 2 p 2. (2.9) Note that we are using a vertical spanning majority rule [1]. Note also that this transformation is not exact. Why? Non trivial fixed points of the RG transformation correspond tothephase fixed transition. This can be seen from the fact that the correlation length is points reduced by a RG transformation as Rξ = λξ; atafixedpointwehave: phase transition λξ = ξ ξ =0,. (2.10)

8 CHAPTER 2. GEOMETRIC PHASE TRANSITIONS: PERCOLATION25 ξ =0corresponds to the p =0and p =1states, while ξ = corresponds to the phase transition at p = p c.thus: or using (2.9): Rp c = p c, (2.11) p c = p 4 c +4(1 p c)p 3 c +2(1 p c) 2 p 2 c. (2.12) Equation (2.12) has three solutions in the range 0 p 1: p c =0 p c = 1 ( ) p c =1. (2.13) 2 The non trivial solution corresponds to the critical point: wepredictedthe existence of the phase transition. Under a RG transformation the correlation length transforms as: or λξ λ p p c ν = Rp p c ν, (2.14) ν = log λ log p pc Rp p c For p near p c we can expand the RG transformation:. (2.15) Rp = R(p c (p p c )) p c (p p c )Lp c + O ( (p p c ) 2), where L is the linearized RG transformation (in this case we simply have L = R )andweusedrp c = p c.relation(2.15)becomes: ν = Inserting the values (2.13) in (2.16) gives the estimate: log 1/λ log Lp c. (2.16) ν =1.64, (2.17) which is to be compared with the known exact result ν exact = Why 3 this value is not exact? Can we do better? The scaling relation (2.7) and the value (2.5) allow an estimate of the order parameter critical exponent: which compares well with β exact = β = ν(2 d f )=0.15, (2.18) computing the critical exponent ν

9 Bibliography [1] H. Gould, J. Tobochnik and W. Christian, An Introduction to Computer Simulation Methods (2005) (Chapter 12). 26