Percolation and the Potts Model

Transcription

1 Chapter 7 Percolaton and the Potts Model c 2012 by W. Klen, Harvey Gould, and Jan Tobochnk 5 November Introducton To ntroduce bond dluton correctly and use bond dluton to map the percolaton model onto the thermal, we need more powerful tools. These tools are provded by the Kasteleyn- Fortun mappng [Kasteleyn and Fortun 1969], whch relates the percolaton model to a partcular lmt of the Potts model [Stauffer 1979]. 7.2 Random Bond Percolaton and the Potts Model The Potts model s a generalzaton of the Isng model. As n the Isng model we have a lattce wth a spn at each vertex. In the Potts model however, the spns can be n one of s states rather than the two states of the Isng model. The Hamltonan βh s ( δσ σ j 1 ) ( + h p δσ 1 1 ). (7.1) βh = J j The σ specfy the drecton of the Potts spn, δ σ σ j s the Kronecker delta and s one f the two spns at and j are n the same state and zero otherwse. The term δ σ 1 s zero unless the Potts spn at ste s n the state we have desgnated as σ = 1 and h p s the Potts appled feld. The values of J and h p are always taken to be greater than or equal to zero. 81

2 CHAPTER 7. PERCOLATION AND THE POTTS MODEL 82 The partton functon z s z = {σ} exp( βh), (7.2) where the sum s over the confguratons of the Potts spns or Potts states. The free energy per spn f(j, h p, s) s f(j, h p, s) = k BT ln(z), (7.3) N where N s the number of spns on the lattce and we have made the s dependence explct. The Kasteleyn-Fortun mappng states that the dervatve of f(j, h p, s) wth respect to s evaluated at s = 1 s the mean number of fnte clusters per ste (generatng functon) n the random bond percolaton model. To obtan ths result we frst rewrte z as { [ e J [ (1 δ σ σ j ) + δ σ σ j }{ e h p ] } (1 δ σ 1) + δ σ 1. (7.4) z = {σ} j We regroup terms and wrte { [ (1 e J )δ σ σ j + e J]}{ [ (1 e h p )δ σ 1 + e hp]}. (7.5) z = {σ} j If s = 1 we have from Eq. (7.4) that z = 1. Consequently df(j, h p, s) ds = k BT dz N ds. (7.6) To nvestgate the dervatve of the partton functon wth respect to s we start wth the smpler case of h p = 0. From Eq. (7.5) the partton functon becomes { [ (1 e J )δ σ σ j + e J]}. (7.7) z = {σ} j We expand ths product and look at the term that results from all factors comng from the e J term, whch we call z 1 z 1 = e JN b, (7.8) {σ} where N b s the number of possble bonds n the lattce. That s, N b equals the number of places where a bond could be. The sum over {σ} n Eq. (7.8) results n a factor of s N, where N s the number of lattce stes or spns. The reason for ths factor s that there are no delta functons to restrct the sum. From these consderatons t follows that dz 1 ds = Ne JN b. (7.9)

3 CHAPTER 7. PERCOLATION AND THE POTTS MODEL 83 We can nterpret Eq. (7.9) as follows. Because J 0, the quantty e J can be thought of as the probablty 1 p that a bond s not occuped or flled. Then the term e JN b s the probablty that there are no flled bonds on the lattce. Although we have defned a bond percolaton problem, we count the clusters by countng stes, so f there are no bonds each ste s an solated one ste cluster and N s the number of clusters when all bonds are unoccuped. What happens f h p 0? The factor z hp = [ (1 e h p )δ σ 1 + e hp] (7.10) can be expanded to gve a sum of terms. Each term connects a subset n of the N Potts spns to the ghost, and each spn connected to the ghost s no longer free to contrbute a factor of s n the sum over {σ} because t s fxed by the δ σ 1 term to be n the state σ = 1. Because h p 0, the factor 1 e hp can be nterpreted as the probablty that a spn s attached to the ghost. If we multply the two factors z 1 and z hp, dfferentate wth respect to s and set s = 1, we obtan d(z 1 z hp ) ds = e JN b N N! (N n) n!(n n)! e hp(n n) (1 e hp ) n. (7.11) n=0 Equaton (7.11) s the mean number of clusters wth no bonds present; the sum s over all confguratons of the ghost bonds. Consder the case n whch we take from the product n Eq. (7.7) only those terms wth one factor of (1 e J ). We label ths contrbuton z 2. We have z 2 = N b e J(N b 1) (1 e J ) {σ} δ σaσ b. (7.12) The sum over {σ} gves a factor s N 1 ; whch s a factor s N 2 for the one ste clusters tmes a factor s for the sngle two ste cluster created by the δ σaσb lnkng the Potts states of the par of spns at {a, b}. Wth the prevous nterpretaton of e J, the factor e J(Nb 1) (1 e J ) s the probablty that only one bond s present and N b s the number of places that one bond can be placed. As before, we can nclude the ghost feld by multplyng z 2 by z hp n Eq. (7.10), takng the dervatve of the product wth respect to s as n Eq. (7.11) and settng s = 1. d(z 2 z hp ) N ds = e J(Nb 1) (1 e J ) n=0 mn{2,n} m=α (N 2)!(1 + δ m1 ) (N n 1 + δ m2 ) (n m)!(n n + m)!. (7.13)

4 CHAPTER 7. PERCOLATION AND THE POTTS MODEL 84 The upper bound on the second sum mn{2, n} means that the sum over m goes from α to the mnmum of the two values, 2 or n. The lower bound on the second sum s α = 0 f n N 2. Or α = 1 f n = N 1 and α = 2 f n = N. We could contnue ths process but at ths stage the pont should be clear. Namely, the dervatve of f(j, h p, s) wth respect to s evaluated at s = 1 s the mean number of clusters per ste n the random bond percolaton model. 7.3 Correlated Ste-Random Bond Percolaton and the Dlute Potts Model We can generalze the mappng dscussed n Sec. 7.2 to the other percolaton models dscussed n Chapter 6. To do so we use the dlute s state Potts model. Before ntroducng ths model we need to express the Isng Hamltonan Eq. (1.4) n terms of lattce gas varables. Ths rewrtng reflects the assocaton of a down spn wth an occuped ste and an up spn wth an empty ste. We ntroduce the occupaton number n, whch s 1 f a ste s occuped and 0 f t s not. The n are related to the Isng spn varable s, whch can take on the values ±1, by The Isng Hamltonan βh I = K j n = 1 s. (7.14) 2 s s j + h s (7.15) can be rewrtten usng Eq. (7.14) as βh I = (ck + h)n βh LG, (7.16) where the coordnaton number c = N b /N. and βh LG = K LG j n n j K LG 2 (n + n j ) 2h j n. (7.17) The term K LG = 4K and the lattce gas Hamltonan s often wrtten n the form βh LG = K LG n n j n, (7.18) j where the chemcal potental s = ck LG + 2h. (7.19)

5 CHAPTER 7. PERCOLATION AND THE POTTS MODEL 85 We now defne the dlute Potts model [Murata 1979, Conglo and Klen 1980] by the Hamltonan βh DP = J (δ σ σ j 1)n n j + h p (δ σ 1 1)n + H LG. (7.20) j Ths model has occuped stes (n = 1) and empty stes (n = 0). Each occuped ste has a Potts spn whch can, as wth the random pure Potts model, take one of s states. The partton functon s z DP = e βh DP, (7.21) {σ} {n} and the free energy per ste s gven by f DP (J, K LG, h, h p, s) = k BT N ln(z DP), (7.22) where we have agan made the s dependence explct. The dstrbuton of the occuped stes s governed by H LG. All of the percolaton models we dscussed n Chapter 6 can be obtaned by usng the Kasteleyn-Fortun mappng and takng the approprate lmts of the couplng constants J and K LG. A straghtforward adaptaton of the technque used to obtan random bond percolaton from the pure Potts model results n the relaton df DP (J, K LG, h, h p, s) ds = G(J, K LG, h, h p ), (7.23) where G(J, K LG, h, h p ) s the mean number of clusters per ste n a percolaton model where the occuped stes are dstrbuted accordng to the lattce gas Boltzmann factor and the bonds between occuped stes at and j wth a probablty p b(j) = 1 e J. (7.24) If we set K LG = 0, the stes are dstrbuted at random and the Kasteleyn-Fortun mappng produces the mean number of clusters per ste for the random ste-random bond percolaton model. If K LG = 0 and J =, t follows from Eq. (7.24) that a bond s present between two occuped stes wth probablty one. We would then have random ste percolaton. To obtan the random bond model we can ether set K LG = and let h 0 or let K LG be any nonzero value and set h = so that all sghts are flled wth probablty one. 7.4 Mappng of Thermal to Percolaton Models: The Crtcal Pont For the purpose of mappng the correlated ste-random bond percolaton model onto the crtcal pont of the Isng model we requre the full dlute state Potts model. We wll frst

6 CHAPTER 7. PERCOLATION AND THE POTTS MODEL 86 consder the smpler case of h = h p = 0. The dlute Potts Hamltonan from Eqs. (7.15) and (7.20) s βh DP = j { J[δ σ σ j 1]n n j + K LG n n j K } LG 2 [n + n j ]. (7.25) We now chose so that J = K LG 2, (7.26) p b = 1 e K LG/2. (7.27) The Potts Hamltonan for a par of stes and j wth J gven n Eq. (7.26) s βh DP(j) = K LG 2 [δ σ σ j 1]n n j + K LG n n j K LG 2 [n + n j ]. (7.28) Ths Hamltonan s equvalent to an s + 1 state pure Potts model wth the s + 1 states beng b 0 n = 0; that s the state s empty, and b α, α = 1 s correspond to n = 1 and σ at the ste takng on the possble states 1 s. To see that ths s a proper Potts model we check Eq. (7.26) to see f the s + 1 states of the Potts varable b have the rght nteracton. Consder the state n = n j = 0. From Eq. (7.26), βh DP(j) = 0. The state n = n j = 1 and σ = σ j also has H DP(j) = 0, whch corresponds to the varables b = b j n the s + 1 pure Potts model. If n = 1 and n j = 0 or n = n j = 1 and σ σ j, then βh DP(j) = K LG /2. Ths corresponds to b b j. These results can be summarzed as follows: If J = K LG /2, then H DP (s) = H P (s + 1) and βh P (s + 1) = βk LG 2 [δ b b j 1]. (7.29) We stll need to show that the percolaton model s somorphc to the thermal model. We wll not qute do that. Instead, we argue that at the thermal crtcal pont the percolaton model also has a transton and that all the percolaton exponents are the same as the Isng model. Ths latter result can be shown n several ways. A renormalzaton group argument has been used and s outlned n Conglo and Klen [1980] and n more detal n Monette s thess [1990]. We wll nvestgate the relaton between the par connectedness functon and the par correlaton functon for T > T c and below the percolaton threshold (fnte cluster regon). The par connectedness functon plays the same role n percolaton that the par correlaton functon plays n thermal models [Essam 1982, Klen and Stell 1985]. For j

7 CHAPTER 7. PERCOLATION AND THE POTTS MODEL 87 example, the ntegral of the par connectedness functon g p (r) s related to the mean sze of the fnte clusters. That s, 0 g p (r)d r χ p. (7.30) In the same way the par correlaton functon g(r) s related to the susceptblty. 0 g(r)d r χ T. (7.31) The par connectedness functon s defned as the probablty that occuped stes at a and b belong to the same cluster. It s smple to see that for the dlute s state Potts model [ g p (r) = d {σ} {n} [1 δ ] σ aγ][1 δ σb γ]n a n b e βh DP, (7.32) ds where γ s a fxed but arbtrary Potts state and z DP s gven n Eq. (7.21). The way to see ths s to note that f the Potts spns at a and b are n dfferent clusters, there s no addtonal restrcton on the sum over ther confguratons other than they cannot be n the state γ, whch s mposed by the factors (1 δ σaγ)(1 δ σb γ). Such terms produce a factor (s 1) 2 and hence vansh n the lmt s 1. However, f there s a connecton between the spns at a and b so that they are n the same cluster, then the sum over the Potts states produces only a factor of (s 1) whch s elmnated by dfferentaton wth respect to s. Ths factor of (s 1) makes t unnecessary to dfferentate the denomnator n Eq. (7.32) so that t becomes equal to z LG n the lmt s 1. These consderatons mply that the rght-hand sde of Eq. (7.32) s the sum of confguratons, weghted by the dlute Potts Boltzmann factor, that have a dagram connectng the spns at a and b. From our mappng Eq. (7.32) can also be wrtten as g p (r) = d ds [ {σ} z DP {n} [1 δ ] σ aγ][1 δ σb γ]e βh P (s+1) z P (s + 1). (7.33) We are nterested n the asymptotc form of g p (r) whch s known [Essam 1972] to scale n the same way as the par correlaton functon. Namely, lm g p(r) e r/ξp. (7.34) r d 2+ηp r Consequently, we can wrte [ ] g p (r) = d δ σaγδ σb γe βh P (s+1) ds z P (s + 1) {σ} {n}, (7.35)

8 CHAPTER 7. PERCOLATION AND THE POTTS MODEL 88 and obtan the same scalng form as Eq. (7.34). Ths results holds because the terms we have neglected n Eq. (7.33) are ndependent of r. Now consder the functon {b} g(r) = δ b aγδ bb γe βh P (s+1). (7.36) z P (s + 1) Equaton (7.36) s the correlaton functon of the Isng model n the lmt s 1. We know that ths functon s also composed of only connected graphs (see Chapter 2). Moreover, they are the same graphs as n Eq. (7.35) weghted by the same Boltzmann factor. Hence the scalng law should be the same up to a constant. Ths result mples T = T c, ν = ν p, and η = η p. From two exponent scalng we conclude that all the exponents are equal. Ths mappng s however not an somorphsm. The crtcal exponents of the percolaton and thermal quanttes are the same, but t s known numercally that the ampltudes are dfferent [Kertesz et al. 1983]. We can not only make the mappng nto an somorphsm, but make the mappng work at any value of the magnetc feld (not just h = 0) by the smple technque of symmetrzng the percolaton model [Hu 1984]. If we add to the dlute Potts Hamltonan n Eq. (7.25) a term of the form J[δ σ σ j 1](1 n )(1 n j ) + h p [δσ 1 1](1 n ), (7.37) then there are clusters between empty as well as full stes. In Isng language there are up spn as well as down spn clusters. We could also add a term to the Hamltonan n Eq. (7.15) of the form K LG [1 n ][1 n j ], (7.38) whch changes the relaton between K LG and the Isng couplng constant K to K LG = 2K. The relaton between the bond probablty p b and the lattce gas couplng constant also changes to p b = 1 e KLG rather than the relaton n Eq. (7.27). The meanng behnd the mathematcs s that the percolaton clusters are a physcal realzaton of the fluctuatons. At the crtcal pont the clusters assocated wth the dvergent connectedness length are the fluctuatons assocated wth the dvergent susceptblty n the thermal model. We can use a smlar mappng to generate a percolaton model for the spnodal. 7.5 Mappng of Thermal to Percolaton Models: The Spnodal To map the spnodal onto the percolaton model [Klen 1980] t s more convenent to use the Landau-Gnsburg free energy. We can use ths form of the free energy because the

9 CHAPTER 7. PERCOLATION AND THE POTTS MODEL 89 spnodal s ntrnscally a mean-feld construct. We begn by wrtng down the Landau- Gnsburg free energy for the dlute s state Potts model derved by Conglo and Lubensky [1980]. A dervaton of ths form of the dlute Potts free energy f p can be found n the thess of L. Monette [1990]. f p = r 1 2 (s 1)ψ2 p (s 1)h p ψ p w 1s(s 1)(s 2)ψ 3 p w 2s(s 1)φψ 2 p + f(φ). (7.39) The parameters r 1 and w 2 are gven by ( r 1 = c 1 c J 2 and ), (7.40) w 2 = c3 2 JK1/2. (7.41) Note that we have used the Isng couplng constant K. The parameter w 1 s of no nterest n ths dscusson except that t does not vansh at the spnodal. The term f(φ) s the standard Landau-Gnsburg free energy. We use the Hubbard-Stratonovch transformaton [Amt 1984] to express the parameters n f(φ), Eq. (1.9), n terms of the Isng couplng constant. In partcular, the value of φ at the spnodal s [Monette 1990] (ck 1)c φ s = ± c 2. (7.42) K We have obtaned φ s as n Chapter 5 and used the relaton between the Landau-Gnsburg parameters and the Isng or lattce gas parameters obtaned from the Hubbard-Stratonovch transformaton. The order parameter φ s proportonal to the magnetzaton m. To obtan the proportonalty constant we note that as T 0 or K, the value of the magnetzaton at the spnodal must approach ±1. Remember that h s the appled feld dvded by k B T and consequently ether no spn wll flp f 2cK > h or they all do f 2cK h. Note that 2c s the number of bonds out of a vertex. The spnodal value of φ n Eq. (7.42) does not approach 1 as K but nstead φ s approaches zero. Therefore the relaton between φ and m must be m = ck 1/2 φ s, (7.43) whch approaches 1 as K. To obtan the percolaton generatng functon we agan use the Kastaleyn-Fortun mappng. We dfferentate f p n Eq. (7.39) wth respect to s and obtan G(J, K, h) = 1 2 ( r 1 + w 2 φ)ψ 2 p h p ψ p w 1ψ 3 p. (7.44)

10 CHAPTER 7. PERCOLATION AND THE POTTS MODEL 90 The percolaton transton occurs when the coeffcent of the ψ 2 p term vanshes. If we use Eqs. (7.40) and (7.41), Eq. (7.44) mples that ( c 1 c J ) = c 3 2JK 1/2 φ s, (7.45) 2 where we have set φ = φ s because we are nterested n the value of the bond probablty that makes the percolaton transton concde wth the spnodal. We use Eq. (7.43) and reduce Eq. (7.45) to 2 cj =, (7.46) 1 + m s where m s s the value of the magnetzaton at the spnodal. We can rewrte Eq. (7.46) n a more famlar form by multplyng the numerator and denomnator of the rght-hand sde of Eq. (7.46) by 1 m s and usng Eqs. (7.42) and (7.43) to express m 2 s as a functon of K. We have J = 2(1 m s )K. (7.47) We have assumed that the up spns are metastable and the down spns are the stable phase. If we want to express J n terms of the densty of the up spns ρ, we have or Ths value of J mples the bond probablty 1 m s = 2(1 ρ s ), (7.48) J = 4(1 ρ s )K = K LG (1 ρ s ). (7.49) whch reduces to the value n Eq. (7.27) at the crtcal pont. p b = 1 e K LG(1 ρ s), (7.50) Suggestons for Further Readng P. W. Kastaleyn and C. M. Fortun, J. Phys. Soc. Jpn. (Suppl.) 26, 11 (1969). D. Stauffer, Scalng theory of percolaton clusters, Phys. Reports 54, 1 7 (1979). K. K. Murata, Hamltonan formulaton of ste percolaton n a lattce gas, J. Phys. A 12, (1979). A. Conglo and W. Klen, Clusters and Isng crtcal droplets: a renormalsaton group approach, J. Phys. A 13, (1980).

11 CHAPTER 7. PERCOLATION AND THE POTTS MODEL 91 J. Essam, Percolaton and cluster sze, n Phase Transtons and Crtcal Phenomena, Vol. 2, edted by C. Domb and M. S. Green, Academc Press (1972). W. Klen and G. Stell, Integral herarches and percolaton, Phys. Rev. B 32, (1985). J. Kertesz, D. Stauffer, and A. Conglo, Clusters for random and nteractng percolaton, Ann. Israel Phys. Soc. 5, (1983). C. K. Hu, Percolaton, clusters, and phase transtons n spn models, Phys. Rev. B 29, (1984). A. Conglo and T. Lubensky, epslon expanson for correlated percolaton: applcatons to gels, J. Phys. A 13, (1980). L. Monette, Ph.D. thess, Boston Unversty (1990). D. Amt, Feld Theory, the Renormalzaton Group and Crtcal Phenomena, second edton, World Scentfc (1984). W. Klen, Smulaton studes of classcal and non-classcal nucleaton, n Computer Smulatons n Condensed Matter Physcs III, edted by D. P. Landau, K. K. Mon, and H. B. Schütter, Sprnger-Verlag (1990).