The annealed Ising magnet on percolation clusters is studied by means of a mapping into a Potts-Ising model and with the Migdal-Kadanoff renormalization-group method. The phase diagram is determined in the three-dimensional parameter space of the Ising coupling K, the bond-occupation probability p, and the fugacity q, which controls the number of clusters. Three phases are identified: percolating ferromagnetic, percolating paramagnetic, and nonpercolating paramagnetic. For large q the phase diagram includes a multicritical point at the intersection of the Ising critical line and the percolation critical line. In the case of random bond percolation (q = 1) the Ising critical line is: p(1 - e-2K) = 1 - exp(- 2L(C)), where Lc is the pure-Ising-model critical coupling

Berger, T.

Bowman, D.

Gujrati, P. D.

Kaufman, Miron

English

Cleveland-Marshall College of Law

Cleveland State UniversityEngagedScholarship@CSUPhysics Faculty Publications Physics Department4-1-1994Phase Diagram of the Ising Model on PercolationClustersMiron KaufmanCleveland State University, m.kaufman@csuohio.eduT. BergerP. D. GujratiD. BowmanFollow this and additional works at: https://engagedscholarship.csuohio.edu/sciphysics_facpubPart of the Physics CommonsHow does access to this work benefit you? Let us know!Publisher's StatementCopyright 1994 American Physical Society. Available on publisher's site at http://link.aps.org/doi/10.1103/PhysRevB.49.9583.Repository CitationKaufman, Miron; Berger, T.; Gujrati, P. D.; and Bowman, D., "Phase Diagram of the Ising Model on Percolation Clusters" (1994). Physics FacultyPublications. 5.https://engagedscholarship.csuohio.edu/sciphysics_facpub/5This Article is brought to you for free and open access by the Physics Department at EngagedScholarship@CSU. It has been accepted for inclusion inPhysics Faculty Publications by an authorized administrator of EngagedScholarship@CSU. For more information, please contactlibrary.es@csuohio.edu.Original CitationKaufman, Miron, T. Berger, P. D. Gujrati, and D. Bowman. "Phase Diagram of the Ising Model on Percolation Clusters." Physical ReviewB 49 (1994): 9583-9585.PHYSICAL REVIE%' B VOLUME 49, NUMBER 14Phase diagram of the Ising model on percolation clusters1 APRIL 1994-IIMiron Kaufman and Jimmy E. Touma*Department ofPhysics, Cleueland State Uniuersity, Cleueland, Ohio 44115(Received 12 November 1993)The annealed Ising magnet on percolation clusters is studied by means of a mapping into a Potts-Isingmodel and with the Migdal-Kadanoff renormalization-group method. The phase diagram is determinedin the three-dimensional parameter space of the Ising coupling E, the bond-occupation probability p,and the fugacity q, which controls the number of clusters. Three phases are identified: percolating fer-romagnetic, percolating paramagnetic, and nonpercolating paramagnetic. For large q the phase diagramincludes a multicritical point at the intersection of the Ising critical line and the percolation critical line.In the case of random bond percolation (q =1) the Ising critical line is: p(1 —e ~)=1—exp( —2L&),where L& is the pure-Ising-model critical coupling.I. INTRODUCTION II. MODEL AND EXACT RESULTSThe Ising model defined on percolation clusters isrelevant to a variety of physical phenomena such asphase separation in a binary fluid mixture in a porousmedium' and amorphous magnets. In this paper we ana-lyze the annealed Ising model on percolation clusters; i.e.,we assume that the relaxation times for spinconfigurations and for percolation configurations areshort compared to the characteristic time of measure-ment. The percolation part of this model is a generaliza-tion of the random-bond percolation which includes afugacity q that controls the number of clusters. This gen-eralized percolation model is employed in Monte Carloand renormalization-group studies of the Potts model.In Sec. II we define the model and present some exactresults. In the particular case of the square lattice the Is-ing critical line for the Ising model on random-bond(fugacit q = 1) percolation clusters is p (1—e «)=2— 2, where E is the Ising coupling constant and p isthe bond-occupation probability.In Sec. III we present a numerical analysis of the mod-el in two-dimensions (2D) based on the Migdal-Kadanoffrenormalization-group technique. Though only approxi-mate for the 2D Bravais lattice this technique is exactfor a hierarchical lattice. Hence the exact resultspresented in Sec. II are satisfied by this technique thusenhancing our confidence that the numerical findings arequalitatively correct. We present the phase diagram inthe Ising coupling E, the bond-occupation probability pplane for fixed fugacity q. For all q we find three phases:the paramagnetic nonpercolating, the paramagnetic per-colating, and the ferromagnetic percolating. If q is smallenough the Ising critical line and the percolation criticalline intersect at zero temperature. If q is sufBciently largethe two lines meet at a multicritical point at finite tern-perature. In this case the ferromagnetic percolatingphase can be accessed direct1y from the paramagneticnonpercolating phase by crossing a line in the universali-ty class of the 2q-state Potts model.The summary of our findings as well as a critique arepresented in Sec. IV.where (i,j ) denotes a pair of sites connected by a bond.The partition function for the annealed Ising magnet onpercolation clusters isZ y ~ cw BZconf (2)where 8 is the number of bonds, C is the number of clus-ters including single-site clusters, and w =p/(1 —p).By using the Kasteleyn-Fortuin expansion for thePotts model we map the model defined in Eqs. (1) and (2)into a Potts-Ising model. At each site i there is the Pottsspin cr; taking q values and the Ising spin s; taking twovalues. The partition function isZ=Tr(, } Tr( } g [1+5(o;,oj)w exp[K(ssi —1)]],(i,j )(3)where 5 is the Kronecker delta and g is a product overall lattice edges. By writing the factor on the right-handside of Eq. (3) as a Boltzmann factor1+5(cr, , tr )w exp[E. (s;s- —1)]=exp[J5(o;,o, )+L5(o;,at )s;s, ], (4)we findJ=—,'ln[(1+ w)(1+ we )],L =—,'ln[(1+w)/(1+we )] .At each site i of a lattice there is an Ising spin s; =+1.The lattice sites are connected by bonds according to thegeneralized bond-percolation model: a bond is presentwith a probability p and percolation configurations withthe same number of bonds are distinguished according tothe number of clusters C as the configuration probabilityis proportional to q . For a given percolationconfiguration the partition function of the Ising model isZl""'=Tr(, } exp E g(,. ,}(s;s, —1)0163-1829/94/49(14)/9583(3)/$06. 00 49 9583 1994 The American Physical Society9584 MIRON KAUFMAN AND JIMMY E. TOUMA 49The model is now given by the HamiltonianH—lke T= g [J5(o;,o~ )+L5(o;,o~ )s;s, ] .(i,j) (6)Z,', =[Tr~ Tr, Za, zk ]The following recursion equations are obtained by usingEqs. (11)and (12):Certain exact results can be obtained by studying theIsing-Potts Hamiltonian of Eq. (6). If K =0, i.e., infinitetemperature, from Eq. (5) it follows that L =0, and themodel reduces to the q-state Potts model. The Pottscritical coupling for the square lattice is (Ref. 5)Jc=ln(1+v q ). Equation (5) then implies wc=&q orthe threshold bond-occupation probability ispc =&q /(1+ &q ). For the random-bond percolationq =1 and thus pc= —,'.If K = ~, or zero temperature, Eqs. (5) imply J =L,and thus the Hamiltonian becomesH/k—AT= g 2J5(o;, oj)5( ;s, s),(i,j )because 1+s;sj=25(s;,sj). This is the 2q-state Pottsmodel. For the square lattice the critical coupling isJc=—,'ln(1+v 2q ) which corresponds to wc=&2q orpc=~2q /(1+&2q ). For the random-bond percolationq =1 and thus pc=2 —~2.If p=1 or w= ~ from Eq. (5) we find J= ~ andL =K. Since 5(o';, oj)=1 if J=ao, the Hamiltonianreduces to (apart from an additive constant) the pure Is-ing model: H/k&T—=QKs;s . The critical couplingfor the square lattice is Kc =—,' ln(1+ &2).For the random-bond percolation q =1, i.e., the Pottsspin takes one value. Consequently 5(o;,o )=1 and theHamiltonian of Eq. (6} becomes the Ising Hamiltonian:HlkttT=Q— (Ls, s, +J). We find by using Eq. (5) theIsing critical line in the parameter w, E plane:exp(2Lc ) =(1+w)/(1+ we ),where Lc is the critical coupling of the pure Ising model.In the parameter plane of bond-occupation probabilityp=w/(1+w) and r =1—e this equation becomesw'=[1+—,'(w +w e )/(q+w+we )] —1,w'e =[1+w e /(q+w+we )] —1 .(13)(14)The phase diagram in the parameter plane ofr =1—e and p =w/(1+w) at fixed q is determined byfollowing the recursion Eqs. (13) and (14). For all q thereare three phases: (i) the nonpercolating paramagneticphase governed by the fixed point at r =0 (K =0 orT= ~) and p =0; (ii) the percolating paramagnetic phasegoverned by the fixed point at r =0 (K =0 or T= co) andp =1; (iii) the percolating ferromagnetic phase governedby the fixed point at r =1 (K = ~ or T =0) and p = l.The ferromagnetic phase is always inside the percolatingphase. In other words, there is no nonpercolating ferromagnetic phase. This result is consistent with our ex-pectation that long-range order (ferromagnetism) requiresan infinite percolating cluster.For all q values the percolating ferromagnetic phase isseparated from the percolating paramagnetic phase bythe Ising critical line governed by the fixed point at p =1and K =0.6094 [exact value for square lattice coupling is—,' ln(1+ ~2)].The percolating paramagnetic phase is separated fromthe nonpercolating paramagnetic phase by the percola-tion critical line which belongs to the universality class ofthe q-state Potts critical point. The fixed point governingthis line is located at r =0 (K =0 or T= oo }.Within the Migdal-Kadanoff renormalization group wefind (see Fig. 1) that for q + 8 the percolation and Isingcritical lines meet at a zero temperature (r =1) pointwhich belongs to the universality class of the 2q-statePotts model. The critical exponent corresponding to theHow along the r =1 axis is the thermal exponent of thepr = 1 —exp( 2LC ) . — (9) 1.0For the square lattice exp(2LC }= 1+&2, and thus Eq. (9)reduces to(10)III. SOLUTION AND PHASE DIAGRAM 0.8—We study the model by using the Migdal-Kadanoffrenormalization-group scheme. The main asset of thisscheme, besides its simplicity, is the fact that it is exactfor the diamond hierarchical lattice. Hence physical re-quirements such as the positivity of the heat capacity andthe exact results discussed in Sec. II above are satisfied bythis renormalization scheme.The Boltzmann weight for bond connecting sites i andJ 1SZ, =1+5(o,,o . )w exp[K(s;sj —1)] .The renormalized Boltzmann weight for a 20 lattice is0.7I0.60.0 0.2I0.4 0.6I0.8 1.0FIG. 1. Phase diagram for q = 1: solid line is Ising criticalline; dotted line is percolation critical line. The two lines meetat T =0 at a point in the universality class of the 2q-state Pottsmodel.49 PHASE DIAGRAM OF THE ISING MODEL ON PERCOLATION. . . 95851.000.98-0.96—exponents determined by linearizing the recursion equa-tions are positive; i.e., this is an unstable fixed point. Thecritical line staring at this multicritical point and endingat the zero temperature (r =1}critical fixed point is inthe universality class of the 2q-state Potts model. Thislatter line separates the percolating ferromagnetic phasefrom the nonpercolating paramagnetic phase.0.94- IV. SUMMARY AND CRITIQUE0.92—0.900.0IONI0.4I0.6I0.8 1.0FIG. 2. Phase diagram for q =20: solid line is Ising criticalline; dotted line is percolation critical line; the double solid lineis in the universality class of the 2q-state Potts model. Thethree lines meet at a multicritical point.2q-state Potts model. This is consistent with the exact re-sult derived in Sec. II according to which at T=O themodel reduces to the 2q-state Potts model. For a smallrange of p values we observe the following reentrancephenomenon. As T is lowered (r increased) at fixed p thefollowing phases are encountered: the percolatingparamagnetic, the nonpercolating paramagnetic, the per-colating paramagnetic, and the percolating ferromagner-tic phase.For q )8 the percolation critical line (universality classof q-state Potts model} meets the Ising critical line at anovel multicritical point, as shown in Fig. 2. The criticalWe presented a study of the phase diagram of the Isingmodel on percolation clusters. The percolation part ofthis problem is a generalization of the random-bond per-colation which includes a fugacity q controlling the num-ber of clusters. By mapping the model on a Potts-Isingmodel and studying the latter we derived a number of ex-act results. In particular for the Ising model on randompercolation clusters (q =1) the Ising critical line isp(1 —e «) =1 exp—( 2Lc), —where Lc is the pure Isingmodel critical coupling.The phase diagram of the model was determined by us-ing the Migdal-KadanofF' renormalization-group scheme.For small q the Ising critical line and the percolation crit-ical line intersect at a 2q-state Potts critical point locatedat T =0. For large q the two lines meet at a novel mul-ticritical point at nonzero temperature. In this case thepercolating ferromagnetic phase can be accessed from thenonpercolating paramagnetic phase across a line in theuniversality class of the 2q-state Potts critical point.The scheme used in this work gives continuous transi-tions for all q while it is known that the 2D Potts modelhas a discontinuous transition for q )4. We do then ex-pect that the multicritical point in Fig. 2 is replaced by acritical end point. ' Numerical simulations are needed toverify the predictions of this work.'Present address: Department of Physics, Auburn University,Auburn, Alabama 36849-5311.D. W. Pohl and W. I. Goldburg, Phys. Rev. Lett. 48, 1111(1982); J. V. Maher, W. I. Goldburg, D. W. Pohl, and M.Lanz, ibid. 53, 60 (1984); L. V. Entov, V. A. Levchenko, andV. P. Voronov, Int. J. Thermophys. 14, 221 (1993).M. Kaufman and D. Andelman, Phys. Rev. B 29, 4010 (1984).3R. H. Spvendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1986).4C.-K. Hu, in Proceedings of the 1986Summer School on Statistical Mechanics, edited by C.-K. Hu (Institute of Physics,Academia Sinica and Physical Society of the Republic of Chi-na, Taipei, 1987), p. 91; T. A. Larsson, J. Phys. A 19, 2383(1986).5F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982).sA. A. Migdal, Zh. Eksp. Teor. Fiz. 69, 1457 (1975) [Sov. Phys.JETP 42, 743 (1976)];L. P. Kadanoff, Ann. Phys. (N.Y.) 100,359 (1976).7A. N. Berker and S. Ostlund, J. Phys. C 12, 4961 (1979).R. B. Griffiths and M. Kaufman, Phys. Rev. B 26, 5022 (1982);M. Kaufman and R. B.Griffiths, ibid. 30, 244 (1984).P. W. Kasteleyn and C. M. Fortuin, J. Phys. Soc. Jpn. 26, 11(1969).M. Kaufman, R. B. Griffiths, J. M. Yeomans, and M. E. Fish-er, Phys. Rev. B 23, 3448 (1981);M. E. Fisher and P. J. Up-ton, Phys. Rev. Lett. 65, 2402 (1990); R. R. Netz and A. N.Berker, Phys. Rev. B 47, 15019 (1993).

Phase Diagram of the Ising Model on Percolation Clusters

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Phase Diagram of the Ising Model on Percolation Clusters

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