Phase Diagrams of the Dilute Ashkin-Teller Model

A. BENYOUSSEF and M. LOULIDI: Phase Diagrams of the Dilute Ashkin-Teller Model 20 1 phys. stat. sol. (b) 182, 201 (1994) Subject classification: 75.10 and 75.30 Luborutoire de Mugnitisme et Physique des Huutes Energies, Dkparternent de Physique, FacultC des Sciences, Rabat â1 Phase Diagrams of the Dilute Ashkin-Teller Model BY A. BENYOUSSEF and M. LOULIDI The phase diagram of the dilute Ashkin-Teller model is investigated within a real-space renormalization scheme. Using two probability distributions which correspond to the two-spin coupling, K *, dilution and the four-spin coupling, K,, dilution, a rich variety of phase diagrams are obtained. The percolation phase diagram is obtained at T = 0. It describes a system oftwo types oflinks with different concentrations. 1. Introduction Spin systems composed by variables possessing a Z(q) symmetry are quite important for their richness and generality and have attracted the attention of many workers [l]. They generalize the universal class of the Clock model and as q + + co the model coincides with the X Y planar model [2] . The Z(q) model includes several models like the q-state Potts model, the cubic model, etc. For q = 4 the model coincides with the Ashkin-Teller (AT) model [3]. The phase diagram of the pure isotropic ferromagnetic AT model on the square lattice was studied by several authors [4 to 81. It is known to present three different phases, namely, paramagnetic (P, Z(4) symmetry), partially ordered (F,, Z(2) symmetry), and ferromagnetic (F, completely broken symmetry). The F-F, and P-F, critical lines describe Ising-like phase transitions and are related to each other by duality. The P-F critical line is known exactly. It is the only self-dual line. The three critical lines meet at the four-state Potts ferromagnetic critical point. Mariz et al. [9] have studied, within a real-space renormalization group (RG) framework based on the self-dual Wheatstone bridge cluster, the effects of a quenched bond dilution on the ferromagnetic AT model on the square lattice. Since the model describes two coupled king models with two-spin coupling K , and four-spin coupling K, , it is interesting to study the effects of dilution of these reduced interactions on the model. The main purpose of this paper is to investigate within the Migdal-Kadanoff renormalization group (MKRG) approximation [lo] the influence of the dilution of the two independent random variables, K , and K,, distributed according to two different probabilities on the AT model. In Section 2 we introduce the model and develop the MKRG method, in Section 3 we present our main results and finally, in Section 4 we conclude. 2. The Model and the MKRG Method The AT model is defined by the following Hamiltonian: â) B.P. 1014, Rabat, Marocco. 202 A. BENYOUSSEF and M. LOULIDI where K!â,j) = pJ!â,j) ( r = 2,4) are the reduced couplings, oi and zi are king variables (= l), and the sum runs over all first-neighbour sites of the square lattice. The AT model reduces to the Ising model (K2.j) = 0, or K2.j) = 0, or K2.j) 4 a) or to the four-state Potts model (K2.j) = K2.J)). In this paper we restrict ourselves to the ferromagnetic case The first application of a RG method to a dilute magnet is due to Young and Stinchcombe [ l l ] . Complete phase diagrams have then been determined by various authors ([12] and references therein). To study the dilute AT model we shall use an extension of the MKRG method which takes into account the two-peak approximation of Young and Stinchcombe [ l l ] . This approach has been described in detail by Benyoussef and Boccara [13] who studied the validity of the two-peak approximation. The dilute AT model can be located as two coupled dilute Ising models with random four-spin coupling, K2.j). Then, as mentioned above the two independent random variables, K2.j) and Kt . j ) , are distributed according to the following probability laws: (K2.j) 2 0 and K2.j) + K 2 - J ) 2 0). The RG recursion relations are obtained using the usual bond-moving Migdal-Kadanoff approximation, which consists in successive contractions by a scale factor b along each of the d Cartesian directions, resulting in a volume contraction by an overall factor bââ. Each contraction involves a bond shift perpendicular to the contraction and a decimation along the contraction. In what follows we specialize to the case which consists in first performing the decimation and then moving the bonds. Then to write the recursion relations we introduce the convenient variables t f * j ) and t2.j) defined as where y takes only two integer values, y = 0, 1. The variables t l ( l , q), t l(q, O), and t2 (q , 1 ) are defined by t6(t, q ) = tbâJ (6 = 1, 2 ) , such that Phase Diagrams of the Dilute Ashkin-Teller Model 203 since after decimation along the x-direction we obtain intermediate couplings K"kk') and such that b - 1 + pb c Q(l ) &(K"$J) - F ? ( l ) ) , I = 1 1 = 1 I = 1 b - 1 b - 1 + c p'Q"(E) c P",(k) S(l?$.j) - Fi(1, k)), I = 1 k = 1 where b - 1 P",(k) = ( )pb-'*(l - P ) k , R = 1 - (4 + (1 - 4 ) P ) ~ + (1 - q)b p b , and the functions F!, F{(l), and Fi(1, k) are given by F! = Fi(tb,(l, I), t",l, 1)) (i = 1, 2) , F: = F,(tb,(l, 0), t:'(l, O)) , Fi(tb,-'(l, 1) t:(l, 0), t",-'(l, 1) tf'(1,O)) for for (i, j ) = (1, 3), (2 ,2) . (i, j) = (2, 3 ) , i F,(O, t",-'(l, 1) t:co> 1)) F{(1) = Fi(I, k) = Fz(O, t i- l-k(l , 1) ti(0, 1) t:'(l, O ) ) , where with When bd-' of these intermediate couplings are added together in the succeeding y contraction, the probabilities to have m(n) present links of K,(K,) are the following: 204 A. BENYOUSSEF and M. LOULIDI Then the probability distributions of the appropriate variables tfJ: defined by tfi = &(KC, K:) ; 6 = 1, 2 , where are given by I 2 I3 F:(yj) + - I1 - ' 2 - '3) F: + ? Ft(q i ) + F ; ( q k , :.I)). i = 1 j= 1 k = 1 (2.8) P; F p' and Pi = q' are the renormalized probability distributions of p and q. As usual, to render the computations tractable we make an additional approximation at each iteration by forcing the full distributions (2.8) back to the initial form in order to obtain a two-peak distribution for the renormalized couplings K$ and KC, such that p;,p,(tf.j~') = C ,p(1 - q ' ) l - v &(t';.j)' - t > ( l , y)) + (1 - p') G(tYJ'), v (2.9) + (1 - 4') p'"1 - p y ' I 6(tyJ' - (t;(y, 0))2), 'I where p', q', t i , and t k are quantities to be determined as functions of p, q, t,, and t,. However, by equating the lowest-order moments of t f % j ) and t y , j ) calculated with both distributions (2.8) and (2.9), (2.10) Phase Diagrams of the Dilute Ashkin-Teller Model 205 ( E , and E , indicate the moments of order 2 and 3, respectively), and since f,(O, y ) = 0 and f2(0, 0) = 0 we obtain the following recursion relations for p' and q': (2.11) The recursion relations for the renormalized variables t i and t i are not given explicitly. However, to generate flux lines in the parameter space ( p , q, t,, t,), from which one can find the critical frontiers, we compute (2.10) numerically. 3. Results The MKRG recursion relations (2.10) and (2.11) have been established for the dilute AT model for arbitrary dimension d and scale factor b. In this paper we restrict ourselves to the case d = 2 and b = 2. Since the recursion relations (2.11) do not depend upon the variables t , and t,, we at first discuss the percolation phase diagram represented in Fig. 1. Then we shall distinguish two types of links, type 1 and type 2 which occur, respectively, with concentrations p and q. From (2.11) we see that the renormalized variable p' depends only upon p , while q' depends upon both p and q. Consequently the recursion relation for p gives the usual two-dimensional percolation threshold, p , = 0.618 (within the MKRG method with b = 2). The percolation phase diagram has got two critical lines which divide it into three different regions (see Fig. 1): (i) The region of attraction which is described by the fixed point L(0, 0), represents a system with only a finite cluster. So there is no percolation. (ii) In the region of attraction described by the fixed point H,(O, 1) the percolation threshold depends upon both variables p and q. Consequently the system percolates at the Fig. 1. The percolation phase diagram. The dashed lines connect different regions of the dilute Ashkin-Teller model (see text) 206 A, BENYOUSSEF and M. LOU LID^ critical value q,, = f ( p ) . Since in this region p < pc, the percolation is realized with an infinite cluster which consists of a finite cluster of type 1 put together with a cluster of type 2. (iii) In the region of attraction described by the fixed point H3(1, 1) we have p > p c . The system has got an infinite cluster which consists at least of type 1 because p c is independent of the value of q. (iv) C,(O, 0.618), C2(0.618,0), and C,(0.618, 1) are instable fixed points. They indicate the critical concentrations which correspond to the threshold percolations in the invariant subspaces p = 0, q = 0, and p = 1, respectively. As usual the renormalized concentrations p' and q' given by (2.11) depend only upon p and q. Therefore, in order to determine the coordinates (p , q, t,, t 2 ) of each fixed point we shall, after having got the coordinates of the fixed points ( p * , q*), determine the remaining coordinates (t:, t;) from (2.10). Below we discuss our main results. (i) As expected, the model exhibits three different phases: (o i ) = ( t i ) = (nizi) = 0 , paramagnetic (P): ferromagnetic (F): (oi) = ( t i ) + 0 , (n i t i ) += 0 , partially ordered (FJ: (oi) = ( t i ) = 0 , (oizi) + 0 . Each of them is characterized by an attractor in the ( p , q, t , , t 2 ) space, which are (1, 1,0,0), (1, 1, 1, l), and (1, 1,0, l), respectively. (ii) The invariant subspace p = 0 corresponds to the bond dilute Ising model (Fig. 2). The percolation concentration is given by type 2 couplings: qc = 0.618. The critical line q = q, in the (q , t z ) plane presents two fixed points: D, (semi-stable) and D, (unstable) located, respectively, at (t , , t 2 ) = (0, 0), (0, 1). The stable fixed points PI,(O, 1,0,0) and FI, (0, 1,0, 1) describe, respectively, the paramagnetic and the ferromagnetic phases of this model. The critical line separating ferromagnetic and paramagnetic phases is described by the unstable fixed point 11, (0, 1,0,0.544). In Fig. 2 we have reproduced the phase diagram of this model which is in the universality class of the pure model. Phase Diagrams of the Dilute Ashkin-Teller Model 207 Fig. 3. Phase diagrams for different values of p at a fixed value of q (q = 0.8). The dashed line is the boundary of the ferromagnetic region (iii) For q = 0 the model reduces to the bond dilute Ising model ( K , = 0). As discussed above in this invariant subspace we have found three fixed points PI,(l, 0, 0,O) (stable), F12(l, 0, 1, 1) (stable), 112(1, 0, 0.544, 0.296) (semi-stable) which describe the pure case for the king model and two fixed points at criticality p = p c , D,(0.618, 0, 0,O) (semi-stable) and D,(0.618, 0, 1, 1) (unstable). (iv) At the fixed point C, we obtain three fixed points D,(semi-stable), D,(unstable), and D,(unstable) located at (t l , t2) = (0, O), (0, l), (1, l), respectively. We observe also that the partially ordered phase F, can be stable only in the subspace t , = 1. a / g. 4. Phase diagrams for different values of p at a fixed value of q (q = 0.45). The dashed / / 1 line is the boundary of the ferromagnetic I j region / / / / /- 0 I tl - 208 A. BENYOUSSEF and M. LOULIDI Fig. 5. Phase diagrams for different values of p = q. The linear dashed line indicates that the bifurcation line P-F does not occur at the diluted four-state Potts critical point ( t l = t 2 ) . The dashed line is the boundary of the ferro- magnetic region 0 I tl - (v) The pure case ( p , q) = (1, 1) reproduces the usual phase diagram of the AT model. It exhibits three different phases which are the region of attraction of three phase sinks P(0, 0), F2(0, l), and F(1, 1). All fixed points are reproduced: a) The Ising fixed points I,(0.544, l), IT(0, 0.544), and I(0.544, 0.296). They generate Ising-like critical lines, F-F,, P-F,, and P-F, respectively. b) All critical lines meet at four-state Potts fixed point P,(0.469, 0.469). (vi) For T + 0 the percolation plane is divided by two dashed lines, as shown in Fig. 1, into several parts: The model remains in the paramagnetic phase if q < q, although the system percolates while it exhibits a phase transition only between F, and P if q > 4,. 1. p < p,: 2. p > p , : a) Below a dashed line the model exhibits only a phase transition between P and F. b) Above, it presents all phase transitions which occur in the pure AT model. Namely P-F, F-F,, and P-F,. (vii) In Fig. 3 for a fixed value of q, such that q > q,, with increasing p we have shown different phase transitions which occur for different values of p . F-P and F-F, phase transitions appear only for p > p c . Fig. 4 shows that for a fixed value of q < q, the dilute AT model, with increasing p , exhibits at first only a F-P phase transition while for a large value of p all phase transitions occur. (viii) Since the subspace p = q in the percolation ( p , q) plane is not invariant, Fig. 5 shows that the bifurcation of the critical line P-F into the P-F, and F -F, critical lines does not occur at the dilute four-state Potts critical line (tl = t2). 4. Conclusion We have studied the phase diagram of the dilute ferromagnetic Ashkin-Teller model on the square lattice by means of a real-space renormalization group technique. However, we have used two probability distributions P(K,) and Q(K4) for the two-spin coupling K , and four-spin coupling K4, respectively. Phase Diagrams of the Dilute Ashkin-Teller Model 209 At temperature T = 0 we obtain the percolation phase diagram. It can be interpreted as describing a system with two different types of links with concentrations p and q. For p < p , the critical concentration q,, depends upon the value of p so the system percolates at q,, 2 f ( p ) while for p > p , the system percolates for all values of q. The bond diluted ferromagnetic AT model was studied by Mariaz et al. [9]. They found that above a critical concentration of links p , the model has got the three distinct phases P, F, and F, and the whole phase diagram for a fixed value of p > p , is qualitatively similar to the one obtained in the pure case. Since we have used two probability distributions which correspond to K , and K , couplings, a rich variety of phase diagrams is obtained. They depend upon the values of p and q. We observe that for p < p , there exists a line q = q, (dashed line in Fig. 1) such that the model exhibits only a phase transition between F and F, for q > q,, while for q,, = f ( p ) < q < q, the model remains in its disordered phase (P) although the system percolates. For p > p , there exists a value qo of q which depends upon p (dashed line in Fig. 1) such that for q > qo we obtain all the critical frontiers which indicate the existence of the three phases F, F,, and P while for q < qo the system undergoes only the P-F phase transition. References [l] S. ELITZUR, R. B. PEARSON, and J. SHIGEMITSU, Phys. Rev. D 19, 3698 (1979). J. L. CARDY, J . Phys. A 13, 1507 (1980). F. C. ALCARAZ and R. KOBERL, J. Phys. A 13, L153 (1980). F. C. ALCARAZ and R. KOBERL, J. Phys. A 14, 1169 (1981). E. DOMANY and E. K. RIEDEL, Phys. Rev. B 19, 5817 (1979). F. C. ALCARAZ, J . Phys. A 25, 251 1 (1987). V. A. FATEEV and A. B. ZAMALODCHIKOV, Zh. eksper. teor. Fiz. 89, 380 (1985); Soviet Phys. - J. exper. theor. Phys. 62, 215 (1985). V. A. FATEEV and A. B. ZAMALODCHIKOV. Phys. Letters A 92, 37 (1982). F. C. ALCARAZ and A. SANTOS, Nuclear Phys. B 275 [FS17], 436 (1986). P. RUJAN, G. 0. WILLIAMS, H. L. FRISH, and G. FORGACS, Phys. Rev. B 23, 1362 (1981). H. H. ROOMANY and H. W. WYLD, Phys. Rev. B 23, 1357 (1981). H. H. ROOMANY, H. W. WYLD, and L. E. HOLLOWAY, Phys. Rev. D 21, 1557 (1980). [2] J. M. KOSTERLITZ and D. J. THOULESS, J. Phys. C 6, 1181 (1972). J. V. JOSE, L. P. KADANOFF, S. KIRPATRICK, and R. D. NELSON, Phys. Rev. B 16, 1217 (1977). J. M. KOSTERLITZ, J. Phys. C 7, 1046 (1974). [3] J . ASHKIN and E. TELLER, Phys. Rev. 64, 178 (1943). [4] H. J. F. KNOPS, J. Phys. A 8, 1508 (1975). [5] F. IGLO'I' and J. ZITTARTZ, Z. Phys. B 73, 125 (1988). [6] R. V. DITZIAN, J. R. BANNAVAR, G. S. GREST, and L. P. KADANOFF, Phys. Rev. B 22, 2542 (1980). [7] P. M. C. DE OLIVERA and F. C. Sd, BARRETO, J. statist. Phys. 57, 53 (1989). [ X I A. BENYOUSSEF, L. LAANAIT, and M. LOULIDI, submitted for publication to J. statist. Phys. N. BENAYAD, A. BENYOUSSEF, N. BOCCARA, and A. ELKENZ, J. Phys. C 21, 5747 (1988). [9] A. M. MARIZ, F. D. NOBRE, and E. S. DE SOUZA, Physica (Utrecht) 178A, 364 (1991). [lo] A. A. MIGDAL, Soviet Phys. - J. exper. theor. Phys. 42, 413, 743 (1975). [ I I] A. P. YOUNG and R. B. STINCHCOMBE, J. Phys. C 9, 4419 (1976). [12] R. B. STINCHCOMBE, Phase Transitions and Critical Phenomena, Vol. 7, Ed. C. DOMB and [13] A. BENYOUSSEF and N. BOCCARA, J. Phys. C 16, 1901 (1983). L. P. KADANOFF, Ann. Phys. (USA) 100, 59 (1976). J. LEBOWITZ, Academic Press, New York (pp. 155 to 280). (Received October 6, 1993) 14 physica (b) 182jl