Brazilian Journal of Physics, vol. 27, no. 4, december, with Aperiodic Interactions. Instituto de Fsica, Universidade de S~ao Paulo

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1 Brazilian Journal of Physics, vol. 27, no. 4, december, Critical Behavior of an Ising Model with periodic Interactions S. T. R. Pinho, T.. S. Haddad, S. R. Salinas Instituto de Fsica, Universidade de S~ao Paulo Caixa Postal 66318, , S~ao Paulo, SP, Brazil Received pril 22, 1997 We write exact renormalization-group recursion relations for a ferromagnetic Ising model on the diamond hierarchical lattice with an aperiodic distribution of exchange interactions according to a class of generalized two-letter Fibonacci sequences. For small geometric uctuations, the critical behavior is unchanged with respect to the uniform case. For large uctuations, the uniform xed point in the parameter space becomes fully unstable. We analyze some limiting cases, propose a heuristic criterion to check the relevance of the uctuations. I. Introduction The experimental discovery of quasi-crystals motivated many investigations of the eects of geometric uctuations produced by dierent types of aperiodic structures. There are several specic results for phonon electronic spectra of linear problems[1]. There are also some results for more dicult non-linear problems, as the analysis of the critical properties of a ferromagnetic Ising model on a layered lattice with aperiodic exchange interactions along an axial direction[2]. In a recent publication[3], Luck presented a detailed study of the critical behavior in the ground state of the quantum Ising chain in a transverse eld (which is known to be related to the transition at nite temperatures of the two-dimensional Ising model). The nearest-neighbor ferromagnetic exchange interactions are chosen according to some (generalized) Fibonacci sequences. The geometric uctuations are gauged by a wering exponent! associated with the eigenvalues of the substitution matrix of each sequence. The critical behavior remains unchanged (that is, of Onsager type) for bounded uctuations (small values of!). Large uctuations induce much weaker singularities, similar to the case of a disordered Ising ferromagnet. In this paper, we takeadvantage of the simplications introduced by a hierarchical lattice to give another example of the eects of geometrical uctuations on the critical behavior of a ferromagnetic model. It should be pointed out that hierarchical lattices have been widely used as a toy model to test critical properties. In some cases the (approximate) Migdal-Kadano renormalization-group transformations on a Bravais lattice are identical to the (exact) transformations on a suitable hierarchical structure[4]. There are rather detailed studies of these exact transformations for the Ising model on a variety of hierarchical structures[5]. There are also some recent investigations of an Ising spin-glass on the diamond hierarchical lattice[6]. This paper is organized as follows. In Section II, we make some comments on Fibonacci sequences introduce a nearest-neighbor ferromagnetic Ising model on a diamond hierarchical lattice with q branches. To simulate a layered system, the exchange interactions (J > J B > ) are distributed along the branches according to the same aperiodic rule. For a large class of two-letter Fibonacci sequences, we write exact renormalization-group recursion relations in terms of two parameters, x = tanh K,x B =tanhk B, where K = J, K B = J B, is the inverse of the temperature. In Sections III IV, we analyze two typical distinct examples. We obtain the xed points the ows of the recursion relations for This paper is dedicated to Prof. Roberto Luzzi in his 6th anniversary.

2 568 S.T.R. Pinho et al. these specic cases. In the rst example, for small!, the critical behavior is unchanged with respect to the uniform case (J = J B ) the physical xed point in parameter space is a saddle point. In the example of Section IV, however, the uctuations turn the physical xed point fully unstable. We also analyze some limiting cases. For q = 1 (Ising chain), there is no transition at nite temperatures. The xed point at zero temperature, however, may change its character depending on the value of the wering exponent. In a particular innite-branching limit (q!1), we obtain rather simple results. In Section V, we present some conclusions, as well as a heuristic adaptation for the diamond hierarchical lattice of Luck's criterion[7] for the relevance of geometric uctuations. From these equations, we can write the asymptotic expression with the wering exponent N N! (7)! = ln j 2j ln 1 : (8) II. Denition of the model Consider a particular two-letter generalized Fibonacci sequence given by the substitutions! B B! : (1) If we start with letter, the successive application of this ination rule produces the sequences! B! B! BBB!: (2) t each stage of this construction, the numbers N N B, of letters B, can be obtained from the recursion relations N N B = M N N B with the substitution matrix 1 2 M = 1 (3) : (4) The eigenvalues of this matrix, 1 = 2 2 = ;1, govern most of the geometrical properties. In a more general case (that is, for a more general rule), at a large order n of the construction, the total number of letters is given by the asymptotic expression N n = N n + N n B n 1 (5) where 1 > j 2 j. The smaller eigenvalue governs the uctuations with respect to these asymptotic values, N n N n N n B j 2 j n : (6) Figure 1. Some stages of the construction of a diamond hierarchical lattice with q = 2 branches. Letters B indicate the exchange interactions (J JB). Now we consider a nearest-neighbor Ising model, given by the Hamiltonian H = ; X (i j) J i j i j (9) with the spin variables i = 1 on the sites of a diamond hierarchical structure. In Fig. 1, which is suitable for the period-doubling Fibonacci rule of Eq.(1), we draw the rst stages of the construction of a diamond lattice with a basic polygon of four bonds (rami- cation q = 2). s indicated in this gure, we simulate alayered system by the introduction of the interactions J > J B > along the branches of the structure. Considering the elementary transformations of Fig.2, using the rules of Eq. (1), it is straightforward to establish the recursion relations tanh K = tanh 2 tanh ;1 (tanh K tanh K B ) (1) tanh K B = tanh 2tanh ;1 ; tanh 2 K (11) where K B = J B. In this particular case, these equations can also be written as x = 2x x B 1+x 2 x2 B (12)

3 Brazilian Journal of Physics, vol. 27, no. 4, december, where 2x2 x B = 1+x 4 (13) x B =tanhk B : (14) In Section 3, we analyze the xed points associated with these simple recursion relations. where k l 1 are two integers. From the substitution matrix, k k + l M = (16) l we have the eigenvalues, 1 = k + l 2 = ;l, the wering exponent,! = ln l= ln (k + l). Now we consider a diamond lattice with q branches a basic polygon of q (k + l) bonds. The exchange interactions are chosen according to the general Fibonacci rule of Eq. (15). We then write the recursion relations tanh K =tanh q tanh ;1 ; tanh k K tanh l K B (17) tanh K B = tanh q tanh ;1 ; tanh k+l K : (18) III. Irrelevant uctuations Consider again the Fibonacci ination rules given by Eq. (1). From the eigenvalues of the substitution matrix, 1 = 2 2 = ;1, we have! =. For the branching number q = 2, the recursion relations are given by Eqs. (12) (13). The xed points some orbits of the second iterates of this map are shown in Fig. 3. Besides the trivial xed points, there is also a non-trivial xed point, given by x = x B =:543689::: (19) which comes from the solution of the polynomial equation x 8 +2x 4 ; 4x 2 +1=: (2) Figure 2. Basic graphs to obtain the recursion relations for an Ising model on a diamond hierarchical lattice with q = 2 exchange interactions given by the generalized Fibonacci rules,! B, B!. We can use similar procedures to consider much more general Fibonacci rules. To avoid any changes in the geometry of the hierarchical lattice, in this paper we restrict the analysis to period-multiplying Fibonacci in- ation rules of two letters (the largest eigenvalue of the substitution matrix, 1,gives the multiplication factor of the period). For example, let us consider the substitutions! k B l B! k+l (15) The linearization about this uniform xed point yields the asymptotic expression x x B where M T = = CM T x x B is the transpose of the substitution matrix, C = 1 ; (x )4 2x (21) (22) =:839286:::: (23) The diagonalization of this linear form gives the eigenvalues 1 = C 1 =2C =1:678573::: (24)

4 57 S.T.R. Pinho et al. 2 = C 2 = ;C = ;:839286:::: (25) s 1 > 1;1 < 2 <, the xed point isa saddle point with a ipping approximation (in Fig. 3, we draw the trajectories of some second iterates of this map). Moreover, if we make J = J B, it is easy to see that the same eigenvalue 1 characterizes the (unstable) xed point of the uniform model (see the diagonal ow in Fig. 3). Therefore, in this particular example the geometric uctuations are unable to change the critical behavior of the uniform system. where y = q 2 K y B = q 2 K B. The linearization about the uniform xed point, y = y B = 1, yields the relations y y = M T y (28) B y B which correspond to a limiting case of Eq. (21), with C! 1. s the limiting eigenvalues are given by 1 = 1 = 2, 2 = 2 = ;1, a linear analysis is not enough to check the (ipping saddle-point) character of this marginal case. nother particular case of interest is the simple Ising chain (q = 1). The recursion relations are given by x = x x B (29) x B = x 2 (3) with the same form of Eqs. (26) (27), but with the parameters x x B given by Eq. (14). s shown in Fig. 4(a), the zero-temperature xed point displays the character of a saddle point. s there is no phase transition at nite temperatures, it cannot be reached from physically acceptable initial conditions. Figure 3. Second iterates xed points (black circles) of the recursion relations in the parameter space, x = tanh K versus xb = tanh KB, for an Ising model on a diamond hierarchical lattice with q = 2 exchange interactions given by the generalized Fibonacci rules,! B B! (irrelevant uctuations). We draw the stable trivial xed points (at zero innite temperatures) the physical saddle point. The dashed lines the black arrows indicate the ows of the second iterates of the map. The uniform model is recovered along the diagonal, x = xb. The light arrows indicate the stable direction in the neighborhood of the physical xed point. It is not dicult to check that the same sort of behavior (saddle point largest eigenvalue associated with the uniform system) still holds for all nite values of the branching number q of the diamond structure. In the limit of innite branching (q!1, K K B!, with q 2 K q 2 K B xed), the recursion relations are particularly simple, y = y y B (26) IV. Relevant uctuations To give an example of relevant uctuations, consider the generalized Fibonacci substitutions,! BB B! : From the substitution matrix, 1 3 M = 2 (31) (32) we have the eigenvalues, 1 = 3 2 = ;2, the wering exponent,! =ln2= ln 3 = :63929:::. For a general branching number q, we can use Eqs. (17) (18), to write the recursion relations tanh K = tanh q tanh ;1 ; tanh K tanh 2 K B (33) y B = y 2 (27) tanh K B =tanh q tanh ;1 ; tanh 3 K : (34)

5 Brazilian Journal of Physics, vol. 27, no. 4, december, Figure 4. Second iterates xed points for the Ising chain: (a) exchange interactions according to the rules! B, B! (irrelevant uctuations) (b) exchange interactions according to the rules! BB, B! (relevant uctuations). For the particular case q = 2, Eqs. (33) (34) reduce to the simple relations x = 2x x 2 B 1+x 2 x4 B 2x3 (35) x = B 1+x 6 (36) where the parameters x x B are given by Eq. (14). In Fig. 5, we show the xed points some second iterates associated with these recursion relations. The nontrivial xed point, given by x = x B =:786151::: (37) comes from the physical solution of a polynomial equation. The linearization about this uniform xed point yields the relations x x B = CM T x x B (38) where the substitution matrix is given by Eq. (32), C = 2(x )2 h 1 ; (x )6 i h1+(x )6i 2 =:61833:::: (39) From the diagonalization of this linear form we have the eigenvalues 1 = C 1 =1:85411::: (4) 2 = C 2 = ;1:23667:::: (41) The absolute values of 1 2 larger than unit indicate the relevance of the geometric uctuations. The uniform xed point is fully unstable ( there might be no transition for J 6= J B ). gain, it is not dicult to check that the uniform xed point remains unstable for all values of the branching number q. In particular, for q!1, K K B!, with q 1=2 K q 1=2 K B xed, we have the limiting recursion relations, y = y y 2 B (42) y B = y 3 (43) where y = q 1=2 K y B = q 1=2 K B. The linearization about the uniform xed point, y = y B = 1, yields the relations y = M T y y (44) B y B with the substitution matrix given by Eq. (32). We then have the limiting eigenvalues, 1 = 1 =3,

6 572 S.T.R. Pinho et al. 2 = 2 = ;2, which conrm the unstable character of this xed point. uniform system. In another example, however, stronger geometric uctuations turn the physical xed point unstable in the parameter space. Even in one dimension, although there is no phase transition at nite temperatures, we show that the geometric uctuations change the stability of the uniform xed point at zero temperature. s in the work of Luck[7], it should be interesting to devise a general criterion to gauge the inuence of the geometric uctuations on the critical behavior of the Ising model on the diamond hierarchical lattice. For a large lattice, with q branches, the total uctuation J in the exchange interactions should be proportional to N. Thus we can write the asymptotic relation J N N! L q! (47) Figure 5. Second iterates xed points (black circles) of the recursion relations in the parameter space, x = tanh K versus xb = tanh KB, for an Ising model on a diamond hierarchical lattice with q = 2 exchange interactions given by the generalized Fibonacci rules,! BB, B! (relevant uctuations). We draw the stable trivial xed points (at zero innite temperatures) the unstable node. The uniform model is recovered along the diagonal, x = xb. The dashed lines are indications of the ow associated with the second iterates of the map. The light arrows indicate the directions of the eigenvectors in the neighborhood of the physical xed point. For the particular case of the linear chain (q = 1), we have x = x x 2 B (45) x B = x 3 : (46) In agreement with the lack of a phase transition, the xed point at zero temperature is unstable [see Fig. 4(b)]. V. Conclusions From the analysis of the exact renormalizationgroup recursion relations associated with an Ising model on a variety of hierarchical diamond structures, we show that aperiodic uctuations of the ferromagnetic exchange interactions may change the character of a (uniform) xed point. In a particular example, with a small wering exponent, the uctuations are irrelevant. In this case, the critical behavior is still characterized by the same exponents of the corresponding where L is a measure of the total length. The critical temperature, T c, should be proportional to the total value of the exchange (that is, to L q ). We can then dene a reduced temperature, t =(T ; T c ) =T c, whose uctuations are given by the asymptotic form t Lq! L q = L q(!;1) : (48) In the neighborhood of the critical point, we have L t ;, where is a correlation length, is a critical exponent. Thus, we can write t with the exponent t t ;q(!;1) = t (49) t = ;q(! ; 1) ; 1: (5) If >, the uctuations are irrelevant. If <, however, the critical behavior is changed drastically. To calculate a suitable value of, we consider the largest eigenvalue of the diagonal form about the physical xed point, write the usual renormalization-group expression 1 = C 1 = b y1 = b 1= : (51) s the largest eigenvalue 1 of the substitution matrix gives the multiplication factor of the Fibonacci rule, we write b =( 1 ) 1=q : (52)

7 Brazilian Journal of Physics, vol. 27, no. 4, december, From Eqs. (51) (52), we have = ln 1 q ln (C 1 ) : (53) Inserting into Eq. (5), we nally have = ; ln 1 (! ; 1) ; 1: (54) ln (C 1 ) The uctuations are relevant for!>; ln C ln 1 (55) where the prefactor C can be obtained from the linear analysis of the unstable xed point of the uniform (J = J B ) system. lthough the existence of some counterexample should not be ruled out, the validity of this criterion has been conrmed by the application to a fair number of cases (including the examples of Sections 3 4). In the limit of innite branching, the prefactor C tends to unit, the criterion of relevance is reduced to the Pisot condition[3],!>. cknowledgments We thank a suggestion of D. Mukamel several discussions with R. F. S. ndrade. STRP is on leave from Instituto de Fisica, Universidade Federal da Bahia, thanks a fellowship from the program CPES/PICD. TSH SRS are partially supported by fellowships from CNPq. This paper is dedicated to the sixtieth birthday of Prof. Roberto Luzzi. References [1] See, for example, the reviews by T. Janssen J. Loos, Phase Transitions 32, 29 (1991), by H. Hiramoto M. Kohmoto, Int. J. Mod. Phys. B6, 281 (1992). [2] See, for example, C.. Tracy, J. Phys. 21, L63 (1988) V. Benza, M. Kolar, M. K. li, Phys. Rev. B41, 9578 (199). [3] J. M. Luck, J. Stat. Phys. 72, 417 (1993). [4]. N. Berker S. Ostlund, J. Phys. C12, 4961 (1979) P. M. Bleher E. Zalys, Comm. Math. Phys. 67, 17 (1979). [5] M. Kaufman R. B. Griths, Phys. Rev. B24, 496 (1981) R. B. Griths M. Kaufman, Phys. Rev. B26, 522 (1982) M. Kaufman R. B. Griths, Phys. Rev. B28, 3864 (1983). [6] E. Nogueira Jr., S. Coutinho, F. D. Nobre, E. M. F. Curado, J. R. L. de lmeida, Phys. Rev. E55, 3934 (1997). [7] J. M. Luck, Europhys. Lett. 24, 359 (1993).