Bifurcations and convergence of the Martinelli-Parisi expansion in the 2d Potts model

Nuclear Physics B251 [FS13] (1985)50-60 © North-Holland Publishing Company

BIFURCATIONS AND CONVERGENCE OF THE M A R T I N E L L I - P A R I S I E X P A N S I O N I N T H E 2d P O T F S M O D E L S. CARACCIOLO Scuola Normale Superiore, Pisa, and INFN, Sezione di Pisa, Pisa, Italy

S. PATARNELLO* Dipartimento di Fisica, Universitit di Pisa, Pisa, and INFN, Sezione di Pisa, Pisa, Italy

Received 29 June 1984 We compute the second order in the Martinelli-Parisi expansion for the 2d Potts model for every number of states q. While we obtain improved estimates for the critical temperatures, the results concerning the fixed-point properties are not satisfactory. We relate the difficulties of the expansion to the presence of a bifurcation in the fixed-point solutions and introduce a rearrangement of the expansion in order to take into account nonlinear effects. As a consequence a better evaluation of qc, where a crossover to first-order transition occurs, is obtained.

I. Introduction T h e m o d e r n theory of critical phenomena, originated by the formulation of the scaling laws [1], has received great m o m e n t u m under Wilson's ideas about the renormalization group (see [2] for a review). But already in lattice theories an explicit construction of the real space renormalization g r o u p is a formidable task. F o r the time being the M o n t e Carlo methods seem the m o s t promising approach to computations [3], although various approximate techniques have been introduced with some success [4]. As it w o u l d be extremely important to develop analytical methods for which errors are u n d e r control, we consider that a more detailed analysis of the k n o w n perturbative expansions is of interest. A m o n g t h e m we have focused on the Martinelli-Parisi expansion (MPE) [5]. It reproduces at zeroth order the M i g d a l - K a d a n o f f approximate recursive relations [6, 7], which for their simplicity have found a wide field of application. Furthermore, in the M P E the complexity of the new interactions generated b y the renormalization p r o c e d u r e is increased only step by step in the expansion parameter. A n y h o w , in o r d e r to i m p l e m e n t the method, as the degree of feasibility of calculation decreases rapidly to zero when the order of the expansion increases, it is crucial to k n o w the * Present address: IBM Italia, Ricerca Scientifica e Technologica, via del Giorgione 129, Roma, Italy. 50

S. Caracciolo, S. Patarnello / 2d Potts model

51

properties of convergence of the expansion, so as to suitably extrapolate toward the right results from the first few terms (see [5, 8-12] for the first applications of the MPE). Recently we have found that the MPE can become a singular perturbation theory in the sense that it may drive a zeroth-order fixed point to infinity, and we have shown that this mechanism can be connected to a crossover from second- to first-order phase transition [12]. The example we have considered is the q-state Potts model [13] defined on a two-dimensional triangular lattice. For this model Baxter rigorously proved the appearance of a latent heat when the number of states q is greater than four [14]. Many approximate renormalization group approaches to this problem, and among them the Migdal-Kadanoff formulas, fail to detect the known crossover, showing a second-order transition for every finite q. A reconciliation with [14] had been obtained only through a somehow artificial introduction of vacant sites whose density is governed by a chemical potential which enlarges the space of allowed effective hamiltonians [15]. In this space no fixed point exists at finite couplings for high q, whereas by decreasing q a bifurcation occurs giving rise to a pair of fixed points, one of which governs the critical behavior of the usual Potts model. An interesting feature of this calculation is that this pair of fixed points has critical indices in substantial agreement with the so-called den Nijs conjecture [16]. More naturally we found that at first order in the MPE a three-spin operator is generated. It acts as a potential for effective vacancies. There is a q¢ = 5.02 when this operator becomes marginal, and for q > qc, starting from the canonical Potts model, no finite non-trivial fixed point can be approached. Two questions soon arise: (i) Can the evaluation of qc be improved at higher order in the MPE? (ii) Can a second fixed point appear at higher order and merge with the first one at qc showing the same bifurcation phenomena as in [15]? Unfortunately one soon realizes, as we shall see, that the answer to the preceding questions is negative if one naively follows the formal expansion rules. Then, as a consequence, at least for 4 < q < q~, the MPE should not converge. Can one find by inspection of the first terms an indication for such behaviour? Is it possible to reorder the expansion suitably so as to obtain a positive answer to questions (i) and (ii)? Moreover, it could be of some interest to understand whether, in the context of a perturbative method like MPE, the convergence properties might be different according to the quantities analyzed (critical temperature, fixed points, etc.) or are to be regarded as a specific feature of the method itself. Motivated by these considerations we have computed the second-order terms in the MPE. In sect. 2 we present the results obtained from the usual formal expansion. In sect. 3 we propose a rearrangement of the expansion which is sensitive to the possibility of a bifurcation which actually occurs for q < q~.

52

s. Caracciolo, S. Patarnello / 2d Potts model 2. The second order in the M P E expansion T h e m o d e l we c o n s i d e r is the u s u a l Potts m o d e l d e f i n e d o n a t r i a n g u l a r lattice,

w h o s e h a m i l t o n i a n % is given b y the r e l a t i o n

%

k T =- fl £ 8,,,oj,

(2.1)

(o)

where k is the Boltzmann constant, T is the temperature, (0") denotes a couple of nearest neighbors, o~ is the spin located at the i site of the lattice and takes q different values. The interaction distinguishes only the aligned configurations in which neighbouring spins are in the same state. The Migdal-Kadanoff renormalization procedure can be seen as a decimation. If one chooses the new lattice as a triangular sublattice whose spacing is twice the original one, the approximation consists in shifting the interactions between the decimated spins toward interactions between a decimated and a fixed one. In our case it amounts to moving the inner bonds of an elementary triangle on the new scale to the border links, which leads to double the strength of the latter, when the contributions of the near triangles are taken into account (see fig. lb). In this way renormalization produces only a rescaling of the coupling fl according to the relation

Y'=

yn+q_l , 2y 2 + q - 2

(2.2)

A

/\/\ a

b

A\

,/,

Fig. 1. Illustration of the MPE on the triangular lattice: (a) original lattice, with nearest-neighbour couplings represented by links; (b) Migdal-Kadanoff approximation; (c) Martinelli-Parisi scheme, in which the dotted finks represent an eft fraction of the couplings and the continuous ones are equal to (2 - e)#.

S, Caracciolo, S. Patarnello / 2d Potts model

53

where y - e ~ and the prime denotes the rescaled quantities. This recursive relation has a fixed point y * when (y*-

1)2(y * + 1 ) = q .

(2.3)

If we restrict ourselves to the physical region y * > 0 eq. (2.3) has always a solution in the ferromagnetic regime ( y * > 1) for every q>_-0 and has also an antiferromagnetic solution (y*~< 1) when 0 ~
3 " = [1 + B ( f l ) ] 3' + e A ( / 3 ) ,

(2.4)

which shows that 3" is of order e, when starting from y = 0, justifying the perturbative treatment of the y-term. In a similar way an e-correction appears in the equation for/3'. These recursion relations can be formally solved by writing

# = / 3 o + e/31 + e2/32+ . . - ,

(2.5)

and similar forms for/3', 3', 3". The equation for/30 is nothing but (2.2). At first order the equation for/31 is also determined and shows a dependence on 3'1. In contrast, (2.4) reveals that 3'1 does not depend on /31, and has a very simple fixed-point solution:

a

where a = A (/3o*) and b -- B (/3o*).

(2.6)

54

S. Caracciolo, S. Patarnello / 2d Potts model

We observe that, as

O~'/Off

is of order e,

O),'

= 1+ b

(2.7)

/3", "r*

is an eigenvalue of the linearized flow near the fixed point at zeroth order. The largest eigenvalue is, for all q's, Oil'/Off. An interesting feature is that b = 0 for q = qc---5.02, where the "r-interaction becomes marginal: it is always negative for q < qc and positive for q > qc where this three-spin interaction induces a new relevant direction in the space of effective hamiltonians. As a is always positive we recover an attractive positive fixed point when q < qc and a repulsive negative fixed point when q > qc. It is well known that the order of the phase transition for the model (2.1) changes f r o m second to first when the number of states q becomes greater than four. But we have already observed that the fixed point at zeroth order given by (2.3) exists for all q's, so that there is no evidence for a crossover to a first-order transition when q increases. The first term in the MPE provides a richer structure. When q > qc, starting f r o m "t = 0 and fl = ff~, a non-trivial fixed point at infinity is approached and a discontinuity in the derivative of the free energy is quite well simulated

I

20.

0.

!

2.

6.

Fig. 2. Plots of the fixed point value y* (attractive branch) as a function of q: t-expansion at first order (continuous) and 7t-~-expansion (dashed).

S. Caracciolo, S. Patarnello / 2d Potts model

55

numerically [12]. At q~, b = 0 and •x* goes to infinity. This raises serious doubts about the possibility of whether MPE could converge here. Moreover, the derivative with respect to q of b at q~ is very small and in fact b is smaller than a over a large region of q-values (see fig. 2). At second order five more interactions are generated. If the ~k and jkl sites define two adjacent elementary triangles of the lattice they are

~oiojo I ,

8~,ojo,ok , 6o, oflo,o~ , ~o,o~oj,~,

(2.8)

and those which are obtained by permutations of the indices. We will denote by 7/i with i = 1 . . . . . 5, the corresponding coupling constants. At first order in the ~/'s their recursion relations have the form 5

~1~= Y'~j Aij( /3 )~lg + ezJi( /3 ) + eTKi( ~ ) + yZLi(/3),

(2.9)

1

showing that it is consistent to assume the ~/'s of order e 2 when y is of order e. Furthermore, they depend only on fl0 and "/1The explicit calculation is already involved at this order. In order to improve our computational efficiency we have made use of the SCHOONSCHIP program for algebraic manipulations even though many of the formulas have been tested analytically. For q < qc the eigenvalues of the A-matrix of eq. (2.9) computed at the fixed point as a function of fl0* are much smaller than one. Therefore at second order no new singularity appears in the fixed-point values, which means that qc is not moved toward the correct value that is four. We report in table 1 the values of/30*,/31,/32 * * for different q ' S . Of * * and )'1,3'2 course, since higher-order corrections are of increasing magnitude it is difficult to extrapolate the fixed-point structure when t goes to 1. The results are much more reasonable for the inverse critical temperature where the corrections are smaller than the zeroth order which is/30*- Since one knows that the error on the critical quantities is of order ( 1 - e)2, one can impose, order by order, the condition of zero derivative with respect to e at e = 1 for their power

S. Caracciolo, S. Patarnello / 2d Potts model

56

TABLE 1 Fixed-point values of coupfings in M P E for some values of q: the expansion is fl0* + ill*e(1 - ½e) + fl~'e2(1 - 2e) for the two-spin coupling and -/1"e(1 - ½e) + "y2*e2(1 - 2e) for the three-spin coupling q

rio*

flf

fl~'

Y7

~'2"

2 3 4 5 6

0.609 0.693 0.756 0.807 0.851

- 1.296 - 2.52 - 5.41 - oo 8.703

3.912 10.65 21.32 + oo - oo

0.834 2.000 4.809 23.942 - 9.06

- 4.758 - 18.156 - 66.46 - oo 308.3

+ oo and - o0 indicate respectively large positive and negative numerical values.

expansion or for example for their [1,1] Pad6 approximant, obtaining the expressions fie(e) = fl0 + flae( 1 - ½e) + (f12 + 1 ~ 1 ) E 2 ( 1

--

3~8),

1 + (flx/Bo - flz,/fl,) e + e2, 1

+

2

which for e = 1 give rise to the results of table 2 where they are compared with the known exact results. 3. On the existence of fixed-point solutions The formal expansion (2.5) used in the preceding section to analyze the recursion relations obtained in the MPE is heavily based on the assumption of the existence of the fixed point established at zeroth order, at least for q 4= qc. Since the fixed-point equations are always linear in the terms to be determined, this property cannot be altered.

TABLE 2

Inverse critical temperature: first order in M P E (a), second order in M P E (b), [1', 1] Pad6 a p p r o x i m a m (c), exact results (d) q

(a)

(b)

(c)

(d)

2 3 4 5 6

0.478 0.565 0.628 0.682 0.727

0.558 0.626 0.655 0.690 0.703

0.533 0.613 0.666 0.711 0.739

0.549 0.631 0.693 0.744 0.787

S. Caracciolo, S. Patarnello / 2d Potts model

57

Moreover, the new couplings generated in the MPE always appear in polynomial forms like (2.4) or (2.9) and the order of these polynomials increases with the order of the expansion. This means that new fixed-point solutions are introduced although their formal expansion like (2.5) can never give rise to a bifurcation with the original solution. We observe that the discussion of the reality of the solutions of a polynomial equation is per se very delicate when its coefficients are known only approximately. In order to discuss these features let us consider a toy version of the equation for 7 in which we forget about the contribution from other couplings. At second order it will be of the form 7 ' = C ( q ) 7 2 + [1 + B ( q ) ] 7 + A ( q ) e ,

(3.1)

where A, B, C, here considered independent of e, are continuous functions of the number of states. When e = 0 (3.1) has two fixed-point solutions: 0

vg=

_B/C,

which are the zeroth-order results in a perturbative expansion in e. The first-order corrections are given by

if'=

-A/B +A/B"

Even though a solution is always found in this way, it could be reliable only when B2 >> A C e ,

(3.2)

i.e. when extrapolating toward e = 1, the two solutions do not tend to cross. Of course in this simple case the exact solutions for every value of e are known: - B :T- v / B 2 _ 4A Ce

7*(t) =

2C

(3.3)

Far from the region (3.2) the correct solutions are completely different from 7~' + e~/~* since the two solutions collapse and eventually disappear in the complex field. But, in analogy with what happens in the renormalization group calculation performed in the enlarged space of the dilute Potts model [15], this could be the mechanism which drives into the first-order transition regime. It is worth noticing that since 07" = 1 T riB z - 4 A C e , O't

(3.4)

58

S. Caracciolo, S. Patarnello / 2d Potts model

it is once more the marginality condition of the "t-operator to give the signal for the crossover. We observe that troubles arise in extrapolating toward e = 1 for e c = B2/AC and in order to overcome this difficulty we construct an expansion in which B is formally replaced by vre-B. In this way the restriction to a linear analysis is overcome, all the terms for the fixed-point equation are formally of the same order and the expansion can be rearranged in series of ¢7. Here at zeroth order there is only the .to* = 0 solution but it is double and at first order in vce-: (.t~,/=) T =

-B-T-v/B2-4AC 2C

reproduces the exact solutions for e = 1. Remark that 1

(.tl*j2) +

1

B

(3.5)

A

(.t:j2)

is the inverse of the extrapolation for the solution recovered at first order in e with .to* = O. Of course since the two solutions are complex conjugates, the sum in (3.5) is real also when they are not. When C goes to zero one of the solutions diverges so that the other coincides with that obtained in the e-expansion. In our actual calculations the condition similar to (3.2) is fulfilled only in a very narrow region of the q-interval between four and five. This fact explains why the e-expansion results for the fixed point are not satisfactory. Switching to the ¢~-eexpansion the lowest-order corrections to the fixed-point equations are of the form B,*/2 - Xrx*/2 = 0 , 5

E j A i j ( ~ j*) l - - ( ~ : ) l 1

"~1-( ~ 1 */ 2 )

2

Li=O,

i = 1 . . . . . 5,

5

a+by~,/2+c ( Y1/2) • 2 +dfl~/2Y~/2+ E , e j ( r l ~ ) l

=0"

1

When the first two equations are solved in terms of 71/2,* we are left with the third, which becomes a second-order equation for .t/*/2, like in the previous simplified example, with

h=a, B=b, 5

C=c+ •d+ ~.,ijei(1-A)~jILj. 1

S. Caracciolo, S. Patarnello / 2d Ports model

59

We find that for q = 4.56 the discriminant of the equation vanishes and the two solutions become complex when q increases. Since b is very small, the vanishing of the discriminant is equivalent to that of coefficient C; therefore the bifurcation appears in the region where the two solutions vary most rapidly. In fig. 2 we have plotted the attractive branches of T* 's for the two expansions. We consider it remarkable that in the v~--expansion a bifurcation of the solutions really takes place, giving also a better estimate for the value of q where the crossover occurs. Again, the divergence of one of the solutions is a byproduct of the truncation performed in the expansion, since a small cubic term would remove it. In other words, it is difficult to reproduce the correct behaviour for an attractive fixed point at o0 - which should be present for every q - by means of a perturbation around T=0.

4. Conclusion The Martinelli-Parisi expansion should, in principle, provide a systematical improvement of the Migdal-Kadanoff recursion relations. From the model considered h e r e w e learned that while the critical temperature behaves smoothly, essentially two limitations of the applicability of this expansion arise in relation to the fixed-point properties. First, the expansion uses a fixed point whose existence is established at zeroth order and this property cannot be altered further. Secondly, the behavior for small e of the recursion relations can be qualitatively changed when e is extrapolated to one: that is, a singularity may appear in the interval [0,1] of e. In a study in which the number of states q is regarded as a continuous parameter, i.e. a control variable, these two features are related by the appearance of a bifurcation of the fixed-point solutions. In order to take into account such a phenomenon we considered a very simple rearrangement of the expansion as a series which allows us to bypass the limits of a linear analysis. The resulting picture reveals that when q varies the manifold of fixed points is folded, so that when q arrives at a critical value a discontinuous change appears and gives an origin to the crossover to first-order phase transition. This behavior is in agreement with what had been obtained in a different context and the evaluation of the critical value of q is better than that recovered in the usual e-expansion. One of the authors (S.P.) wishes to thank the Scuola Normale Superiore for kind hospitality and access to computer facilities.

References [1] L.P. Kadanoff, Physics 2 (1966) 263 [2] K.G. Wilson, Rev. Mod. Phys. 55 (1983) 583

60 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

S. Caracciolo, S. Patarnello / 2d Potts model

R.H. Swendsen, Phys. Rev. Lett. 42 (1979) 859 T.W. Burkhardt and J.M.J. van Leeuwen (eds.), Real space renormalization (Springer, Berlin, 1982) G. Martinelli and G. Parisi, Nucl. Phys. B180[FS2] (1981) 201 A.A. Migdal, Soy. Phys. JETP 42 (1976) 413,773 L.P. Kadanoff, Ann. of Phys. 100 (1976) 359 S. Caracciolo, Nucl. Phys. B180[FS2] (1981) 405 G. Martinelli and P. Menotti, Nucl. Phys. B180[FS2] (1981) 483 S. Belforte and P. Menotti, Nucl. Phys. B205[FS5] (1982) 325 L.E. Roberts, Nucl. Phys. B230[FS10] (1984) 385 S. Caracciolo and S. Patarnello, Scuola Normale Preprint (Dec. 1983), J. Phys. A., in press R.B. Potts, Proc. Cambridge Philos. Soc. 48 (1952) 106 R.J. Baxter, J. Phys. C6 (1973) L445 B. Nienhuis, A.N. Berker, E.K. Riedel and M. Schick, Phys. Rev. Lett. 43 (1979) 737 M.P.M. den Nijs, J. Phys. A12 (1979) 1857