Physica A 281 (2000) 112–128 www.elsevier.com/locate/physa Monte Carlo investigations of phase transitions: status and perspectives Kurt Bindera , Erik Luijtena; ∗, Marcus Muller a , Nigel B. Wildinga; 1, Henk W.J. Bloteb aInstitut fur Physik, Johannes Gutenberg-Universitat, Staudinger Weg 7, D-55099 Mainz, Germany bDepartment of Physics, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands Abstract Using the concept of ÿnite-size scaling, Monte Carlo calculations of various models have become a very useful tool for the study of critical phenomena, with the system linear dimension as a variable. As an example, several recent studies of Ising models are discussed, as well as the extension to models of polymer mixtures and solutions. It is shown that using appropriate cluster algorithms, even the scaling functions describing the crossover from the Ising universality class to the mean-ÿeld behavior with increasing interaction range can be described. Additionally, the issue of ÿnite-size scaling in Ising models above the marginal dimension (d∗ = 4) is discussed. c 2000 Elsevier Science B.V. All rights reserved. Keywords: Critical phenomena; Ising model; Crossover scaling; Polymers; Finite-size scaling 1. Introduction It is a common belief that at the present time, about 30 years after the renormalization- group theory of critical phenomena was invented [1], static critical behavior of sys- tems in thermal equilibrium is rather well understood. In particular, this is expected to be true for the most intensively studied case, the Ising universality class [2,3], to which systems such as uniaxial ferromagnets, binary alloys, simple uids, uid mix- tures, polymer solutions and polymer blends belong [4]. However, in the present work, we shall draw attention to some aspects of critical behavior in Ising-like spin systems which are, even today, still incompletely understood. The ÿrst of these concerns the ∗ Correspondence address: Max-Planck-Institut fur Polymerforschung, Postfach 3148, 55021 Mainz, Germany. E-mail address: [email protected] (E. Luijten) 1 Present address: Department of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3JZ, UK. 0378-4371/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII: S 0378-4371(00)00025-X K. Binder et al. / Physica A 281 (2000) 112–128 113 problem of crossover between the Ising universality class and mean-ÿeld critical be- havior. This crossover occurs, for instance, when the interaction range (and hence the “Ginzburg number” G entering the Ginzburg criterion [5]) is varied [6–12]. A closely related crossover is found for symmetrical polymer mixtures when the chain length N of the polymers is varied [4,13–23]. A part of this crossover (though typically not the full extent of the crossover scaling function) can be probed experimentally near the critical point of uids and uid binary mixtures [24–27]. While the Ginzburg criteria [5,14–16] provide a qualitative understanding of this crossover, the quantitatively ac- curate theoretical prediction of the crossover scaling function is a challenging problem [28–36], and hence Monte Carlo studies [6–12,17–19] are of great potential beneÿt. In particular, the question as to what extent (if at all) such crossover scaling functions are universal is an intriguing one [10–12,24–26,36]. Another very interesting crossover which can also be studied is that which occurs near the critical point of unmixing for polymer solutions in a bad solvent [13,37–46]. For chain length N →∞ the critical temperature Tc(N) moves towards the -temperature, where a single coil undergoes a transition from a swollen coil to a collapsed globule. This limit corresponds to a tricritical point [13]. Monte Carlo analyses of critical phenomena typically apply ÿnite-size scaling con- cepts [47–52]. However, care is necessary in the proper application of these methods in the mean-ÿeld limit. In fact, the standard formulation of ÿnite-size scaling (“linear dimensions L scale with the correlation length ”) implies that the hyperscaling rela- tion [2,3] between critical exponents should hold [48,53], which is not the case for mean-ÿeld exponents (apart from d = d∗ = 4 dimensions). This problem already arises for Ising models with short-range interactions for d¿d∗ [54–66], and some disagree- ments between Monte Carlo results [54,55,57] and theoretical predictions [56,65] have stimulated a long-standing debate (see Ref. [66] for a detailed review). 2. Mean-ÿeld to Ising crossover We consider the Hamiltonian [6,7] X X X H=kBT = − K(ri − rj)sisj − h0 si ; (1) i j¿i i −d with si =1 and an interaction K(r) ≡ cR for |r|6R and zero elsewhere. The critical behavior of this model on d-dimensional lattices can be studied eciently with a new cluster algorithm adapted for long-range interactions [67]. To analyze the crossover it is instructive to consider the associated Ginzburg–Landau ÿeld theory in continuous space, Z ( Z 1 h c i 1 H()=k T = − dr dr (r)(r0) − v2(r) B d V 2 |r−r0|6R R 2 ) 4 −u0 (r)+h0(r) ; (2) 114 K. Binder et al. / Physica A 281 (2000) 112–128 Fig. 1. Qualitative picture of the renormalization trajectory describing the crossover from the Gaussian ÿxed ∗ ∗ ∗ ∗ point 0 =(0; 0) to the Ising ÿxed point =(r0 ;u ). where (r) is the single-component order-parameter ÿeld, v is a temperature-like parameter and u0 is a constant. After Fourier transformation and suitable rescaling this can be rewritten as (here N is the total number of lattice sites) 1 X h r i H =k T = k2 + 0 B 2 R2 k −k k r u X X X h N + − ; (3) 4R4N k1 k2 k3 −k1−k2−k3 R 2 k=0 k1 k2 k3 where u is related to u0 and h to h0 [7], and r0 in mean-ÿeld theory is the deviation of the temperature from its critical-point value. We are now interested in identifying the crossover scaling variable associated with the crossover from the Gaussian ÿxed point u = 0 and r0 = 0 to the nontrivial Ising ÿxed point (Fig. 1). Because of the trivial character of the Gaussian ÿxed point and the fact the crossover scaling description should hold all the way from the Ising ÿxed 4=(4−d) point to the Gaussian ÿxed point, one can infer the crossover length scale l0 = R exactly! This is done by considering a renormalization by a length scale l, such that the wave number changes from k to k0 = kl, the number of degrees of freedom is 0 −d 0 −1 reduced from N to N = Nl , and k changes into k0 = l k to leave H invariant. From inspection of the terms in the Hamiltonian one can conclude that the singular part of the free energy must satisfy the scaling relation r u h r u h f˜ 0 ; ; = l−df˜ 0 l2; l4−d; l1+d=2 : (4) s R2 R4 R s R2 R4 R ˜ We see that a ÿnite and nonzero value for the second argument of fs is retained exactly when l takes the value of the crossover scale l0. Thus, we conclude that the singular K. Binder et al. / Physica A 281 (2000) 112–128 115 Fig. 2. E ective susceptibility exponent ∗ above T for the three-dimensional Ising model with variable e c interaction range R (numbers in the key) plotted versus t=G, along with three theoretical calculations for this quantity; due to Refs. [34] (BK), [30] (BB), and [28,32] (SF), respectively. From Ref. [10]. part of the free energy scales with R as follows: ˜ −4d=(4−d) ˆ 2d=(4−d) 3d=(4−d) fs = R fs(˜r0R ; u;˜ hR ) ; (5) where a natural choice of coordinates (Fig. 1) is to measurer ˜0 andu ˜ 0 as distances from the Ising ÿxed point, unlike in the original Hamiltonian, where r0 and u0 are distances from the Gaussian ÿxed point. Eq. (5) describes how the temperature distancer ˜0 from criticality and the magnetic ÿeld h scale with the range of interaction R: Note that the crossover exponent is known exactly (unlike other cases of crossover, e.g., between the Ising and Heisenberg universality class in isotropic magnets with varying uniaxial anisotropy [68]). The same result for the crossover exponent follows [6], of course, from simple-minded arguments using the Ginzburg criterion. However, the location of the nontrivial ÿxed point u∗ (Fig. 1), the associated other exponents, and the explicit form of the scaling function ˜ fs cannot be obtained exactly. The calculation of the scaling function for the free-energy density or its derivatives, such as the susceptibility, is a nontrivial task for both renormalization-group and Monte Carlo calculations. This is demonstrated in Fig. 2 where the e ective critical exponent + e of the susceptibility for T¿Tc is plotted versus the thermal crossover scaling −6 variable t=G, with t =(T − Tc)=Tc being the reduced temperature and G = G0R the Ginzburg number in d = 3, for which G0 ≈ 0:277. Note that e ective exponents are deÿned as e ≡−dlnˆ=dln|t|; ˆ ≡ kBTc(R)(@M=@h)T ; (6) 116 K. Binder et al. / Physica A 281 (2000) 112–128 where refers to T ? Tc, respectively, M = hsiT;h, and the range R is deÿned from the second moment of the interaction (z being the e ective coordination number) , X X 2 2 R = |ri − rj| K(ri − rj) K(ri − rj) j6=i j6=i 1 X = |r − r |2 with |r − r |6R : (7) z i j i j m j6=i 2 Here the second equality holds only for a square-well potential and values Rm = 1; 2; 3; 4; 5; 6; 8; 12; 18; 28; 60; 100, and 160 were studied.