The 3D Ising model at the ferromagnetic transition line
We study the critical behavior of the three-dimensional Ising model [with a random-exchange probability ] at the transition line between the paramagnetic and ferromagnetic phase, which extends from to a multicritical (Nishimori) point at . By a finite-size scaling analysis of Monte Carlo simulations at various values of in the region , we provide strong numerical evidence that the critical behavior along the ferromagnetic transition line belongs to the same universality class as the three-dimensional randomly-dilute Ising model. We obtain the results and for the critical exponents, which are consistent with the estimates and at the transition of randomly-dilute Ising models.
pacs:75.10.Nr, 75.40.Cx, 75.40.Mg, 64.60.Fr
The Ising model has played an important role in the study of the effects of quenched random disorder and frustration on Ising systems. It is defined by the lattice Hamiltonian
where , the sum is over the nearest-neighbor sites of a simple cubic lattice, and the exchange interactions are uncorrelated quenched random variables, taking values with probability distribution
For we recover the standard Ising model, while for we obtain the usual bimodal Ising spin-glass model.
The phase diagram of the three-dimensional (3D) Ising model is sketched in Fig. 1. The high-temperature phase is paramagnetic for any . The low-temperature phase depends on the value of : it is ferromagnetic for small values of , while it is a spin-glass phase with vanishing magnetization for sufficiently large values of . The different phases are separated by transition lines, which meet at a multicritical point located along the so-called Nishimori line. Nishimori-81 (); LB-88 (); KR-03 () The spin-glass transition has been mostly studied at the symmetric point , see, e.g., Refs. KR-03, ; KKY-06, and references therein. The spin-glass transition line extends up to the Nishimori multicritical point,LB-88 () located at ON-87 (); Singh-91 (); OI-98 (); pnest () . For larger values of , the transition is ferromagnetic, up to where one recovers the pure Ising model, and therefore a transition in the Ising universality class. At the ferromagnetic transition line, for , the critical behavior is expected to belong to a different universality class.
An interesting hypothesis, which has already been put forward in Refs. Hukushima-00, ; KR-03, , is that the ferromagnetic transition of the Ising model belongs to the 3D randomly-dilute Ising (RDIs) universality class (see, e.g., Refs. PV-02, ; FHY-03, for reviews on randomly-dilute spin models). A representative of the RDIs universality class is the randomly site-dilute Ising model (RSIM) defined by the lattice Hamiltonian
where are uncorrelated quenched random variables, which are equal to with probability
For and above the percolation threshold of the spins ( on a cubic lattice BFMMPR-99 ()), the RSIM undergoes a continuous phase transition between a disordered and a ferromagnetic phase, whose nature is independent of . This transition is definitely different from the usual Ising transition: for instance, the correlation-length critical exponent HPPV-07 (); CMPV-03 (); PV-00 (); BFMMPR-98 () differs from the Ising value CPRV-02 (); DB-03 () . The RDIs universality class is expected to describe the ferromagnetic transition in generic diluted ferromagnetic systems. For instance, it has been verified that also the randomly bond-diluted Ising model (RBIM) belongs to the RDIs universality class.HPPV-07 (); Janke-POS () These results do not necessarily imply that also the Ising model has an RDIs ferromagnetic transition line. Indeed, while the RSIM (3) has only ferromagnetic exchange interactions, the Ising model is frustrated for any value of . Therefore, the ferromagnetic transition in the Ising model belongs to the RDIs universality class only if frustration is irrelevant, a fact that is not obvious and should be carefully investigated.
Reference Hukushima-00, investigated the issue by means of a Monte Carlo (MC) renormalization-group (RG) study, claiming that the Ising model belongs to the same RDIs universality class as the RSIM and the RBIM. It should be noted however that the quoted estimate for the correlation-length exponent at the ferromagnetic transition, , is close to but not fully consistent with the RDIs value .HPPV-07 () Another numerical MC work IOK-99 () investigated the nonequilibrium relaxation dynamics of the Ising model and showed an apparent nonuniversal dynamical critical behavior along the ferromagnetic transition line. These results are not conclusive and further investigation is called for to clarify this issue.
In this paper we focus on the transition line of the 3D Ising model between the paramagnetic and the ferromagnetic phase. We investigate the critical behavior by means of MC simulations at various values of in the region . Our finite-size scaling (FSS) analysis provides a strong evidence that the critical behavior of the 3D Ising along the ferromagnetic line belongs to the 3D RDIs universality class. For example, we obtain and , which are in good agreement with the presently most accurate estimates HPPV-07 () and for the 3D RDIs universality class.
The paper is organized as follows. In Sec. II we summarize some FSS results which are needed for the analysis of the MC data, and describe our strategy to check whether the transition belongs to the RDIs universality class. In Sec. III we describe the MC simulations. In Sec. IV we report the results of the FSS analysis. Finally, in Sec. V we draw our conclusions. In App. A we report the definitions of the quantities we compute.
Ii Strategy of the finite-size scaling analysis
In this work we check whether the ferromagnetic transition line in the 3D Ising models belongs to the RDIs universality class. For this purpose, we present a FSS analysis of MC data for various values of in the region . We follow closely Ref. HPPV-07, , which studied the ferromagnetic transition line in the 3D RSIM and RBIM and provided strong numerical evidence that these transitions belong to the same RDIs universality class. We refer to Ref. HPPV-07, for notations (a short summary is reported in App. A) and a detailed discussion of FSS in these disordered systems.
According to the RG, in the case of periodic boundary conditions and for , where is the lattice size, a generic RG invariant quantity at the critical temperature behaves as
where is the universal infinite-volume limit and and are the leading and next-to-leading correction-to-scaling exponents. In RDIs systems scaling corrections play an important role,BFMMPR-98 (); CPPV-04 () since is quite small. Indeed we have and in the 3D RDIs universality class.HPPV-07 () These slowly-decaying scaling corrections make the accurate determination of the universal asymptotic behavior quite difficult.
Instead of computing the various quantities at fixed Hamiltonian parameters, we keep a RG invariant quantity fixed at a given value .Has-99 () This means that, for each , we determine the pseudocritical inverse temperature such that
All interesting thermodynamic quantities are then computed at . The pseudocritical inverse temperature converges to as . The value can be specified at will, as long as is taken between the high- and low-temperature fixed-point values of . The choice (where is the critical-point value) improves the convergence of to for ; indeed for generic values of , while for . This FSS method has already been applied to the study of the critical behavior of -vector spin models, Has-99 (); CHPV-06 () and of randomly-dilute Ising models.HPPV-07 ()
As in Ref. HPPV-07, , we perform a FSS analysis at fixed , which is very close to the fixed-point value of at . Given any RG invariant quantity , such as the quartic cumulants and , we consider its value at fixed , i.e., . For , behaves as :
where the coefficients depend on the Hamiltonian. The derivative with respect to of a generic RG invariant quantity behaves as
Finally, the FSS of the magnetic susceptibility is given by HPPV-07 ()
where represents the background contribution.
A standard RG analysis, see, e.g., Ref. HPPV-07, , shows that the amplitudes of the scaling corrections are proportional to (with a universal coefficient), where is the leading irrelevant scaling field with RG dimension . Hamiltonians such that —we call them improved Hamiltonians—have a faster approach to the universal asymptotic behavior, because the scaling corrections vanish: in Eqs. (7), (8), and (9). In this case the leading scaling corrections are proportional to , where is the next-to-leading irrelevant scaling field and is its RG dimension. In Ref. HPPV-07, it was shown that the RSIM for and the RBIM for are improved. Since scaling fields are analytic functions of the Hamiltonian parameters, must be proportional to close to , i.e. . Therefore, since the coefficients , , and that appear in Eqs. (7), (8), and (9) are proportional to , we have
Beside the quantities defined in App. A, we also consider observables—in analogy with the previous terminology, we call them improved quantities—characterized by the fact that the leading scaling correction proportional to (approximately) vanishes in any model belonging to the RDIs universality class.HPPV-07 () We consider the combination of quartic cumulants
and improved estimators of the critical exponent defined as
( is defined in App. A). In Ref. HPPV-07, we showed that, if the transition belongs to the RDIs universality class, the leading scaling correction proportional to of these improved observables is suppressed. More precisely, we showed that the universal ratio of the amplitudes of the leading scaling correction in and satisfies
while the one for the quantities and is bounded by
The remaining scaling corrections are of order and . These improved observables are particular useful to check whether the transition in a given system belongs to the 3D RDIs universality class.
To summarize: in order to check whether the ferromagnetic transition of the 3D Ising model belongs to the RDIs universality class, we perform a FSS analysis at fixed , and check if the results for the critical exponents and other universal quantities are consistent with those obtained for the RDIs universality class, which is characterized by HPPV-07 () critical exponents and , by the leading and next-to-leading scaling-correction exponents and and by the universal infinite-volume values of the quartic cumulants , , and . Notice that the fact that we fix does not introduce any bias in our FSS analysis.
Iii Monte Carlo simulations
We performed MC simulations of Hamiltonian (1) with for , close to the critical temperature on cubic lattices of size with periodic boundary conditions, for a large range of lattice sizes: from to for , to for , and to for . We chose values of not too close to : indeed, as we expect crossover effects due to the presence of the Ising transition for and, therefore, that the asymptotic behavior sets in only for large values of . We return to this point later.
We used a Metropolis algorithm and multispin coding. multispin () In the simulation systems evolve in parallel, where or depending on the computer that is used. For each of these systems we use a different set of couplings . This allows us to perform 64 parallel simulations on a 64-bit machine, and therefore to gain a large factor in the efficiency of the MC simulations. We used high-quality random-number generators, such as the RANLUX ranlxd () or the twister twister () generators. footnote1 () Using the twister random-number generator, we need about seconds for one Metropolis update of a single spin on an Opteron processor running at 2 GHz. Our simulations took approximately 3 CPU years on an Opteron (2 GHz) processor.
It is worth mentioning that cluster algorithms, such as the Swendsen-Wang cluster SW-87 () and the Wolff single-cluster Wolff-89 () algorithm, show significant slowing down in the Ising model. At the earlier stage of this work we performed some simulations of the Ising model at using the algorithm used in Ref. HPPV-07, to simulate the RSIM and the RBIM. There we used a combination of Metropolis, Swendsen-Wang cluster,SW-87 () and Wolff single-cluster Wolff-89 () updates. More precisely, each updating step consisted of 1 Swendsen-Wang update, 1 Metropolis update, and single-cluster updates. In all cases the exponential autocorrelation times of the magnetic susceptibility was small: in units of the above updating step, even for the largest lattice sizes considered, i.e. . In the Ising model at autocorrelation times are much larger. In Fig. 2 we plot estimates of as obtained from the magnetic susceptibility. They show a clear evidence of critical slowing down: with . Such a value of should be compared with the dynamic exponent of Swendsen-Wang and Wolff cluster algorithms in the RSIM, which is much smaller: IIBH-06 () . These results show that cluster algorithms behave differently in the Ising model, likely due to frustration. They suggest that frustration is relevant for the cluster dynamics.
Taking also into account the computer time required by the cluster algorithms, we then turned to a multispin Metropolis algorithm. This turns out to be much more effective at the lattice sizes considered, although it has a larger dynamic exponent , see, e.g., Ref. IIBH-06, and references therein. We also mention that the autocorrelation time significantly increases with decreasing (keeping fixed). For example, for it increases by approximately a factor of 10 from to . This represents a major limitation to perform simulations for large lattices close to the multicritical point.
For each lattice size we considered disorder samples, with decreasing with increasing , from for to for the largest lattices. For each disorder sample, we collected a few hundred independent measurements at equilibrium. The averages over disorder are affected by a bias due to the finite number of measures at fixed disorder.BFMMPR-98-b (); HPPV-07 () A bias correction is required whenever one considers the disorder average of combinations of thermal averages. We used the formulas reported in App. B of Ref. HPPV-07, . Errors were computed from the sample-to-sample fluctuations and were determined by using the jackknife method. footnote1 ()
Our FSS analysis is performed at fixed . In order to determine expectation values at fixed , one needs the values of the observables as a function of in some neighborhood of the inverse temperature used in the simulation. In Ref. HPPV-07, we used the reweighting method for this purpose. This requires that the observables and, in particular, the values of the energy are stored at each measurement. For the huge statistics like those we have for the smaller values of , this becomes unpractical. Therefore, we used here a second-order Taylor expansion, determining from . The coefficients and are obtained from appropriate expectation values as in Ref. CHPV-06, . Since their computation involves disorder averages of products of thermal averages, we have implemented in all cases an exact bias correction, using the formulas of Ref. HPPV-07, . Derivatives with respect to are then obtained as . Of course, this method requires to be sufficiently small. We have carefully checked the results by performing, for each and , runs at different values of .
Iv Finite-size scaling analysis
In this section we present the results of our FSS analysis of the MC data at fixed .
iv.1 Renormalization-group invariant quantities
In Fig. 3 we show the MC estimates of versus with , which is the leading scaling exponent of the RDIs universality class. The data vary significantly with and . This and dependence is always consistent with the existence of the expected next-to-leading scaling corrections, i.e. with a behavior of the form
where , and are fixed to the RDIs values:HPPV-07 () , and . The fits corresponding to are shown in Fig. 3. Note that in most of the cases it is crucial to include a next-to-leading correction. Only for the data are well fitted by taking only the leading scaling correction.
An unbiased estimate of can be obtained from the difference of data at different values of , i.e. by considering
Linear fits of the logarithm of these differences give results in reasonable agreement with the RDIs estimate , especially when only data corresponding to are used. For , we obtain  from the data at and ,  from those at and , and  from the results at and . We also fitted the difference at to (for this value of next-to-leading corrections are apparently very small, see Fig. 3). We obtain  for .
The results of the above-reported fits of show that the leading scaling corrections proportional to vanish for . Note that, close to , the relevant next-to-leading scaling corrections should be those proportional to with . Indeed, according to Eq. (10), the coefficient of those proportional to is of order , i.e. . Therefore, is small if (we checked this numerically). This applies to the FSS at and 0.90, where the corrections can be neglected, although in these two cases we cannot neglect the leading correction whose coefficient is proportional to . An analysis of the leading scaling corrections at , assuming the RDIs values and (we perform combined fits to (15) with ) gives the estimate
which is approximately in the middle of the ferromagnetic line, i.e. . We performed a similar analysis for , obtaining a consistent estimate of . Thus, the Ising model for is approximately improved. Therefore, at , fits of the data assuming leading scaling corrections should provide reliable results.
As discussed in Sec. II, a useful quantity to perform stringent checks of universality within the RDIs universality class is the combination of quartic cumulants reported in Eq. (11). For this quantity the scaling corrections proportional to are small, cf. Eq. (13), and thus the dominant corrections should behave as , with . As already discussed, for values of close to , such as , also the term is expected to be small and thus the dominant corrections should scale as with . In Fig. 4 we show the MC results for for various values of . Fig. 5 shows results of fits to
with . We obtain respectively for , fixing (the error in brackets is related to the uncertainty of ) and using data with ; moreover we obtain respectively for , fixing and using data with . For all values of the results are in good agreement with RDIs estimate HPPV-07 () . They provide strong support to a RDIs critical behavior along the ferromagnetic line.
A further stringent check of universality comes from the analysis of the data at , because the data of are very precise due to a cancellation of the statistical fluctuations.HPPV-07 () Since the model is improved, the scaling corrections are negligible and the large- behavior is approached with corrections of order , . We thus fit the data to Eq. (18) with . In Fig. 6 we show the results. We obtain (the error in brackets is related to the uncertainty of ) for respectively. Moreover, by fitting the data to with and , we obtain for respectively. These results are in perfect agreement with the RDIs estimate HPPV-07 () . Such an agreement is also confirmed by the analysis of the data of , for example a fit to Eq. (18) with gives for , to be compared with the RDIs estimate HPPV-07 () .
We have not shown results for values of too close to 1, for say, because they are affected by crossover effects due to presence of the Ising transition for , as it also occurs in randomly dilute Ising models. BFMMPR-98 (); CPPV-04 (); footnotecrossover () For instance, for the data are not compatible with a behavior of the form (15) with fixed to the RDIs value. Our data that correspond to lattice sizes apparently converge to a smaller value, consistently with the expected crossover from pure to random behavior (in pure systems . The same quantitative differences are observed in the RSIM and in the RBIM close to the Ising transition. This suggests that in FSS analyses up to the asymptotic RDIs behavior can only be observed for .
iv.2 Critical exponents
The correlation-length exponent can be estimated by fitting the derivative of and to the expression (8). Accurate estimates are only obtained for improved Hamiltonians. For generic models, as shown in Ref. HPPV-07, , good estimates are only obtained by using improved estimators, such as those reported in Eq. (12).
We analyze the data at , which is a very good approximation of the improved value . In Fig. 7 we report several results for the critical exponent , obtained by analyzing , , and their improved versions and . We show results of fits of their logarithms to
fixing to several values. Since the Hamiltonian is approximately improved, scaling corrections are expected to decrease as with . Since is only approximately equal to , one may be worried of the residual leading scaling corrections that are small but do not vanish exactly. Improved estimators should provide the most reliable results since the leading scaling corrections are additionally suppressed.
As can be seen in Fig. 7, the results obtained by using and and are perfectly consistent with those obtained from their improved versions. This confirms that the Hamiltonian is improved. Fits of to (19) with do not provide stable results. The results approach the values obtained in the other fits only when increasing the minimum size allowed in the fit. This is expected, since the corrections should be negligible with respect to the ones. In conclusion, our final estimate of the correlation-length exponent is
Estimate (20) is also confirmed by the analysis of the data at the other values of . Fig. 8 shows results obtained by fitting the logarithm of to the function (19) for other values of . They are definitely consistent with the result obtained at . Results for are not shown because the available data are not sufficient to get reliable results.
In order to estimate the critical exponent , we analyze the FSS of the magnetic susceptibility , cf. Eq. (9). We fit it to (where represents a constant background term), to , and to (more precisely, we fit to the logarithm of the previous expressions). The results at are shown in Fig. 9, versus the minimum size allowed in the fits. We obtain the estimate
which includes all results obtained for . This estimate agrees with the most precise RDIs estimate . Fig. 10 shows results for the other values of . Again, they are in good agreement.
iv.3 The critical temperature
The critical temperature can be estimated by extrapolating the estimates of at , cf. Eq. (6). Since we have chosen , HPPV-07 () we expect in general that . For , since the model is approximately improved, the leading scaling corrections are related to the next-to-leading exponent . Thus, in this case . This behavior is nicely observed in Fig. 11, which shows at vs with . A fit to gives .
For the other values of we expect with . Linear fits of (for with sufficiently large to give an acceptable ) give the estimates for , for , for , for , for . We finally recall that DB-03 () for (the standard Ising model), and that OI-98 () at the multicritical Nishimori point at . In Fig. 12 we plot the available estimates of the critical temperature in the region .
The estimates of shown in Fig. 12 hint at a smooth linear behavior for small values of , close to the Ising point at . This can be explained by some considerations on the multicritical behavior around the Ising point at . The Ising critical behavior at is unstable against the RG perturbation induced by quenched disorder at , Harris-74 () which leads to the RDIs critical behavior. Indeed such a perturbation has a positive RG dimension at the Ising fixed point: Aharony-76 (); PV-02 () where and are the Ising specific-heat and correlation-length critical exponents, and therefore CPRV-02 () . Thus, in the absence of an external magnetic field, beside the scaling field related to the temperature, there is another relevant scaling field associated with the quenched disorder parameter . General RG scaling arguments LS-84 (); CPPV-04 () show that the singular part of the free energy for behaves as
where is the crossover exponent, and is a crossover scaling function which is universal (apart from normalizations). The scaling fields and depend on the parameters of the model. In general, we expect
where , is the critical temperature of the Ising model, and is a constant. No such mixing between and occurs in , since vanishes for . Hence, we can take . The system has a critical transition for at . Since the singular part of the free energy close to a critical point behaves as ( is the specific-heat exponent of the RDIs universality class), we must have , where is the value of obtained by setting (see, e.g., Ref. PV-07, and references therein). Hence, we obtain
where the dots indicate higher-order terms. This expression provides the dependence of the critical temperature for small. Note that the nonanalytic term in Eq. (25) is suppressed with respect to the analytic ones, because . Thus, . Since , we can also infer that . From the results for we estimate for the Ising model.
In this paper we have studied the critical behavior of the 3D Ising model at the transition line between the paramagnetic and the ferromagnetic phase, which extends from to a multicritical (Nishimori) point at . We presented a FSS analysis of MC simulations at various values of in the region . The results for the critical exponents and other universal quantities are consistent with those of the RDIs universality class. For example, we obtained and , which are in good agreement with the presently most accurate estimates HPPV-07 () and for the 3D RDIs universality class. Therefore, our FSS analysis provides a strong evidence that the critical behavior of the 3D Ising along the ferromagnetic line belongs to the 3D RDIs universality class.
We also note that the random-exchange interaction in the Ising model gives rise to frustration, while the RDIs universality class describes transitions in generic diluted Ising systems with ferromagnetic exchange interactions. This implies that frustration is irrelevant at the ferromagnetic transition line of the 3D Ising model. Moreover, the observed scaling corrections are consistent with the RDIs leading and next-to-leading scaling correction exponents and . This indicates that frustration does not introduce new irrelevant perturbations at the RDIs fixed point with RG dimension .
Appendix A Notations
We define the two-point correlation function
where the overline indicates the quenched average over the probability distribution. Then, we define the corresponding susceptibility and the correlation length
where , , and is the Fourier transform of . We also consider quantities that are invariant under RG transformations in the critical limit. Beside the ratio
we consider the quartic cumulants , and defined by
We also define corresponding quantities , , and at fixed . Finally, we consider the derivative of , and of , with respect to , which allow one to determine the critical exponent .
- (1) H. Nishimori, Prog. Theor. Phys. 66, 1169 (1981).
- (2) P. Le Doussal and A.B. Harris, Phys. Rev. Lett. 61, 625 (1988).
- (3) N. Kawashima and H. Rieger, in Frustrated Spin Systems, edited by H.T. Diep (World Scientific, Singapore, 2004); cond-mat/0312432.
- (4) H. Katzgraber, M. Körner, and A.P. Young, Phys. Rev. B 73, 224432 (2006).
- (5) Y. Ozeki and H. Nishimori, J. Phys. Soc. Japan 56, 3265 (1987).
- (6) R.R.P. Singh, Phys. Rev. Lett. 67, 899 (1991).
- (7) Y. Ozeki and N. Ito, J. Phys. A 31, 5451 (1998).
- (8) Refs. ON-87, ; Singh-91, ; OI-98, report the estimates , , and respectively.
- (9) K. Hukushima, J. Phys. Soc. Japan 69, 631 (2000).
- (10) A. Pelissetto and E. Vicari, Phys. Rept. 368, 549 (2002).
- (11) R. Folk, Yu. Holovatch, and T. Yavors’kii, Uspekhi Fiz. Nauk 173, 175 (2003) [Phys. Usp. 46, 175 (2003)].
- (12) H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo, J. Phys. A 32, 1 (1999).
- (13) M. Hasenbusch, F. Parisen Toldin, A. Pelissetto, and E. Vicari, J. Stat. Mech.: Theory Exp. P02016 (2007).
- (14) P. Calabrese, V. Martín-Mayor, A. Pelissetto, and E. Vicari, Phys. Rev. E 68, 036136 (2003).
- (15) A. Pelissetto and E. Vicari, Phys. Rev. B 62, 6393 (2000).
- (16) H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, and J.J. Ruiz-Lorenzo, Phys. Rev. B 58, 2740 (1998).
- (17) M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. E 65, 066127 (2002).
- (18) Y. Deng and H.W.J. Blöte, Phys. Rev. E 68, 036125 (2003).
- (19) W. Janke, in Proceedings of the XXIII International Symposium on Lattice Field Theory, Dublin, July 2005, POS(LAT2005)018
- (20) N. Ito, Y. Ozeki, and H. Kitatani, J. Phys. Soc. Jpn. 68, 803 (1999).
- (21) P. Calabrese, P. Parruccini, A. Pelissetto, and E. Vicari, Phys. Rev. E 69, 036120 (2004).
- (22) M. Hasenbusch, J. Phys. A 32, 4851 (1999).
- (23) M. Campostrini, M. Hasenbusch, A. Pelissetto, and E. Vicari, Phys. Rev. B 74, 144506 (2006); Phys. Rev. B 63, 214503 (2001).
- (24) See, e.g., S. Wansleben, J.B. Zabolitzky, and C. Kalle, J. Stat. Phys. 37, 271 (1984); G. Bhanot, D. Duke, and R. Salvador, Phys. Rev. B 33, 7841 (1986).
- (25) M. Lüscher, Comput. Phys. Commun. 79, 100 (1994).
- (26) The SIMD-oriented fast Marsenne twister random number generator has been introduced by M. Matsumoto and M. Saito. Details can be found in M. Saito, Master Thesis (2007) and at http://www.math.sci.hiroshima-u.ac.jp/ m-mat/MT/emt.html.
- (27) In order to make the use of these expensive (in terms of CPU-time) generators affordable, we employed the same sequence of random numbers for the update of all systems (for the initialization of the configurations at the beginning of the simulation we used independent random numbers for each of the systems). This may give rise to a statistical correlation among the systems. This effect is probably small and we have not detected it. Anyway, in order to ensure a correct estimate of the statistical error, all systems that use the same sequence of random numbers have been put in the same bin in our jackknife analysis.
- (28) R.H. Swendsen and J-S. Wang, Phys. Rev. Lett. 58, 86 (1987).
- (29) U. Wolff, Phys. Rev. Lett. 62, 361 (1989).
- (30) D. Ivaneyko, J. Ilnytskyi, B. Berche, and Yu. Holovatch, Physica A 370, 163 (2006).
- (31) H.G. Ballesteros, L.A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, and J.J. Ruiz-Lorenzo, Nucl. Phys. B 512, 681 (1998).
- (32) The crossover exponent from pure Ising to RDIs critical behavior is the Ising specific-heat exponent Aharony-76 () , see also Sec. IV.3. This implies that the crossover scaling variable in the FSS at is given by the combination , where , , and is a normalization constant. When , strong crossover effects are expected for , which corresponds to . The RDIs asymptotic critical behavior is observed for .
- (33) A. B. Harris, J. Phys. C 7, 1671 (1974).
- (34) A. Aharony, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz (Academic Press, New York, 1976), Vol. 6, p. 357.
- (35) I.D. Lawrie and S. Sarbach, in Phase Transitions and Critical Phenomena, Vol. 9, edited by C. Domb and J. Lebowitz (Academic, London, 1984).
- (36) A. Pelissetto and E. Vicari, cond-mat/0702273.