Fat-tailed and compact random-field Ising models on cubic lattices
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov RESEARCH REPORT
Fat-tailed and compact random-field Ising models oncubic lattices
Nuno Crokidakis and S´ılvio M. Duarte Queir´os Instituto de F´ısica - Universidade Federal FluminenseAv. Litorˆanea s/n, 24210-340 Niter´oi - RJ, Brazil Centro de F´ısica do PortoRua do Campo Alegre 687, 4169-007 Porto, Portugal e-mail: [email protected],[email protected]
Using a single functional form which is able to represent several different classes ofstatistical distributions, we introduce a preliminary study of the ferromagnetic Ising modelon the cubic lattices under the influence of non-Gaussian local external magnetic field.Specifically, depending on the value of the tail parameter, τ ( τ < τ < /
3, such distributions have finite standard deviation and they are either the Student- t (1 < τ < /
3) or the r -distribution ( τ <
1) extended to all plausible real degrees offreedom with the Gaussian being retrieved in the limit τ →
1. Otherwise, the distributionhas got the same asymptotic power-law behaviour as the α -stable L´evy distribution with α = (3 − τ ) / ( τ − τ → ∞ . Our resultspurport the existence of ferromagnetic order at finite temperatures for all the studied valuesof τ with some mean-field predictions surviving in the three-dimensional case. Keywords : Random-Field Ising Model, Fat tails, Compact distributions
PACS: 02.70.-c, 05.50.+q, 05.70.Fh, 64.60.-i, 75.10.Nr, 75.50.Lk
Dated : 20th Obtober 2010 at-tailed and compact random-field Ising models on cubic lattices I. Introduction and motivation
Disordered magnetic systems have long ceased being a “mere” theoretical exerciseto find a limelight place in the class of problems which enables the reasoning aboutthe equilibrium and out-of-equilibrium behaviour of several compounds that can beexperimentally studied. Accordingly, the Sherrington-Kirkpatrick spin-glass model [1]and the random field Ising model (RFIM) [2] are quintessential systems that have soonfound a correspondence with condensed matter and materials science case-studies [3].For uncertain reasons, perhaps related to the nature of the replica method and thetook for granted universality of the Gaussian, most of the studies have been devotedto the analysis of systems with n -modal or n -Gaussian disorder in either the couplingconstant or the local external magnetic field [4, 5, 6, 7, 8, 9, 10, 11]. However, thelast decade has been particularly prolific in examples of the ubiquity of non-Gaussiandistributions in natural systems and man-made phenomena [12, 13]. Therefore, it wascompletely upheld the introduction of generalized disordered magnetic systems in whichfat tailed and compact distributions in the local magnetic field [14] and fat tailed ( α -stable L´evy) distributions in the coupling constant were considered [15, 16, 17, 18]. Asa matter of fact, earlier experimental studies on organic charge-transfer compoundslike N-methyl-phenazium tetra-cyanoquinodimethanide (NMP-TCNQ), quinolinium-(TCNQ) , acridinium-(TCNQ) and phenazine-TCNQ suggested the experimentalvalidity of non-Gaussian (fat-tailed and uniform) distributions. [19] Furthermore, inalternative quantities and systems the importance of non-Gaussian distributions hasbeen also demonstrated. For instance, in Refs. [16, 17], it is shown that for the α -stableL´evy spin-glass the local fields are not Gaussian distributed.Concerning the non-Gaussian RFIM, a recent work [14] analyzed the caseof distributions with power-law asymptotic behaviour, namely the generalized r -distribution and the Student- t [20] in the limit of infinite-range interactions. In this case,interesting results were found, namely the appearance of a change of the concavity of thecritical transition line for τ > i.e. , exhibiting critical behaviour. In this particular case, it isknown that the RFIM cannot undergo a discontinuous phase transition for d ≤ i.e. , nophase transition at all. Alternatively, the field-theory approach based on the existenceof the dimensional reduction does not hold for the RFIM [22], thus leaving the existenceof critical behaviour for 3D systems under dispute.In order to better understand this problem, we herein report preliminary results onthe study of the critical frontier of the RFIM on cubic lattices with short-range (nearest- at-tailed and compact random-field Ising models on cubic lattices II. Model and Monte Carlo Simulation
We have considered the following Hamiltonian on a cubic lattice of linear size L : H = − J X S i S j − X i H i S i , (1)where S i = ± P being performed over the nearest-neighbour (NN)pairs of spins. J is the ferromagnetic interaction constant between NN and the randomfields { H i } are quenched variables ruled by a PDF that is defined by a parameter τ (generic degree of freedom). For τ < P i ( H i ) = r − τπ B τ Γ h − τ − τ ) i Γ (cid:2) − τ − τ (cid:3) [1 − B τ (1 − τ ) H i ] − τ , (2)(with | H | ≤ [ B τ (1 − τ )] − / ) which is the generalized r -distribution, and for τ >
1, wehave P s ( H i ) = r τ − π B τ Γ (cid:2) τ − (cid:3) Γ h − τ τ − i [1 − B τ (1 − τ ) H i ] − τ , (3)which is the generalized Student- t distribution also widely named q -Gaussian withinstatistical mechanics [13]. By generalized we mean that the degrees of freedom, m and n , of t - and r -distributions are extended to the entire domain of feasible real valuesaccording to the relations τ = ( m + 3) / ( m + 1) [ m ≥
0] and τ = ( n − / ( n − n ≥ . ] is the Gamma function and B τ is given by B τ = 1(3 − τ ) σ τ , (4)where σ τ is the width of the PDF. For τ < / σ , are naturally related by,(5 − τ ) σ = (3 − τ ) σ τ . (5)In Fig. 1, we show the PDFs (2) and (3) for some values of τ , in the linear-linear(left panel) and log-log scales (right panel). We can see the power-law behaviour for τ > J = 1 hereinafter. We studied systemsof L = 6 , , ,
12 and 16, with periodic boundary conditions and a random initialconfiguration of the spins by means of Monte Carlo simulations. The algorithm is at-tailed and compact random-field Ising models on cubic lattices - - P H H L Τ= (cid:144) Τ= (cid:144) Τ= Τ= (cid:144) Τ=- - - - - - P H H L Figure 1: (Color online) Random-field probability distributions for some values of theparameter τ (from bottom to top: τ = 5 / , / , , / − σ τ = 1 in all cases. as follows: a configuration of the random field { H } is generated at the beginning ofthe simulation according to the corresponding probability P i,s ( H ), and remains frozenduring the whole dynamics; then, every lattice site is visited and a spin flip occursaccording to the standard Metropolis rule. The results for all values of the magneticfield parameters τ and σ τ show that finite-size effects are less-pronounced for L ≥
10. Wehave used 2 × MC steps to equilibrate the system and 10 MC steps for averaging.Notice that these averages are MC or time averages. Nonetheless, in systems withquenched disorder we need to average over random-field realizations [23], i.e. , a certainquantity A has the mean value given by, h A i = 1 n r n r X i =1 ¯ A i , (6)where ¯ A i stands for MC average of the i -th sample and n r is the number of random-fieldrealizations. Thus, in addition to the MC steps used for time averaging, we have usednumber of samples than went up to 2000 samples of the distributions P i,s ( H ), althoughwe afterwards statistically verified by means of the resampling procedure [24] that thisnumber can be substantially diminished without jeopardizing the results regarding theanalysis of the critical nature and frontiers. However, the reader must be aware that fora survey on the universality class of the problem, which is defined by the set of criticalexponents, a whopping number of samples is mandatory.To obtain the quenched random fields we have used standard procedures forgenerating t -Student and r -distribution variables [25]. III. Numerical Analysis
As previously discussed [14], the PDFs of Eqs. (2) and (3) may be platykurtic , for τ < leptokurtic , for τ >
1. To study the critical properties of the model we performed at-tailed and compact random-field Ising models on cubic lattices T < m > block 1block 2block 3block 4block 5block 6 T < m > average τ = 2 Figure 2: (Color online) Magnetization per spin as a function of temperature for τ = 2, σ τ = 1 . L = 16 and some blocks with 200 samples each one. In the inset we showthe corresponding average magnetization per spin, considering all the 2000 samples. MC simulations for τ = −∞ , 0.2, 3/2, 5/3 and 2.0, in the range 0 . < T < . . ≤ σ τ ≤ .
4. Notice that for τ > /
3, we must use the PDF width σ τ instead of thestandard deviation σ . Next, we will present our results for the order parameter h m i ,the magnetization per spin, and the magnetic susceptibility χ . The latter is obtainedfrom the simulations by making use of the fluctuation-dissipation relation, χ = h m i − h m i kT (7)where k is the Boltzmann constant (we set k = 1) and h i stands for averages givenby Eq. (6). Before moving forwards we discuss the choice of our sampling. To thataim, we considered a quite broad distribution, τ = 2 which is equivalent to the Lorentzdistribution, and performed a total of 2000 samples in a system with L = 16 on whichwe have applied a jackknife-like procedure, which belongs to the established class ofresampling methods in Statistics [26]. Explicitly, we regrouped our magnetization datainto smaller groups. The smallest group size we considered was 200 elements. Alreadyat this group size we could achieve statistical significance in Student t -tests for a p -valueequal to 0 .
05 when we appraised the likelihood of having groups of samples of equaland unequal size (with non-coincident elements obtained from the same τ , σ τ and T parameters) with the same statistical moments. Bear in mind that if we consider 200samples of a L = 8 system, we are actually taking into reckoning over than 10 generatedquenched random fields, which are sufficient to have a worthy description of the fat tailbehaviour, namely the effect of very unlikely random fields. Such a validity can still beintuitively grasped when we plot the magnetization curve considering different groupsizes in Fig. 2. These data are results for simulations for τ = 2, L = 16 and differentsub-samples of 200 samples each. It is evident the concurrence between the sub-sets. Inthe inset of the same figure we plot the mean magnetization averaged over all the 2000samples which presents the same behaviour as the smaller sets. Once more, we wouldlike to emphasize that we are not setting our sights on the critical universality of the at-tailed and compact random-field Ising models on cubic lattices T < m > σ τ = 0.1 σ τ = 0.2 σ τ = 0.4 σ τ = 0.8 σ τ = 1.2 σ τ = 2.4 T U L L = 6L = 8L = 10L = 12L = 16 σ τ = 2.4 Figure 3: (Color online) Magnetization per spin as a function of temperature for aplatykurtic case corresponding to the uniform distribution with L = 16 and typicalvalues of σ τ , showing that ferromagnetic order occurs at finite temperatures. Forvalues of σ τ at least up to 2.4, we observe a continuous transition between the orderedand the disordered phases (left side). Observe the small variation of the transitiontemperatures for increasing values of σ τ as it happens in the mean-field limit [14]. Itis also shown the Binder cumulant for σ τ = 2 . T ∼ = 3 . system, but in the definition of critical regions instead, as done in previous cases (seee.g. Ref. [27]).As made in the mean-field case [14], we will study the platykurtic ( τ <
1) and theleptokurtic ( τ >
1) cases separately.
III.1. Platykurtic case: τ < τ → ∞ (see Fig. 3). The numerical results suggest thatthe transition temperature is not much affected by the values of σ τ , in agreement withmean-field predictions [14]. In addition, plotting the data for the Binder cumulant (leftpanel of Fig. 3)), obtained by U L = 1 − h m i h m i , (8)we can observe the signature of a phase transition, i.e., the crossing of the curves fordifferent lattice sizes at the critical temperature. For 3D, comparing the latter resultswith mean-field estimates, we are still able to perceive a phase transition for widths ofthe distribution greater than 1, which is the maximum value of σ τ wherein symmetry-breaking exists in the uniformly distributed external random-field case.In Fig. 4, we present our results for the magnetization per spin versus thetemperature for τ = 0 . L = 16 and typical values of σ τ . One can observe thatferromagnetic order occurs at low temperatures. We have continuous phase transitionsbetween the ordered ferromagnetic phase ( m →
1) and the disordered paramagnetic one at-tailed and compact random-field Ising models on cubic lattices T < m > σ τ = 0.1 σ τ = 0.2 σ τ = 0.4 σ τ = 0.8 σ τ = 1.2σ τ = 2.4 T U L L = 6L = 8L = 10L = 12L = 16 σ τ = 1.2 Figure 4: (Color online) Magnetization as a function of temperature for anotherplatykurtic case ( τ = 0 .
2) and typical values of σ τ (left side). It is also shown theBinder cumulant for σ τ = 1 . T ∼ = 4 . ( m →
0) for all studied values of σ τ . In the right panel of the same figure, we depictthe scale-behaviour of the Binder cumulant for a particular value of width, σ τ = 1 . τ wesoar the disorder. As mentioned for the uniform case, the cubic lattice presents a largerinterval of values of σ τ in which we can verify a phase transition.In conclusion, if the random fields are generated by platykurtic distributions( τ < σ τ ,with a ferromagnetic phase at low temperatures and a paramagnetic phase at hightemperatures. For large values of σ τ , the system is in the paramagnetic phase for alltemperatures. In addition, the critical temperatures are not much affected by increasingvalues of τ . III.2. Leptokurtic case: τ > τ we simulated. However, the critical temperatures are more sensitive to variationsin σ τ than the platykurtic region, in agreement with mean-field predictions [14].Concomitantly, we have understood a decrease of the maximum value of σ τ , σ max τ , forwhich it is possible to identify the existence of a ferro-paramagnetic transition at finitetemperature. Once again, it is straightforward to verify that the greater τ , the smaller σ max τ , since an augment of the value of τ encloses an increase of the disorder. Forinstance, comparing the results for τ = 5 / τ = 2, we notice that fixing σ τ = 2 . τ while we cannot at-tailed and compact random-field Ising models on cubic lattices T < m > σ τ = 0.1 σ τ = 0.2 σ τ = 0.4 σ τ = 0.8 σ τ = 1.2 σ τ = 2.4 Figure 5: (Color online) Magnetization as a function of temperature for the leptokurticcase ( τ = 5 / L = 16 and typical values of σ τ . The results suggest that for σ τ > . descry the existence of critical behaviour at finite temperature for the latter (see Figs. 5and 6). Comparing with high-dimensional result (mean field), we have the limit widthat T = 0 for σ τ =5 / = 1 / √ σ τ =2 = 2 /π [14]. We must note that the τ = 2corresponds to the sub-domain of values of the tail index we obtain a divergent freeenergy at T = 0 in the mean-field limit was calculated.One of the points that called our attention is the fact that for large values of thedisorder, particularly when there is an obvious tail effect, the magnetization we foundat low temperatures is significantly different from 1. In order to check this result wemade use of the mean-field treatment. Explicitly, we compared the numerical resultsfrom our Monte Carlo simulations with the solutions to the mean-field equation [14], m = Z + ∞−∞ d H P s ( H ) tanh β ( J m + H ) . (9)The results of two cases are depicted in Fig. 7. For a very small value of σ τ (0.1),the low-temperature magnetization per spin is equal to 1, as in the three-dimensionalcase, but it is visible that for σ τ = 0 . m < . .
4% in the numerical integration.Moreover, for large values of the temperature, which imply a paramagnetic phase, weobtained values of | m | ∼ = 0 . T versus τ for a fixed value σ τ = 1 . τ = 1). The computational results are ingood agreement with the heuristic expression, T ( τ ) = a (cid:2) − ( τ − τ c ) − (cid:3) − b , (10) at-tailed and compact random-field Ising models on cubic lattices T < m > σ τ = 0.1 σ τ = 0.2 σ τ = 0.4 σ τ = 0.8 σ τ = 1.2 σ τ = 2.4 T χ L = 6L = 8L = 10L = 12L = 16 σ τ = 0.4 Figure 6: (Color online) Magnetization as a function of temperature for anotherleptokurtic case case ( τ = 2) σ τ = 1 . τ = 2) σ τ = 0 . T < m > σ τ = 0.1 σ τ = 0.4 Figure 7: (Color online) Mean-field magnetization as a function of temperature for τ = 2 and σ = 0 . .
4. We can observe in this high-dimensional limit the samebehaviour at low temperatures presented by the three-dimensional case. The pointswere obtained numerically and the line is presented as guide to the eye. when a = 5 . ± . b = 0 . ± .
04 and τ c = 2 . ± .
002 ( R = 0 .
998 and χ /n = 0 . τ > τ , but they do notdecrease significantly for τ <
1, as above-discussed. The validity of the adjustmentcan be tested by comparing the asymptotic limit of our ansatz (10), which is equalto a , with the critical temperature of the uniform distribution case, T c = 4 .
41. Thisapproximately corresponds to a 10% difference (considering error) which is acceptabletaking into account the reduced number of parameters we assumed so that the Akaikeinformation criterion is implicitly minimized [30]. at-tailed and compact random-field Ising models on cubic lattices τ T σ τ = 1.2 F P
Figure 8: (Color online) Sketch of the phase diagram of the model in theplane Temperature versus τ , for σ τ = 1 .
2, separating the Ferromagnetic ( F ) andParamagnetic ( P ) phases. The points are Monte Carlo estimates of the criticaltemperatures and and the line is the fit using Eq. (10). IV. Remarks and Outlook
In this work, we have presented a preliminary survey on both the existence and thenature of critical behaviour of the random-field Ising model on cubic lattices withnearest-neighbour interactions by means of Monte Carlo simulations based on thestandard Metropolis algorithm. In order to generate the quenched random fields,we have considered a family of continuous probability density functions, defined by aparameter τ comprising the r -distribution, for τ <
1, and the Student- t , for τ >
1. Thisset of distributions allowed us to study a larger variety of cases that go from the uniformdistribution to fat-tailed distributions with the same asymptotic regime as the L´evydistribution that were already used in the study of vitreous systems [15], and for whichit was recently found that the local fields follow non-Gaussian distributions [16, 17].These models are considered relevant to the description of spin-glasses with Ruderman-Kittel-Kasuya-Yosida interaction and spin glasses exhibiting a wide hierarchy of couplingintensities as well. The current results show that in the cubic lattice, this RFIM modelpresents many of the traits found for the mean-field case [14]. Namely, based on theanalysis of different quantities such as the order parameter, response function and Bindercumulant, we have found evidence for the existence of continuous phase transitions forall values of τ with the critical temperature, T c , depending on the characteristic widthof the distribution, σ τ , and the tail index, τ , itself. As expected, the spell of σ τ in whichwe can detect a ferro-paramagnetic phase transition is greater than the region found inthe mean-field approach. Although we might be tempted to think that the parameter τ represents just a supplementary way of tuning the non-thermal disorder introducedin the system, we must take into account results obtained at T = 0 where the thermaldisorder is quelled. Our previous calculations have shown quite different behaviour ofthermodynamical quantities depending on the parameter τ , particularly for distributions at-tailed and compact random-field Ising models on cubic lattices Acknowledgements
We would like to thank M. Azeinman for exchange of correspondence and I. Giardina forclarifying comments. We would also like to acknowledge the collaboration of D.O. Soares-Pinto during the embryonic phase of this work. NC thanks the financial support from theBrazilian funding agency CNPq.
References [1] D. Sherrington, S. Kirkpatrick, Phys. Rev. Lett. , 1792 (1975)[2] Y. Imry and S.k. Ma, Phys. Rev. Lett. , 1399 (1975)[3] S. Fishman and A. Aharony, J. Phys. C 12, L729 (1979); C. Dekker, A. F. M. Arts, and H. W. deWijn, Phys. Rev. B 38, 8985 (1988); J. A. Mydosh, Spin Glasses: An Experimental Introduction (Taylor & Francis, London, 1994) K. Binder and W. Kob,
Glassy Materials and DisorderedSolids: An Introduction to Their Statistical Mechanics (World Scientific, Singapore, 2005); M.Schechter, Phys. Rev. B , 020401(R) (2008)[4] A. Aharony, Phys. Rev. B , 3318 (1978).[5] T. Schneider and E. Pytte, Phys. Rev. B , 1519 (1977).[6] M. Kaufman, P. E. Klunzinger and A. Khurana, Phys. Rev. B , 4766 (1986).[7] N. Crokidakis and F. D. Nobre, J. Phys. Condens. Matter , 145211 (2008).[8] O. R. Salmon, N. Crokidakis and F. D. Nobre, J. Phys. Condens. Matter , 056005 (2009).[9] A. N. Berker, Phys. Rev. B , 5243 (1984).[10] L. Hern´andez and H. Ceva, Phys. A , 2793 (2008).[11] Y. Wu and J. Machta, Phys. Rev. B , 064418 (2006).[12] N. Mercadier, W. Gerin, M. Chevrollier and R. Kaiser, Nature Phys. , 602 (2009); D. Sornette, Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder (Springer, Berlin, 2006);
Complexity from Microscopic to Macroscopic Scales: Coherence andLarge Deviations , edited by A. T. Skjeltorp and T. Vicsek (Kluwer Academic Publishers,Dordrecht, 2002)[13] C. Tsallis,
Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World (Springer, Berlin, 2009)[14] S. M. Duarte Queir´os, N. Crokidakis and D. O. Soares-Pinto, Phys. Rev. E , 011143 (2009)[15] P. Cizeau and J. -P. Bouchaud, J. Phys. A 26 , L187(1993)[16] K. Janzen, A. K. Hartmann A. and Engel, J. Stat. Mech. P04006 (2008); I. Neri, F. L. Metz andD. Boll´e D., J. Stat. Mech. P01010 (2010)[17] K. Janzen, A. Engel and M. M´ezard, EPL , 67002 (2010)[18] K. Janzen, A. Engel and M. M´ezard, Phys. Rev. E , 021127 (2010)[19] G. Theodorou and M.H. Cohen, Phys. Rev. Lett. , 1014 (1976); G. Theodorou, Phys. Rev. B , 2264 (1977); C. Dasgupta and S. Ma, Phys. Rev. B , 1305 (1980).[20] A. M. C. de Souza and C. Tsallis, Physica A , 52 (1997)[21] M. Aizenman and J. Wehr, Phys. Rev. Lett. , 2503 (1989); M. Aizenman and J. Wehr, Comm.Math. Phys. , 489 (1990).[22] C. de Dominicis and I. Giardina, Random Fields and Spin Glasses - A Field Theory Approach (Cambridge University Press, Cambridge, 2006) at-tailed and compact random-field Ising models on cubic lattices [23] K. Binder and A. P. Young, Rev. Mod. Phys. , 801 (1986).[24] B. Efron, Ann. Stat. , 1 (1979)[25] R. W. Bailey, Math. Comput. , 779 (1994)[26] P. I. Good, Permutation, Parametric and Bootstrap Tests of Hypotheses (Springer, New York,2005)[27] M. Itakura, Phys. Rev. B , 012415 (2001).[28] A.R. Krommer and C.W. Ueberhuber, Computational Integration (SIAM Publications,Philadelphia, 1998)[29] J. Machta, M. E. J. Newman and L. B. Chayes, Phys. Rev. E , 8782 (2000).[30] H. Akaike, IEEE Trans. Aut. Contr.19