WOS: 000379821800005The five-dimensional ferromagnetic Ising model is simulated on the Creutz cellular automaton algorithm using finite-size lattices with linear dimension 4 <= L <= 8. The critical temperature value of infinite lattice is found to be T-chi(infinity) = 8.7811 (1) using 4 <= L <= 8 which is also in very good agreement with the precise result. The value of the field critical exponent (delta = 3.0067 (2)) is good agreement with delta = 3 which is obtained from scaling law of Widom. The exponents in the finite-size scaling relations for the magnetic susceptibility and the order parameter at the infinite-lattice critical temperature are computed to be 2.5080 (1), 2.5005 (3) and 1.2501 (1) using 4 <= L <= 8, respectively, which are in very good agreement with the theoretical predictions of 5/2 and 5/4. The finite-size scaling plots of magnetic susceptibility and the order parameter verify the finite-size scaling relations about the infinitelattice temperature

Aras, N.

Kurkcu, C.

Merdan, Z.

Acta Physica Polonica A

The five-dimensional ferromagnetic Ising model is simulated on the Creutz cellular automaton algorithm using finite-size lattices with linear dimension 4 ≤ L ≤ 8. The critical temperature value of infinite lattice is found to be $T^{χ}$ (∞=8.7811 (1) using 4 ≤ L ≤ 8 which is also in very good agreement with the precise result. The value of the field critical exponent (δ =3.0067(2)) is good agreement with δ =3 which is obtained from scaling law of Widom. The exponents in the finite-size scaling relations for the magnetic susceptibility and the order parameter at the infinite-lattice critical temperature are computed to be 2.5080 (1), 2.5005 (3) and 1.2501 (1) using 4 ≤ L ≤ 8, respectively, which are in very good agreement with the theoretical predictions of 5/2 and 5/4. The finite-size scaling plots of magnetic susceptibility and the order parameter verify the finite-size scaling relations about the infinite-lattice temperature

Kürkçü, C.

Biblioteka Nauki - repozytorium artykuÅÃ³w

The Finite-Size Scaling Study of Five-Dimensional Ising Model

Gazi University Dspace

The finite-size scaling study of five-dimensional ising model

Kırşehir Ahi Evran University Institutional Repository

Vol. 129 (2016) ACTA PHYSICA POLONICA A No. 6The Finite-Size Scaling Study of Five-Dimensional Ising ModelZ. Merdana,∗, N. Arasa and C. KürkçübaFaculty of Arts and Sciences, Department of Physics, Gazi University, Ankara, TurkeybFaculty of Arts and Sciences, Department of Physics, Ahi Evran University, Kirsehir, Turkey(Received October 7, 2015; in final form March 16, 2016)The five-dimensional ferromagnetic Ising model is simulated on the Creutz cellular automaton algorithm usingfinite-size lattices with linear dimension 4 ≤ L ≤ 8. The critical temperature value of infinite lattice is found tobe Tχ(∞) = 8.7811 (1) using 4 ≤ L ≤ 8 which is also in very good agreement with the precise result. The valueof the field critical exponent (δ = 3.0067(2)) is good agreement with δ = 3 which is obtained from scaling law ofWidom. The exponents in the finite-size scaling relations for the magnetic susceptibility and the order parameterat the infinite-lattice critical temperature are computed to be 2.5080 (1), 2.5005 (3) and 1.2501 (1) using 4 ≤ L ≤ 8,respectively, which are in very good agreement with the theoretical predictions of 52and 54. The finite-size scalingplots of magnetic susceptibility and the order parameter verify the finite-size scaling relations about the infinite-lattice temperature.DOI: 10.12693/APhysPolA.129.1100PACS/topics: 05.50.+q, 64.60.Cn, 75.40.Cx, 75.40.Mg1. IntroductionWhile the five-dimensional ferromagnetic Ising modelis not directly applicable to real magnetic systems, it isuseful to investigate the influence of dimensionality onphase transitions [1]. In fact, in Euclidean quantum fieldtheory, the 4-d ferromagnetic Ising model describes thephysical dimension. As the dimensionality and/or thelattice size increases, the simulation of the ferromagneticIsing model by the conventional Monte Carlo methodbecomes impractical and faster algorithms are needed.The Creutz cellular automaton algorithm [2] does not re-quire high-quality random numbers, it is an order of mag-nitude faster than the conventional Monte Carlo methodand compared to the Q2R cellular automaton [3], it hasthe advantage of fluctuating internal energy from whichthe specific heat can be computed.The four-dimensional Ising model in the presence ofexternal magnetic field is simulated [3]. In two dimen-sions, the solution of ferromagnetic Ising model is in-vestigated [4–8]. By considering different approximatemethods, the approximations of the solutions of two-dimensional ferromagnetic Ising model are presented [9–13]. In addition, the four-dimensional ferromagneticIsing model solution is approximated by using Creutzcellular automaton algorithm with nearest neighbor in-teractions and near the critical region [14–23]. The algo-rithm of approximating finite size behavior of ferromag-netic Ising model is extended to higher dimensions [14–32]. It is established that the algorithm has been pow-erful in terms of providing the values of static criticalexponents near the critical region in four and higher di-mensions with nearest neighbor interactions [14–32].∗corresponding author; e-mail: ziyamerdan@gazi.edu.trIn this paper, we present the result of the simulationof the five-dimensional ferromagnetic Ising model withthe Creutz cellular automaton algorithm. The value ofthe field critical exponent (δ) is obtained. The expo-nents(d2 ,d4)for the magnetic susceptibility and the orderparameter at Tc are obtained for 4 ≤ L ≤ 8 which is alsoin very good agreement with the theoretical results.The model is described in Sect. 2, the results are dis-cussed in Sect. 3 and a conclusion is given in Sect. 4.2. ModelIn general, n ≥ 5 binary bits are associated with eachsite of the lattice. The value for each site is determinedfrom its value and from those of its nearest neighborsat the previous time step. The updating rule, which de-fines a deterministic cellular automaton, is as follows: ofthe n binary bits on each site, the first one is the Isingspin Bi. Its value may be “0” or “1”. The Ising spin en-ergy in the presence of an external magnetic field, HI, isdescribed by the Hamiltonian of the formHI = −J∑〈ij〉SiSj − h∑〈i〉Si, (1)by taking into account of the interaction between thenearest neighbors and also the interaction of the spins Siwith external magnetic field h, directed “up” (Si = +1).The spins affected by the field are directed “up” and notchanged during the simulation. Therefore, these spinsplay the role of the magnetic field. In the Hamiltonian,Si = 2Bi − 1 and h is the ratio of the number of “up”spins to the number of all spins. The next n− 2 bits arefor the momentum variable conjugate to the spin (the de-mon). These n − 2 bits form an integer which can takevalues within the interval (0,∑n−2i=1 2i−1). The kinetic en-ergy (in units of J) associated with the demon can takeon four times these integer values. The total energyH = HI +HK, (2)(1100)The Finite-Size Scaling Study of Five-Dimensional Ising Model 1101is conserved; here HK is the kinetic energy of the lat-tice. For a given total energy, the system temperature T(in units of J/kB where kB is the Boltzmann constant)is obtained from the average value of the kinetic en-ergy of a demon. The n-th bit provides a checkerboardstyle updating, therefore it allows the simulation of theIsing model on a cellular automaton. The black sitesof the checkerboard are updated and then their colour ischanged into white; the white sites are changed into blackwithout being updated. The updating rules for the spinand the momentum variables are as follows: For a site tobe updated its spin is flipped and the change in the Isingenergy (internal energy), HI, is calculated. If this energychange is transferable to or from the momentum variableassociated with this site, such that the total energy H isconserved, then this change is made and the momentumis appropriately changed. Otherwise, the spin and themomentum are not changed. As the initial configurationall the spins are taken ordered (up or down). The initialkinetic energy is given to the lattice via the second andthe third bits of the momentum variables in the whitesites randomly, such that the value of the initial kineticenergy for such a demon is 24 (in units of J).Simulations are carried out on simple hypercubic lat-tices L5 of linear dimensions 4 ≤ L ≤ 8 with periodicboundary conditions. The cellular automaton develops9.6 × 105 (L = 4, 6, 8) sweeps for each run, with 7 runsfor each total energy.3. Results and discussionThe temperature dependences of the order parameterand the magnetic susceptibility are illustrated in Fig. 1for 4 ≤ L ≤ 8. The critical temperatures of the finite-sizelattices obtained from the magnetic susceptibility max-ima Tχc (L) for 4 ≤ L ≤ 8 are listed in Table I. Thecomputed values of χc and Mc are listed in Table II.TABLE IThe maximum values and the critical temperaturesof the magnetic susceptibility for 4 ≤ L ≤ 8.L Tχc (L) χmax4 8.576(2) 1541(2)6 8707(3) 4.256(4)8 8745(1) 8767(3)The dependence of the critical temperatures Tχc (L) ob-tained from the magnetic susceptibility maxima of thefinite-size lattices on linear dimension L is given by thefollowing expression [14, 26, 33–37]:Tχc (∞)− Tχc (L) ∝ L−d/2. (3)The value of the infinite-lattice critical temperaturefor the five-dimensional ferromagnetic Ising model,8.7811 (1) is obtained from the straight line fit of the mag-netic susceptibility maxima for 4 ≤ L ≤ 8 (Fig. 2, Ta-ble III). The value obtained from infinite lattice criti-cal temperature Tχ(∞) = 8.7811(1) for 4 ≤ L ≤ 8Fig. 1. The temperature dependence of (a) the orderparameter (M) and (b) the magnetic susceptibility (χ)for 4 ≤ L ≤ 8.agrees with the results obtained previously using differentmethods [14, 26, 37–46].Fig. 2. The value of the infinite-lattice critical tem-perature for the five-dimensional ferromagnetic Isingmodel, Tχc (∞) = 8.7811 (1), obtained by extrapolatingtemperatures of the lattice with the linear dimension4 ≤ L ≤ 8 as L→∞.For a lattice linear dimension L and very small h atT = Tc(L), the order parameter is given byM(L) ∝ h1/δ(L), (4)where δ(L) is the field critical exponent (Fig. 3). Scaling1102 Z. Merdan, N. Aras, C. Kürkçülaw of Widom is the following:γ = β(δ − 1), (5)where γ = 1, β = 1/2 and δ = 3 for d = 5 [29, 30, 41, 42].Fig. 3. The dependence ofM against h for the latticeswith the linear dimension 4 ≤ L ≤ 8 (Tc = 8.7811 (1)).Fig. 4. The log–log plots of M(L, Tc(L, h)) against hwith the slope giving the value of 1/δ = 0.0502, 1/δ =0.0696 and 1/δ = 0.0875 for L = 4, 6 and 8 at Tc =8.7811(1), 8.7856(6), 8.8370(1), respectively.The log–log plots of M(L) at T = Tc(L, h) versus hfor h in the interval 0 ≤ h ≤ 0.0025 yield to 1/δ(L)(Fig. 4). The straight line which fits to the plot of δ(L)against 1/L results in the infinite-lattice critical expo-nents δ = 3.0136(3) (Fig. 5). The result for the δ(∞) iscompared with δ = 3 which is obtained from scaling lawof Widom [35, 47, 48].In d ≥ 5 dimensions the finite-size scaling relation forthe free-energy density is given as [33]:fL = L−dF (tLy∗t , hLy∗h), h→ 0, L→∞, (6)where y∗t = d/(γ + 2β) and y∗h = d(γ + β)/(γ + 2β),with α = 2 − d/y∗t , β = d/y∗t − ∆ and γ = 2∆ − d/y∗t ,t = (T − TC)/TC is the reduced temperature with t > 0for T > TC and t < 0 for T < TC, h is the reduced exter-nal magnetic field, α, β and γ are the critical exponentsfor the specific heat, order parameter and the magneticsusceptibility of the infinite lattice, respectively. Thus,Fig. 5. The plot of δ(L) against 1/L. The extrapola-tion of the fit lines to 1/L→∞ gives δ = 3.0067(2).Fig. 6. Finite-size scaling plot of ML at Tc =8.7811 (1) for 4 ≤ L ≤ 8.fL takes the following form [16, 33, 35, 36, 49]:fL = L−dF (tL1(2−α)/d , hL(γ+β)(2−α)/d ). (7)According to Eq. (6), ML and χL have the scaling formML =∂fL∂h= L−d/4X(tLd/2, hL3d/4), (8)χL =∂2fL∂h2= Ld/2Y (tLd/2, hL3d/4), (9)where the scaling functions X and Y are obtainedfrom fL. These finite-size scaling relations take the fol-lowing form T = Tc [14, 16, 35]:Mc ∝ L−d/4, (10)χc ∝ Ld/2. (11)The relations for 4 ≤ L ≤ 8 can be tested by simula-tions directly. The finite-size scaling plots for |ML(t)| andχL(t) are given in Figs. 6 and 7, respectively. The overlapof the plots of the scaled quantities for different L verifythe finite-size scaling relations given in Eqs. (8) and (9)at Tc = 8.7811(1).The slope of straight lines in Fig. 8 (Eq. (10)) for4 ≤ L ≤ 8 give the results of d4 = 1.2501 (1) whichis also in very good agreement with the theoretical re-The Finite-Size Scaling Study of Five-Dimensional Ising Model 1103Fig. 7. Finite-size scaling plot of χL at Tc = 8.7811 (1)for 4 ≤ L ≤ 8.Fig. 8. The log–log plot of Mc against L at Tc =8.7811 (1) for 4 ≤ L ≤ 8. The slope gives d4= 1.2501 (1).sult(d4 = 1.25). The slopes of straight lines in Figs. 9and 10 (Eq. (11)) for 4 ≤ L ≤ 8 give the results ofd2 = 2.5080 (1) andd2 = 2.5005 (3), respectively, whichare also in very good agreement with the theoretical re-sult(d2 = 2.5)are given in Table IV.Fig. 9. The log–log plot of χmax(L) against L at Tc =8.7811 (1) for 4 ≤ L ≤ 8. The slope gives d2= 2.5080 (1).Fig. 10. The log–log plot of χc against L at Tc =8.7811 (1) for 4 ≤ L ≤ 8. The slope gives d2= 2.5005(3).TABLE IIThe values of χc and Mc obtainedat Tc = 8.7811 (1) for 4 ≤ L ≤ 8.L χc Mc4 1.310(3) 0.170(1)6 3.701(1) 0.102(3)8 7.392(4) 0.071(2)4. ConclusionsThe five-dimensional ferromagnetic Ising model is sim-ulated on the Creutz cellular automaton algorithm byusing the finite-size lattices with the linear dimensionsL = 4, 6, and 8. In our work, the critical temperaturevalue of infinite lattice for 4 ≤ L ≤ 8 is in agreement withthe other simulation results. In this study, the value ofthe field critical exponent (δ = 3.0067(2)) is satisfied byscaling law of Widom. The finite-size scaling relations of|ML(t)| and χL(t) at the infinite-lattice critical temper-ature for 4 ≤ L ≤ 8 are verified. The exponents in thefinite-size scaling relations for the magnetic susceptibil-ity and the order parameter at the infinite-lattice criticalTABLE IIIThe values of the infinite-lattice critical temperaturefor 4 ≤ L ≤ 8.Tχc Method8.7774(35) [31] Monte Carlo8.7812(23) [32, 33] Monte Carlo8.778475(31) [34] Monte Carlo8.780(10) [35] Monte Carlo8.78(1) [36] Monte Carlo8.77832(54) [37] series expansion8.7769(12) [38] series expansion8.7780(5) [39] series expansion8.77886(77) [38, 40] dynamic Monte Carlo8.779(8), 8.7572 [14, 26] Creutz cellular automaton8.7811(1), this work Creutz cellular automaton1104 Z. Merdan, N. Aras, C. 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The Finite-Size Scaling Study of Five-Dimensional Ising Model

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The Finite-Size Scaling Study of Five-Dimensional Ising Model

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