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3D Ising Nonuniversality: a Monte Carlo study 5 0 0 2 Melanie S hulte and Caroline Drope n a J Institute for Theoreti al Physi s, Cologne University, 50923 Köln, Germany. 9 Email: msthp.uni-koeln.de, dropethp.uni-koeln.de. 2 ] h Abstra t c e m Weinvestigate asamember oftheIsinguniversality lasstheNext-Nearest Neighbour Ising model without external (cid:28)eld on a simple ubi latti e by - t a using the Monte Carlo Metropolis Algorithm. The Binder umulant and the t sus eptibility ratio, whi h should be universal quantities at the riti al point, s t. were shown to vary for small negative next-nearest neighbour intera tions. a m keywords: next-nearest neighbour Ising model, Binder umulant, fourth order u- - d mulant. n o c [ 1 Introdu tion 1 v The greatvariety of riti albehaviour isredu ed by dividingallsystems intoa small 1 number of universality lasses. These lasses are hara terized by global properties 3 7 like dimensionality and the number of omponents of the order parameter and have 1 the same set of riti al exponents and s aling fun tions, whi h are independent of 0 5 mi ros opi details[1℄. Here the3D Ising universality lassis onsidered by studying 0 the Binder umulant [2℄ of the Next-Nearest Neighbour Ising model (NNN model). / t a If alluniversal quantities of one lass are known, the asymptoti riti albehaviour is m believedtobealsoknown ompletelyunderthe ondition,thatonlytwononuniversal K K - amplitudes t and h are given. This parti ular feature is alled two-s ale fa tor d f n universality. The free-energy-density , from whi h we an derive thermodynami al k o B quantities and whi h is measured in units of , an be expressed by [3℄: c : f(t,H;L) = f (t,H;L)+f (t;L), v s ns i X t = 1 T/T H f c s r where − and is the ordering (cid:28)eld. The singular part is the part a L fns whi h yields thermodynami al singularities in the → ∞ limit. denotes the f non-singular part of , whi h an be hosen without (cid:28)eld dependen e. For the f s singular part we an write for the Ising model f (t,H;L) = L−dY(K tL1/ν,K HLδ/ν) s t h Y(x,y) is alled the s aling fun tionand isuniversal using system-dependent fa tors K K h t and . Chφe4n and Dohm [4℄ re ently predi ted some deviations from this universality, from a theory. In the following se tion we study the universal value of the Binder umulant for 1 J NNN the NNN model for di(cid:27)erent next-nearest neighbour intera tion by simula- tions. A ording to Privman, Hohenberg and Aharony [3℄, the Binder umulant s ales a ording to U (t,L) G(K tL1/ν) , 4 t ≈ G(x) = [(∂4Y/∂y4)/(∂2Y/∂y2)2] t = 0 T = T U (0,L) y=0 c 4 where . For ∗ (mean∗s ), L U (0, ) U (0,L) 4 4 approa hes with growing a universal onstant ∞ , whi h∗we are U (0, ) 4 going to simulate, shows only small deviations from the universal value ∞ [5∗℄. Consequently, it is possible to give a realisti approximation of the (cid:28)nal value U (0, ) 4 ∞ simulating (cid:28)nite latti es. 2 Applied NNN Models 2.1 Standard Next-Nearest Neighbour Model ∗ U 4 To show the variation of within the 3D Ising universality lass, we hoose the Next-Nearest Neighbour model (NNN Model) without external (cid:28)eld and a simple a ubi latti e. Ea h latti e site has 6 next neighbours in the distan e and 12 next- √2 a a nearest neighbours in the distan e · , if is de(cid:28)ned as the latti e onstant. The J NNN ex hange for e of the next-nearest neighbours an now be hosen to be di(cid:27)er- J NN ent from the ex hange for e of the 6 nearest neighbours as we also would also intuitively expe t, be ause their distan e to the onsidered site is also not identi al. A ferromagneti phase transition is also existent for an antiferromagneti ex hange J NNN for e . This kind of ex hange, ferromagneti for nearest neighbours and anti- ferro∗magneti for next-nearest neighbours, will be treated for our universality test U 4 of . 2.2 Anisotropi Next-Nearest Neighbour Model ∗ U 4 Furthermore we studied the universality of in the zero (cid:28)eld NNN Ising Model J > 0 with an isotropi nearest-neighbour (NN) oupling and an anisotropi next- J < 0 NNN nearest neighbour (ANNN) oupling . We refer to the onsidered model of Chen and Dohm in their publi ation (cid:16)Nonuniversal (cid:28)nite-size s aling in anistropi systems(cid:16) [4℄. The anisotropi NNN Ising Model is established by onsidering only 6 J < 0 NNN of the 12 next-nearest neighbours being e(cid:27)e tive for NNN intera tion and the other 6 NNN having no NNN intera tion. The e(cid:27)e tive next-nearest neighbours (1,0,0) (1,0,1) (0,1,1) have the positions ± , ± and ± of a simple ubi latti e. 3 Simulation Results J J = 1 NNN NN Simulatingthe NNN model anbe varied, whileitis hosen . Forval- J J 0.5 NNN NNN,crit ues of higher than a riti al value ≈ − we observe ferromagneti J NNN,crit phase transitions, while below no ferromagneti phase transition o urs. J = 0 NNN At we obtain the standard 3-dimensional (3D) Ising ferromagnet. Here J < J < 0 NNN,crit NNN the region and thus a small antiferromagneti intera tion J NNN is onsidered. 2 Si∗mulating now the "(cid:28)xed po∗int value" at the riti al point of the Binder umulant U = U (T ) U (J ) 4 4 c 4 NNN , we evaluate to he k its universality within the 3D Ising universality lass. For an Ising Model in zero (cid:28)eld, the Binder umulant simpli(cid:28)es to M4 U = 1 h i , 4 − 3 M2 2 (1) h i ∗ M U 4 where symbols the magnetization per spin. For the determination of ea h U T L 4 was plotted as a fun tion of temperature for three di(cid:27)erent∗latti e lengths = 10 20 40 U (T) 4 ( , , ). To get more pre ise results, several urves of for di(cid:27)erent latti e sizes with di(cid:27)erent seeds were averaged. In (cid:28)gure 1 is shown this method for J NNN two di(cid:27)erent . The errors were al ulated by averaging the values of the three L = (10,20) (10,40) (20,40) di(cid:27)erent rossing points of , , . 0.6 J =0.000 0.6 J =-0.247 NNN NNN 0.55 0.55 0.5 0.5 U4 0.45 U4 0.45 0.4 0.4 0.35 0.35 0.3 0.3 4.5 4.504 4.508 4.512 4.516 4.52 1.22 1.222 1.224 1.226 1.228 1.23 T [J /k ] T [J /k ] NN B NN B ∗ U T J 4 c NNN Figure 1: Lo alization of the (cid:28)xed point ( , ) for di(cid:27)erent by quantifying U (T) L = 10 20 40 4 the rossing point of for latti e sizes (dotted), (dashed) and (solid). 3.1 Results for the Standard NNN Ising Model ∗ U (J ) 4 NNN In (cid:28)gure 2 and table 1 we an see all simulated results as a fun tion and these results in om∗parison with theoreti al results of Chen and Dohm [4℄ plotted 1 U (w)/U(0) w = J /(J + 2J ) 4 NNN NN NNN as a fun tion − with . Chen and Dohm onsidered an anisotropi NNN ∗Ising model, whi h we will dis uss later. It learly U J 4 NNN an be seen in(cid:28)gure and, that isnota (cid:28)xed value fordi(cid:27)erent , parti ularly J = NNN in the interval of [-0.25:-0.22℄. This implies in this region the absen e of universality for the 3D Standard NN∗ N Ising model. U L J = 0 4 NNN Great deviations of the resulting for → ∞ an be ex luded for [5℄. L = 100 J = 0 NNN Observing latti e length up to for 6 we observed no signi(cid:28) ant hange of our simulation results. J NNN We an on lude, that in the region of small negative values for the 3D Next- Nearest Neighbour Ising model shows a di(cid:27)erent universal behaviour than the 3D Nearest Neighbour Ising model. 3 ∗ ∗ ∗ J w U 1 U (w)/U (0) NNN 4 4 4 − 0.000 0.000 0.4853 3.20 10−3 0.0000 1.32 10−2 0.100 0.125 0.4867±4.80·10−3 2.8609 10−3 ±1.98·10−2 − − ± · − · ± · 0.200 0.333 0.4894 6.90 10−3 8.4366 10−3 2.84 10−2 − − ± · − · ± · 0.230 0.426 0.4854 1.68 10−3 2.6201 10−4 6.92 10−3 −0.240 −0.462 0.4753±3.22·10−3 −2.0685· 10−2± 1.33· 10−2 − − ± · · ± · 0.241 0.465 0.4801 6.44 10−3 1.0700 10−2 2.65 10−2 − − ± · · ± · 0.245 0.480 0.4692 1.02 10−2 3.3247 10−2 4.20 10−2 −0.246 −0.484 0.4290±1.40·10−2 0.1161· ±5.77·10−2 −0.247 −0.488 0.3994±3.96·10−3 0.1771 ±1.63·10−2 − − ± · ± · ∗ U 4 Table 1: Simulation results of the universal quantity and its statisti al errors for J NNN di(cid:27)erent next-nearest neighbour intera tion . 0.2 0.5 0.15 0.48 0) 0.46 (4 0.1 U* U*4 0.44 w)/ (4 U* 0.05 0.42 1- 0 0.4 0.38 -0.05 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 -0.5 -0.4 -0.3 -0.2 -0.1 0 JNNN w Figure 2∗: Standard 3D NNN model. Left side: Simulation results of the (cid:28)xed U4 JNNN (cid:4) point as a fun tion of ; Right side: Results of the (cid:28)gure on the left ( ) in omparison with theoreti al results (for a di(cid:27)erent neighbourhood) of Chen and Dohm [4℄ (dashed line). 3.2 Results for the ANNN Ising Model Figure 3 and table 2 show the results for the ANNN Ising Model analogue to se tion 3.1. The simulation results show a signi(cid:28) ant non-universality of the Binder umu- lant. As ompared to the analyti results of Chen and Dohm [4℄ the analyti al urve have a di(cid:27)erent te∗nden y and is not within the errorbars of the simulation results. w 0.5 U 4 For → − , in reases whereas Chen and Dohm obtained a de rease. Thus, the theoreti al results annot be on(cid:28)rmed fully. 4 0.52 0.12 0.515 0.1 0.51 0.08 0.06 0.505 0.5 (0)4 0.04 U 0.02 *U4 0.495 (w)/4 0 0.49 U 1- -0.02 0.485 -0.04 0.48 -0.06 0.475 -0.08 0.47 -0.1 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 w w + U∗(w) 4 Figure 3: Left side: Simulation∗ results∗ ( ) for the ANNN model of . Right 1 U (w)/U (0) 4 4 side: Simulationresultsof − in omparisonwiththetheoreti alresults (for this neighbourhood) of Chen and Dohm [4℄ (solid line). ∗ ∗ ∗ J w U 1 U (w)/U (0) NNN 4 4 4 − 0 0 0.478 3.53 10−3 0 1.04 10−2 ± · ± · 0.143 0.2 0.484 5.80 10−3 1.33 10−2 1.43 10−2 − − ± · − · ± · 0.188 0.3 0.488 6.04 10−3 2.06 10−2 1.47 10−2 −0.222 −0.4 0.498±5.96·10−3 −4.20·10−2 ±1.47·10−2 −0.237 −0.45 0.499±9.42·10−3 −4.50·10−2 ±2.12·10−2 − − ± · − · ± · 0.245 0.48 0.507 7.08 10−3 6.07 10−2 1.68 10−2 − − ± · − · ± · 0.247 0.49 0.507 1.10 10−2 5.82 10−2 2.43 10−2 −0.249 −0.495 0.507±8.34·10−3 −5.98·10−2 ±1.91·10−2 − − ± · − · ± · ∗ U (w) 4 Table 2: Simulation results for the ANNN Model for . 5 3.3 Universality Test of the Sus eptibility for the ANNN Model χ = (N/k T)(cid:0) M2 M 2(cid:1) B Simulating the sus eptibility h i−h| |i for the ANNN Ising J NNN Model we use the same range of NNN intera tion . We onsider two temper- Tc T− = Tc 3%Tc T+ = Tc +3%Tc atures near the riti al temperature , − and . The results show expli itly the absen e of universality. Figure 4 shows the ratio of the T+ T− w sus eptibilityat and asafun tionof . Asigni(cid:28) antdeviationofnearly100% w = 0 w = 0.45 T L c exists here between and − . Varying below the latti e size be- 10 317 MCS 50 5 105 tween and and the number of MoLnt=e C40arlo steMpsCS = 5be1tw05een and · we an ex lude (cid:28)nite-size e(cid:27)e ts with and · . Table 3 shows χ(T+) χ(T−) χ(T+)/χ(T−) JNNN all results of , and for di(cid:27)eren<tM2 >. (W<eMlook>e2d at the (cid:29)u tuations in the absolute value of the magnetization − | | , not of T 1 2/π = 0.3634 c the magnetization. Thus our values above are lower by a fa tor − < M2 > than the usual , assuming Gaussian (cid:29)u tuations [2℄.) 4.5 4 T)- 3.5 χ( )/+ T χ( 3 2.5 2 -0.5 -0.4 -0.3 -0.2 -0.1 0 w χ(T+)/χ(T−) Figure 4: Simulation results of the ratio of sus eptibility as a fun tion J NNN of . JNNN w χ(T+) χ(T−) χ(T+)/χ(T−) 0 0 32.10 0.39 15.03 9.3 10−2 2.14 2.9 10−2 0.143 0.2 34.26±0.70 15.37±4.1·10−2 2.23±4.6·10−2 − − ± ± · ± · 0.188 0.3 35.46 0.59 15.34 11.1 10−2 2.31 4.2 10−2 − − ± ± · ± · 0.222 0.4 38.54 0.47 14.91 12.2 10−2 2.59 3.8 10−2 −0.237 −0.45 39.53±0.55 14.72± 6.6 ·10−2 2.69±3.9·10−2 − − ± ± · ± · 0.245 0.48 57.27 1.36 14.28 12.5 10−2 4.01 10.2 10−2 − − ± ± · ± · T− = Tc 3%Tc T+ = Table 3: Simulation results of the sus eptibility at − and Tc +3%Tc χ(T+)/χ(T−) JNNN and of the ratio as a fun tion of . 6 4 Con lusions This work examined the universality of the Binder umulant for the 3D NNN Ising model in the region of small negative values for the next-nearest neighbour intera - tion. Monte Carlo simulations of the universal quantity of the Binder umulant and the universal sus eptibility ratio with di(cid:27)erent next-nearest neighbour intera tions J NNN showed a variation of both quantities for all applied models. The variation of these universal quantities leads to the on lusion, that in the onsidered region the NNN Ising model does not belong fully to the 3D Ising universality lass. J J NN NNN We assume that for regions, where hosen values of and still ause fer- romagneti phase transitions, but are lose to a region where we will (cid:28)nd no more ferromagneti behaviour, a new de(cid:28)nition for universality lasses for this model has to be found. 5 A knowledgements We thank Dietri h Stau(cid:27)er for ontributing his knowledge and experien es in Monte Carlo simulations. Spe ial thanks also to Volker Dohm from the te hni al university of Aa hen and Xiaosong Chen from the Institute of Theoreti al Physi s, Chinese A ademy of S ien es for their dis ussions and help. Referen es [1℄ D. P. Landau and K. Binder, A Guideto Monte Carlo SimulationsinStatisti al Physi s (Cambridge University Press, 2000). [2℄ K. Binder, Z. Phys. B 43, 119 (1981). [3℄ V. Privman, P. C. Hohenberg and A. Aharony, Phase Transitions and Criti al Phenomena 14, eds. C. Domb and J. L. Lebowitz (A ademi Press Limited 1991). [4℄ X. S. Chen and V. Dohm. Phys. Rev. E 70, 056136 (2004). [5℄ A. M. Ferrenberg, D. P. Landau, Phys. Rev. B 44, 5081 (1991). 7

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