Zero temperature dynamics in two dimensional ANNNI model
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Zero temperature dynamics in two dimensional ANNNI model
Soham Biswas, Anjan Kumar Chandra, and Parongama Sen Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India Theoretical Condensed Matter Physics Division and Center for Applied Mathematics and Computational Science,Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India
We investigate the dynamics of a two dimensional axial next nearest neighbour Ising(ANNNI) model following a quench to zero temperature. The Hamiltonian is given by H = − J P Li,j =1 S i,j S i +1 ,j − J P i,j =1 [ S i,j S i,j +1 − κS i,j S i,j +2 ]. For κ <
1, the system does not reach theequilibrium ground state but slowly evolves to a metastable state. For κ >
1, the system shows abehaviour similar to the two dimensional ferromagnetic Ising model in the sense that it freezes to astriped state with a finite probability. The persistence probability shows algebraic decay here withan exponent θ = 0 . ± .
001 while the dynamical exponent of growth z = 2 . ± .
01. For κ = 1,the system belongs to a completely different dynamical class; it always evolves to the true groundstate with the persistence and dynamical exponent having unique values. Much of the dynamicalphenomena can be understood by studying the dynamics and distribution of the number of domainswalls. We also compare the dynamical behaviour to that of a Ising model in which both the nearestand next nearest neighbour interactions are ferromagnetic. PACS numbers: 64.60.Ht, 75.60.Ch, 05.50.+q
I. INTRODUCTION
Dynamics of Ising models is a much studied phe-nomenon and has emerged as a rich field of present-dayresearch. Models having identical static critical behav-ior may display different behavior when dynamic criticalphenomena are considered [1]. An important dynamicalfeature commonly studied is the quenching phenomenonbelow the critical temperature. In a quenching process,the system has a disordered initial configuration corre-sponding to a high temperature and its temperature issuddenly dropped. This results in quite a few interest-ing phenomena like domain growth [2, 3], persistence[4, 5, 6, 7, 8] etc.In one dimension, a zero temperature quench of theIsing model ultimately leads to the equilibrium configu-ration, i.e., all spins point up (or down). The averagedomain size D increases in time t as D ( t ) ∼ t /z , where z is the dynamical exponent associated with the growth.As the system coarsens, the magnetisation also grows intime as m ( t ) ∼ t / z . In two or higher dimensions, how-ever, the system does not always reach equilibrium [8]although these scaling relations still hold good.Apart from the domain growth phenomenon, anotherimportant dynamical behavior commonly studied is per-sistence. In Ising model, in a zero temperature quench,persistence is simply the probability that a spin has notflipped till time t and is given by P ( t ) ∼ t − θ . θ is calledthe persistence exponent and is unrelated to any otherknown static or dynamic exponents.Drastic changes in the dynamical behaviour of the Isingmodel in presence of a competing next nearest neighborinteraction have been observed earlier [9, 10, 11]. Theone dimensional ANNNI (Axial next nearest neighbour Ising) model with L spins is described by the Hamiltonian H = − J L X i =1 ( S i S i +1 − κS i S i +2 ) . (1)Here it was found that for κ <
1, under a zero tempera-ture quench with single spin flip Glauber dynamics, thesystem does not reach its true ground state. (The groundstate is ferromagnetic for κ < .
5, antiphase for κ > . κ = 0 . κ > L × L lattice is given by H = − J L X i,j =1 S i,j S i +1 ,j − J X i,j =1 [ S i,j S i,j +1 − κS i,j S i,j +2 ] . (2)Henceforth, we will assume the competing interaction tobe along the x (horizontal) direction, while in the y (ver-tical) direction, there is only ferromagnetic interaction.Although the thermal phase diagram of the two dimen-sional ANNNI model is not known exactly, the groundstate is known and simple. If one calculates the mag-netisation along the horizontal direction only, then for κ < .
5, there is ferromagnetic order and antiphase or-der for κ > .
5. Again, κ = 0 . x directionfor different values of κ . T = 0 κ Antiphase ++ −−++ −−
Ferro
Highly degenerate
FIG. 1: The ground state (temperature T = 0) spin config-urations along the x direction are shown for different valuesof κ . In the ferromagnetic phase, there is a two fold degen-eracy and in the antiphase the degeneracy is four fold. Theground state is infinitely degenerate at the fully frustratedpoint κ = 0 . In section II, we have given a list of the quantities cal-culated. In section III, we discuss the dynamic behaviourin detail. In order to compare the results with thoseof a model without competition, we have also studiedthe dynamical features of a two dimensional Ising modelwith ferromagnetic next nearest neighbour interaction,i.e., the model given by eq. (2) in which κ <
0. Theseresults are also presented in section III. Discussions andconcluding statements are made in the last section.
II. QUANTITIES CALCULATED
We have estimated the following quantities in thepresent work:1. Persistence probability P ( t ): As already men-tioned, this is the probability that a spin does notflip till time t .In case the persistence probability shows a powerlaw form, P ( t ) ∼ t − θ , one can use the finite sizescaling relation [13] P ( t, L ) ∼ t − θ f ( L/t /z ) . (3)For finite systems, the persistence probability satu-rates at a value L − α at large times. Therefore, for x << f ( x ) ∼ x − α with α = zθ . For large x , f ( x ) is a constant.It has been shown that the exponent α is related tothe fractal dimension of the fractal formed by thepersistent spins [13]. Here we obtain an estimate of α using the above analysis. I II I(a) IIIIIII(b)
FIG. 2: The schematic pictures of configurations with flatinterfaces separating domains of type I and II are shown: (a)when the interface lies parallel to y axis, we have nonzero f D x (= 2 /L in this particular case) and (b) with interfaces parallelto the x axis we have nonzero f D y (= 4 /L here)
2. Number of domain walls N D : Taking a single stripof L spins at a time, one can calculate the numberof domain walls for each strip and determine theaverage. In the L × L lattice, we consider the frac-tion f D = N D /L and study the behaviour of f D asa function of time. One can take strips along boththe x and y directions (see Fig. 2 where the calcu-lation of f D in simple cases has been illustrated).As the system is anisotropic, it is expected that thetwo measures, f D x along the x direction and f D y along the y direction, will show different dynamicalbehaviour in general. The domain size D increasesas t /z as already mentioned and it has been ob-served earlier that the dynamic exponent occurringin coarsening dynamics is the same as that occur-ring in the finite size scaling of P ( t ) (eq. (3)) [13].Although we do not calculate the domain sizes, theaverage number of domain walls per strip is shownto follow a dynamics given by the same exponent z , at least for κ > P ( f D ) (or P ( N D )) of the fraction (ornumber) of domain walls at steady state: this isalso done for both x and y directions.4. Distribution P ( m ) of the total magnetisation atsteady state for κ ≤ L × L with L =40, 100, 200 and 300 to study the persistence be-haviour and dynamics of the domain walls of thesystem and averaging over at least 50 configura-tions for each size have been made. For estimatingthe distribution N D we have averaged over muchlarger number of configurations (typically 4000)and restricted to system sizes 40 ×
40, 60 × ×
80 and 100 × x and y directions. J = J = 1 has been used in the numerical simu-lations. III. DETAILED DYNAMICAL BEHAVIOUR
Before going in to the details of the dynamicalbehaviour let us discuss the stability of simple config-urations or structures of spins which will help us inappreciating the fact that the dynamical behaviour isstrongly dependent on κ . A. Stability of simple structures
An important question that arises in dynamics is thestability of spin configurations - it may happen that con-figurations which do not correspond to global minimumof energy still remain stable dynamically. This has beentermed “dynamic frustration” [14] earlier. A known ex-ample is of course a striped state occurring in the two orhigher dimensional Ising models which is stable but nota configuration which has minimum energy.In ANNNI model, the stability of the configurationsdepend very much on the value of κ . It has been previ-ously analysed for the one dimensional ANNNI modelthat κ = 1 is a special point above and below which thedynamical behaviour changes completely because of thestability of certain structures in the system.Let us consider the simple configuration of a single upspin in a sea of down spins. Obviously, it will be unstableas long as κ <
2. For κ >
2, although this spin is stable,all the neighbouring spins are unstable. However, for κ <
2, only the up spin is unstable and the dynamicswill stop once it flips. When κ = 2 the spin may or maynot flip, i.e., the dynamics is stochastic.Next we consider a domain of two up spins in a sea ofdown spin. These two may be oriented either along hori-zontal or vertical direction. These spins will be stable for κ > κ <
1, all spins except the up spins are stable. When κ = 1, the dynamics is again stochastic.A two by two structure of up spins in a sea of downspins on the other hand will be stable for any value of κ >
0. But the neighbouring spins along the verticaldirection will be unstable for κ ≥
1. This shows thatfor κ <
1, one can expect that the dynamics will affectthe minimum number of spin and therefore the dynamicswill be slowest here. A picture of the structures describedabove are shown in Fig 3.One can take more complicated structures but theanalysis of these simple ones is sufficient to expect thatthere will be different dynamical behaviour in the regions κ < , κ = 1 , κ > , κ = 2 and κ >
2. However, wefind that as far as persistence behaviour is concerned,there are only three regions with different behaviour: κ < , κ = 1 and κ >
1. On the other hand, when thedistribution of the number of domain walls in the steadystate is considered, the three regions 1 < κ < , κ = 2and κ > − − + + − −− − + + − − − − − − − − − − + − − − − + + − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − (a) (b) (c) FIG. 3: Analysis of stability of simple structures: (a) singleup spin in sea of down spins; here for κ < κ < κ < B. < κ < We find that as in [10], in the region 0 < κ <
1, thesystem has identical dynamical behaviour for all κ . Also,like the one dimensional case, here the system does not goto its equilibrium ground state. However, the dynamicscontinues for a long time, albeit very slowly for reasonsmentioned above. In Figs. 4 - 7, we show the snapshots ofthe system at different times for a typical quench to zerotemperature. As already mentioned, here domains of sizeone and two will vanish very fast and certain structures,the smallest of which is a two by two domain of up/downspins in a sea of oppositely oriented spins can survive tillvery long times. These structures we call quasi-frozenas the spins inside these structures (together with theneighbourhood spins) are locally stable; they can be dis-turbed only when the effect of a spin flip occurring at adistance propagates to its vicinity which usually takes along time.The pictures at the later stages also show that the sys-tem tends to attain a configuration in which the domainshave straight vertical edges, it can be easily checked thatstructures with kinks are not stable. We find a tendencyto form strips of width two (“ladders”) along the verticaldirection - this is due to the second neighbour interac-tion - however, these strips do not span the entire latticein general. The domain structure is obviously not sym-metric, e.g., ladders along the horizontal direction willnot form stable structures. The dynamics stops once theentire lattice is spanned by only ladders of height N ≤ L . FIG. 4: Snap shot of a 40 ×
40 system at time t = 10 for κ < FIG. 5: Same as Fig. 4 with t = 100. The persistence probability for κ < t ) for an appreciable range of time. At later times, it ap-proaches a saturation value in an even slower manner.The slow dynamics of the system accounts for this slowdecay.The fraction of domain walls f D x and f D y along the x direction and y directions show remarkable differenceas functions of time. While that in the x direction satu-rates quite fast, in the y direction, it shows a gradual de-cay till very long times (see Fig. 8). This indicates thatthe dynamics essentially keeps the number of domainsunchanged along x direction while that in the other di-rection changes slowly in time. The behaviour of f D x issimilar to what happens in one dimension. In fact, theaverage number of domain walls N D x at large times isalso very close to that obtained for the ANNNI chain, itis about 0 . L . However, in contrast to the one dimen-sional case where the domain walls remain mobile, herethe mobility of the domain walls are impeded by the pres-ence of the ferromagnetic interaction along the verticaldirection causing a kind of pinning of the domain walls.The distribution of the fraction of domain walls in thesteady state shown in Fig. 9 also reveals some importantfeatures. The distribution for f D x and f D y are both quitenarrow with the most probable values being f D x ≃ . f D y ≃ .
04 (these values are very close to the av-erage values). With the increase in system size, the dis-tributions tend to become narrower, indicating that theyapproach a delta function like behaviour in the thermo-dynamic limit.
FIG. 6: Same as Fig. 4 with t = 500. One of the two bytwo structures has melted while another one has formed. Theladder like structures which have formed are perfectly stable. FIG. 7: Same as Fig. 4 with t = 75000. This snapshotis taken after a very long time to show that the system hasundergone nominal changes compared to the length of thetime interval. The whole configuration now consists of laddersand the dynamics stops once the system reaches such a state. C. κ > It was already observed that κ = 1 is the value at whichthe dynamical behaviour of the ANNNI model changesdrastically in one dimension. In two dimensions, this isalso true, however, we find that the additional ferromag-netic interaction along the vertical direction is able toaffect the dynamics to a large extent. Again, similar tothe one dimensional case, we have different dynamicalbehaviour for κ = 1 and κ >
1. In this subsection wediscuss the behaviour for κ > κ = 1 case isdiscussed in the next subsection.The persistence probability follows a power law decaywith θ = 0 . ± .
001 for all κ >
1, while the finitesize scaling analysis made according to (3) suggests a z value 2 . ± .
01. This is checked for different values of κ ( κ = 1 . , . , . , , θ and z have negligible variations with κ which do not show anysystematics. Hence we conclude that the exponents areindependent of κ for κ >
1. A typical behaviour of theraw data as well as the data collapse is shown in Fig. 10.The dynamics of the average fraction of domain wallsalong the horizontal direction, f D x again shows a fastsaturation while that in the y direction has a power lawdecay with an exponent ≃ .
48 (Fig. 11). This exponentis also independent of κ . As mentioned in section II,we find that there is a good agreement of the value ofthis exponent with that of 1 /z obtained from the finite -2 -1 Time P(t)f D x f D y FIG. 8: Persistence P ( t ) and average number of domain wallsper site, f D are shown for κ < P r obab ili t y o f o cc u r en c e Fraction of domain wall f D x f D y FIG. 9: Steady state distributions of fraction of domain wallsat κ < size scaling behaviour of P ( t ) implying that the averagedomain size D is inversely proportional f D y . This is quiteremarkable, as the fraction of domain walls calculated inthis manner is not exactly equivalent to the inverse ofdomain sizes in a two dimensional lattice; the fact that f D x remains constant may be the reason behind the goodagreement (essentially the two dimensional behaviour isgetting captured along the dimension where the numberof domain walls show significant change in time).Although the persistence and dynamic exponents are κ independent, we find that the distribution of the numberof domain walls has some nontrivial κ dependence.Though the system, for all κ >
1, evolves to a statewith antiphase order along the horizontal direction, theferromagnetic order along vertical chains is in some casesseparated by one or more domain walls. A typical snap-shot is shown in Fig. 12 displaying that one essentiallygets a striped state here like in the two dimensional Isingmodel. Interfaces which occur parallel to the y axis, separatingtwo regions of antiphase and keeping the ferromagneticordering along the vertical direction intact, are extremelyrare, the probability vanishing for larger sizes. Quan-titatively this means we should get f D x = 0 . − − + + − − · · · type and a − − + + − − + + · · · type, which one can call a ‘shifted’antiphase ordering with respect to the first type).It is of interest to investigate whether these stripedstates survive in the infinite systems. To study this, weconsider the distribution of the number of domain wallsrather than the fraction for different system sizes. Theprobability that there are no domain walls, or a perfectferromagnetic phase along the vertical direction, turnsout to be weakly dependent on the system sizes but hav-ing different values for different ranges of values of κ . For1 < κ <
2, it is ≃ . κ = 2 .
0, it is ≃ .
544 whilefor any higher value of κ , this probability is about 0.445.Thus it increases for κ although not in a continuous man-ner and like the two dimensional case, we find that thereis indeed a finite probability to get a striped state.While we look at the full distribution of the number ofdomain walls at steady state (Fig. 13), we find that thereare dominant peaks at N D y = 0 (corresponding to theunstriped state) and at N D y = 2 (which means there aretwo interfaces). However, we find that the distributionshows that there could be odd values of N D y as well.This is because the antiphase has a four fold degeneracyand the and a ‘shifted’ ordering can occur in several wayssuch that odd values of N D y are possible. In any case,the number of interfaces never exceeds N D y = 6 for thesystem sizes considered. -1 -1 -1 P(L,t)/t - Θ L/t κ > 1 L = 40L = 100L = 200L = 30010 -2 -1 P(t) t FIG. 10: The collapse of scaled persistence data versus scaledtime using θ = 0 .
235 and z = 2 .
08 is shown for differentsystem sizes for κ >
1. Inset shows the unscaled data. -3 -2 -1 N o o f D o m a i n W a ll s Time f D x f D y FIG. 11: Decay of the fraction of domain walls with time at κ >
FIG. 12: A typical snapshot of a steady state configurationfor κ > D. κ = 1 Here we find that the persistence probability follows apower law decay with θ = 0 . ± . z value 1 . ± .
01 (Fig. 14).We have again studied the dynamics of f D x and f D y ;the former shows a fast saturation at 0 . ≈ .
515 (Fig. 15). Thisvalue, unlike in the case κ >
1, does not show very goodagreement with 1 /z obtained from the finite size scal-ing analysis. We will get back to this point in the nextsection.The results for f D x and f D y imply that the systemreaches a perfect antiphase configuration as there are nointerfaces left in the system with f D x = 0 . f D y = 0at later times. P r obab ili t y o f o cc u r en c e No. of domain Walls κ > 2.0 κ = 2.0 < 2.0 FIG. 13: Normalised steady state distributions of numberof domain walls for different κ > κ increases. The lines areguides to the eye. -1 -1 -2 -1 P(L,t)/t - Θ L/t κ = 1 L = 40L = 100L = 200L = 30010 -2 -1 P(t) t FIG. 14: The collapse of scaled persistence data versus scaledtime using θ = 0 .
263 and z = 1 .
84 is shown for differentsystem sizes at κ = 1. Inset shows the unscaled data. E. κ ≤ . In order to make a comparison with the purely ferro-magnetic case, we have also studied the Hamiltonian (2)with negative values of κ which essentially correspondsto the two dimensional Ising model with anisotropic nextnearest neighbour ferromagnetic interaction. κ = 0 corresponds to the pure two dimensional Isingmodel for which the numerically calculated value of θ ≃ .
22 is verified. We find a new result when κ is allowed toassume negative values, the persistence exponent θ has avalue ≃ .
20 for | κ | > < | κ | ≤
1, the value of θ has an apparent dependence on κ , varying between 0.22to 0.20. However, it is difficult to numerically confirmthe nature of the dependence in such a range and wehave refrained from doing it. At least for | κ | >>
1, thepersistence exponent is definitely different from that ofat κ = 0. The growth exponent z however, appears tobe constant and ≃ κ ≤
0. A datacollapse for large negative κ is shown in Fig. 16 using -6 -5 -4 -3 -2 -1 Time f D x f D y FIG. 15: Decay of the fraction of domain walls with time at κ = 1 are shown along horizontal and vertical directions. Thedashed line has slope equal to 0.515. θ = 0 .
20 and z = 2 . P(L,t)/t - Θ L/t κ < 0 L = 100L = 200L = 300 0.1 10 P(t) t FIG. 16: The collapse of scaled persistence data versus scaledtime using θ = 0 .
20 and z = 2 . κ < −
1. Inset shows the unscaled data.
The effect of the anisotropy shows up clearly in thebehaviour of f D x and f D y as functions of time (Fig. 17).For κ = 0, they have identical behaviour, both reachinga finite saturation value showing that there may be inter-faces generated in either of the directions (correspondingto the striped states which are known to occur here). Asthe absolute value of κ is increased, f D x shows a fast de-cay to zero while f D y attains a constant value. The sat-uration value attained by f D y increases markedly with | κ | while for f D x the decay to zero becomes faster. Onecan conduct a stability analysis for striped states to showthat such states become unstable when the interfaces arevertical and κ increases beyond 1, leading to the result f D x → z value from the variations of f D x or f D y is not very simple here as the quantities do not showsmooth power law behaviour over a sufficient interval oftime.The fact that f D y and/or f D x reach a finite saturation -4 -3 -2 -1 f r a c t i on o f do m a i n w a ll s Time κ = -1.5 κ = -1.0 κ = -0.7 κ = -0.7 κ = -1.0 κ = -1.5 κ = 0.0 κ = 0.0 FIG. 17: Decay of the fraction of domain walls with timeat κ ≤ f D x ), shown by dottedlines) and vertical ( f D y ), shown by solid lines) directions. value indicates that striped states occur here as well. Thebehaviour of f D x and f D y suggests that in contrast to theisotropic case where interfaces can appear either horizon-tally or vertically, here the interfaces appear dominantlyalong the x direction as κ is increased. Thus the nor-malised distribution of the number of domain walls along y is shown in Fig. 18. We find that as κ is increased inmagnitude, more and more interfaces appear. However,the number of interfaces is always even consistent withthe fact that interfaces occur between ferromagnetic do-mains of all up and all down spins. P r obab ili t y o f o cc u r en c e No. of domain walls κ = 0.0 κ = -0.7 κ = -1.0 κ = -1.5 κ = -2.0 κ = -20.0 FIG. 18: Normalised steady state distributions of numberof domain walls for different κ ≤ | κ | increases. The lines areguides to the eye. Lastly in this section, we discuss the behaviour of themagnetisation which is the order parameter in a fer-romagnetic system. As striped states are formed, themagnetisation will assume values less than unity. Theprobability of configurations with magnetisation equal tounity shows a stepped behaviour, with values changing at | κ | = 1 and 2 and assuming constant values at 1 < | κ | < | κ | = 2 (Fig. 19). P r obab ili t y o f m agne t i s a t i on = κ FIG. 19: Probability that the magnetisation takes a steadystate value equal to unity is shown against κ when κ ≤ IV. DISCUSSIONS AND CONCLUSIONS
We have investigated some dynamical features of theANNNI model in two dimensions following a quench tozero temperature. We have obtained the results that thedynamics is very much dependent on the value of κ , theratio of the antiferromagnetic interaction to the ferro-magnetic interaction along one direction. This is similarto the dynamics of the one dimensional model studiedearlier, but here we have more intricate features, e.g.,that of the occurrence of quasi frozen-in structures for κ < κ ≥
1, but exactly at κ = 1, the exponents θ and z aredifferent from those at κ >
1. The exponents for κ > κ > κ = 0 and κ > θ and z are individually quiteclose for κ = 0 and κ >
1, the product zθ = α are quitedifferent. For κ = 0, α ≃ .
44 while for κ >
1, it is0 . ± . κ = 0and κ > κ = 1 is the special pointwhere the dynamic behaviour changes radically. Herethere appears to be some ambiguity regarding the valueof z ; estimating α from the finite size scaling analysisgives α ≈ . ± .
005 while using the z value from thedomain dynamics, the estimate is approximately equalto 0.51. However, the dynamics of the domain sizes maynot be very accurately reflected by the dynamics of f D y in which case α ≈ .
48 is a more reliable result. Thuswe find that although the values of θ and z are quitedifferent for κ = 1 and κ >
1, the α values are close.We would like to add here that when there is a powerlaw decay of a quantity related to the domain dynamics,it is highly unlikely that it will be accompanied by anexponent which is different from the growth exponent.Thus, even though we get slightly different values of z for κ = 1 from the two analyses, it is more likely thatthis is an artifact of the numerical simulations.Another feature present in the two dimensional Isingmodel is the finite probability with which it ends up in astriped state. The same happens for κ >
1, but here theprobabilities are quite different and also dependent on κ .We find that there is a significant role of the point κ = 2here as this probability has different values at κ = 2, κ > κ < κ < − κ ≥ κ = 1) are in fact very close to that of the two di-mensional Ising model, but simulations done for identi-cal system sizes averaged over the same number of initialconfigurations are able to confirm the difference. Thequalitative behaviour of the domain dynamics is againstrongly κ dependent when κ is negative. Another pointto note is that the probability that the system evolves to apure state is κ dependent in both the ANNNI model andthe Ising model. In both cases in fact, this probabilitydecreases in a step like manner with increasing magni-tude of κ . We also find the interesting result that whilethe distribution of the number of domain walls can havenon-zero values at odd values of N D in the ANNNI modelbecause of the four fold degeneracy of the antiphase, forthe Ising model, odd values of N D are not permissible asthe ferromagnetic phase is two fold degenerate.Finally we comment on the fact that although the dy-namical behaviour, as far as domains are concerned, re-flects the inherent anisotropy of the system (in both theferromagnetic and antiferromagnetic models), the persis-tence probability is unaffected by it. In order to verifythis, we estimated P ( t ) along an isolated chain of spinsalong x and y directions separately and found that thetwo estimates gave identical results for all values of κ .In conclusion, it is found that except for the region0 < | κ | <
1, the dynamical behaviour of the Hamiltonian(2) is remarkably similar for negative and positive κ ; thepersistence and growth exponents get only marginallyaffected compared to the values of the two dimensionalIsing case ( κ = 0) and the domain distributions havesimilar nature. However, the region 0 < κ < κ values also in the sense that exceptfor κ = 1, the system has a tendency to get locked ina “striped state”. However, even in that case, the al-gebraic decay of the persistence probability is observed.Thus algebraic decay of persistence probability seems tobe valid only when the metastable state is a striped state.Although there is no dynamic frustration at κ = 1 in thesense that it always evolves to a state with perfect an-tiphase structure, it happens to be a very special pointwhere the persistence exponent and growth exponentsare unique and appreciably different from those of the κ = 0 case.In this paper, the behaviour of the two dimensionalANNNI model under a zero temperature has been dis- cussed; the dynamics at finite temperature can be in factquite different. At finite temperatures, the spin flippingprobabilities are stochastic, and the dynamical frustra-tion may be overcome by the thermal fluctuautions. Ithas been observed earlier [14] that in a thermal anneal-ing scheme of the one dimensional ANNNI model, the κ = 0 . [1] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys.
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