One dimensional phase-ordering in the Ising model with space decaying interactions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y J. Stat. Phys. manuscript No. (will be inserted by the editor)
Federico Corberi · Eugenio Lippiello · Paolo Politi
One dimensional phase-ordering in the Ising modelwith space decaying interactions
Received: date / Accepted: date
Abstract
The study of the phase ordering kinetics of the ferromagnetic one-dimensional Ising modeldates back to 1963 (R.J. Glauber, J. Math. Phys. 4, 294) for non conserved order parameter (NCOP)and to 1991 (S.J. Cornell, K. Kaski and R.B. Stinchcombe, Phys. Rev. B 44, 12263) for conserved orderparameter (COP). The case of long range interactions J ( r ) has been widely studied at equilibrium buttheir effect on relaxation is a much less investigated field. Here we make a detailed numerical andanalytical study of both cases, NCOP and COP. Many results are valid for any positive, decreasingcoupling J ( r ), but we focus specifically on the exponential case, J exp ( r ) = e − r/R with varying R > J pow ( r ) = 1 /r σ with σ >
0. We find that the asymptotic growthlaw L ( t ) is the usual algebraic one, L ( t ) ∼ t /z , of the corresponding model with nearest neighborginteraction ( z NCOP = 2 and z COP = 3) for all models except J pow for small σ : in the non conserved casewhen σ ≤ z NCOP = σ + 1) and in the conserved case when σ → + ( z COP = 4 β + 3, where β = 1 /T is the inverse of the absolute temperature). The models with space decaying interactions also differmarkedly from the ones with nearest neighbors due to the presence of many long-lasting preasymptoticregimes, such as an exponential mean-field behavior with L ( t ) ∼ e t , a ballistic one with L ( t ) ∼ t , aslow (logarithmic) behavior L ( t ) ∼ ln t and one with L ( t ) ∼ t /σ +1 . All these regimes and their validityranges have been found analytically and verified in numerical simulations. Our results show that themain effect of the conservation law is a strong slowdown of COP dynamics if interactions have anextended range. Finally, by comparing the Ising model at hand with continuum approaches based ona Ginzburg-Landau free energy, we discuss when and to which extent the latter represent a faithfuldescription of the former. Keywords
Ising model · Coarsening · Phase-ordering · Long-range interactions
Federico CorberiDipartimento di Fisica “E. R. Caianiello”, and INFN, Gruppo Collegato di Salerno, and CNISM, Unit`a diSalerno,Universit`a di Salerno, via Giovanni Paolo II 132, 84084 Fisciano (SA), ItalyE-mail: [email protected] LippielloDipartimento di Matematica e Fisica, Universit`a della Campania, Viale Lioncoln 5, 81100, Caserta, ItalyE-mail: [email protected] PolitiIstituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, via Madonna del Piano 10, I-50019 SestoFiorentino, ItalyINFN Sezione di Firenze, via G. Sansone 1 I-50019, Sesto Fiorentino, ItalyE-mail: [email protected]
Phase ordering [1] is the dynamical process of growth of order when a system is quenched from ahigh temperature homogeneous phase to a low temperature broken-symmetry phase. It occurs throughdomain coarsening, with the average size of domains of different phases, L ( t ), which increases in time.Phase ordering is an old research topic and its most unified picture is based on a continuumapproach whose starting point is a Landau-Ginzburg free energy. In an impressive series of papersdating back around twenty five years, Alan Bray and collaborators have constructed a theory whichcovers nonconserved and conserved models, scalar and vector fields, short and long range interactions [1,2,3]. Their results for the growth law L ( t ) do not depend on the spatial dimension d of the physicalsystem, which only appears when defining the limits of applicability of the theory.It is clear that one dimension, the case we focus on in this paper, plays a special role for scalarsystems because the Curie temperature vanishes for short-range interactions, T c = 0. This means, firstof all, that it is not possible to quench the temperature from T i > T c to T f ≡ T < T c . However, we canconsider quenches to a vanishing T = 0, or to a very small finite temperature T ≪ T = 0) the equilibrium state is fully ordered and the dynamics can never increase thesystem energy. This means that zero temperature dynamics can be blocked, which is actually whathappens if the order parameter is conserved. In the latter case ( T >
0) the final equilibrium stateis made of ordered regions of average size equal to the equilibrium correlation length ξ and, hence,coarsening stops when L ( t ) ≃ ξ ( T ). Since ξ is a very fast increasing function of (1 /T ) (for nearestneighbor (nn) interactions, ξ ( T ) ≃ e /T ), low − T coarsening dynamics lasts for a long time.The nn Ising model has been studied decades ago. Nonconserved dynamics proceeds via spin-flips and it is possible to attain the ground state through processes which lower or keep constantthe energy. We will see that dynamics can be easily described in terms of random walks performedby domain walls, so it is not surprising that the average size of domains grows according to thelaw [4] L ( t ) ≃ t / . Conserved dynamics proceeds differently because single spin-flips, which wouldchange the order parameter (the magnetization), are not allowed. In this case we rather have spin-exchange processes with spins that can evaporate from a domain wall and condensate to another dropletafter a diffusion process. It is known that such evaporation-condensation mechanism slows down thedynamics [5] with respect to spin-flip and results in the growth law L ( t ) ≃ t / .In a recent publication [6] we have studied the effects of a coupling constant J ( r ) = e − r/R , decreas-ing exponentially with the distance r between two spins, on the one-dimensional coarsening dynamics.Since this interaction introduces the new length scale R , it is reasonable to expect the regime L ( t ) < R to be physically different from the one with L ( t ) > R . We actually found more than that, becausewe identified different dynamical regimes for large L ( t ). In this paper we go beyond the exponentialcoupling, finding a series of results which are independent of the explicit form of the coupling, providedthat J ( r ) is a positive, decreasing function of r . For definiteness, detailed simulations and specific cal-culations have been done for exponential, J exp = e − r/R , and power-law, J pow ( r ) = 1 /r σ , couplings.We focus on the growth law L ( t ) of the domains’ size, which we compute by means of different ana-lytical approaches along the whole time history, from the instant of the quench up to the asymptoticstages. Our results are successfully compared to the outcome of numerical simulations.Besides addressing the modifications of the kinetics due to a space decaying interaction, in thispaper we also discuss an interesting question which has not been considered previously, namely thecomparison between the coarsening dynamics of the Ising model and the one emerging from a de-terministic continuum description of the same system. This analysis allows us to provide a physicalinterpretation to the various dynamical regimes and to show that some of them, although occurringwith the same coarsening law in the discrete and in the continuum models, are associated to differentphysical mechanisms. We will also comment on a recent preprint [7] studying phase ordering for astrictly related Non Conserved Order Parameter (NCOP) long-range discrete model.This paper is organized as follows: in Sec. 2 we introduce the models, we define the form of theinteractions and discuss the equilibrium structure. We also specify the kinetic rules and discuss theelementary processes driving the evolution and the methods that can be used to perform numericalsimulations. Section 3 is devoted to the definition of the simplified models with only few domains fromwhich most of our analytical results can be deduced. In Sec. 4 we focus on the NCOP case, by derivingour analytical predictions and comparing them with the outcome of numerical simulations. We do the same in Sec. 5 for the Conserved Order Parameter (COP) case. In Sec. 6 we compare our results forthe discrete Ising model with the behavior of the continuum model based on a Ginzburg-Landau freeenergy, discussing to which extent the latter can grasp the physics observed in the former. In Sec. 7 wediscuss the results of this paper on general grounds, and suggest some possible future research lines,while in the Appendix we give some details of numerical simulations. The captions of most figuresshowing our results are preceded by an abbreviation to immediately identify the model in question.For example, (NCOP exp) means we are considering the exponential model of the nonconservedclass. We will consider a general one-dimensional Ising model described by the Hamiltonian H = − + ∞ X i = −∞ X r> J ( r ) s i s i + r , (1)where s i = ± J ( r ) is a positive, decreasing function ofthe distance r .Main formulas will be written for general J ( r ), but specific calculations and simulations will bedone for an exponentially decreasing coupling, J exp ( r ) = e − r/R (2)and for a power-law one, J pow ( r ) = 1 r σ , (3)with σ > J nn ( r ) = δ r, . (4)In all cases, Eqs. (2-4), J ( r ) should be proportional to some energy scale J , which will be assumedto be equal to one throughout all the paper.Since we will use both a spin and a lattice gas language, it is useful to rephrase the Ising Hamiltonian(1) using the variable n i = (1 + s i ) /
2, which takes the values n i = 1 (for s i = +1) and n i = 0 (for s i = − H = − + ∞ X i = −∞ X r> J ( r )(2 n i − n i + r −
1) (5)= − + ∞ X i = −∞ X r> J ( r ) n i n i + r + 4 + ∞ X i = −∞ n i X r> J ( r ) − + ∞ X i = −∞ X r> J ( r ) . (6)The third term on the right-hand side is an irrelevant (extensive) constant. The second term onthe right-hand side is constant as well if the dynamics preserves the order parameter, i.e. if P i s i = P i (2 n i −
1) is a constant of motion. The spin language is appropriate for NCOP systems, like magneticsystems; the lattice gas language is appropriate instead for COP systems, like alloys or fluids. Withinthis language, the coupling energy between two particles at distance r , see Eq. (6), is − J ( r ).As for the equilibrium properties, it is straightforward that there are no qualitative differencesbetween J exp ( r ) and J nn ( r ). The power-law case, instead, requires a few more words. If GS is one ofthe two ground states and we flip one single spin obtaining the (one flip) state 1F, we have the energydifference E − E GS = 4 ∞ X r =1 r σ , (7) which diverges for σ ≤
0. For negative σ we are in the so called strong long-range regime, whereextensivity and (above all) additivity do not hold. We will not consider this case, which is reviewed inRef. [8].If we flip all spins i >
0, therefore obtaining the microscopic state 1DW characterized by a singledomain wall, we have the energy difference E − E GS = 2 ∞ X i =1 0 X j = −∞ i − j ) σ , (8)which is finite for σ >
1. Therefore the standard arguments to explain the absence of long-range [9,10]order at finite temperature still applies. For 0 < σ ≤ weak long-range regime) there is an orderedphase at finite temperature: for 0 < σ < σ = 1there is a Kosterlitz-Thouless phase transition with a jump of the magnetization [11,12,13]. It is alsopossible to argue that mean-field critical exponents are expected for 0 < σ < [14].2.1 Dynamical evolutionAll models will be studied for both nonconserved and conserved order parameter. The simplest NCOPdynamics is based on single spin-flip processes, also called Glauber dynamics: s i → − s i . Instead, thesimplest COP dynamics is implemented by a spin-exchange processes between two opposite spins, alsocalled Kawasaki dynamics: s i ↔ s j . In the lattice gas language, if (say) s i = 1, this move is simply thehopping of a particle on site i towards an empty site j : ( n i , n j ) = (1 , → (0 , i, j ) must be nn: j = i ± W IF between an initial state I and a final state F must satisfy detailed balancein order to ensure relaxation to equilibrium, W IF /W FI = e − β ( E F − E I ) in a canonical ensemble. If weimpose the additional constraint W IF + W FI = 1 we obtain the Glauber transition rates, W IF = 11 + e β ( E F − E I ) . (9)These transition rates will be used in the following for the elementary moves, namely spin flips forNCOP or spin exchanges for COP.Starting with the NCOP case, in Fig. 1(a) we plot the three possible different single spin-flipprocesses. We use here also a language which associates a domain wall (DW) to a pair of nn antiparallelspins. Within such language the three processes correspond to diffusion, annihilation and creation ofDWs.It is important to evaluate for each process the energy difference ∆E = E F − E I between thefinal and the initial state. For a nn Ising model, it is straightforward to conclude that ( ∆E ) nn = 0for DW diffusion, ( ∆E ) nn < ∆E ) nn > J ( r ) = 0 for r > ∆E forany pair of initial and final configurations differing for a single spin flip, since this task involves theknowledge of the whole system configuration. However, as argued in Sec. 4, most of NCOP dynamicscan be understood making reference to a simple configuration, see Fig. 2(a). In this case we stillhave ∆E < ∆E > ∆E = 0 and its sign is such that W IF favors the closing of the smaller domain.Dynamics at T = 0 is therefore trivial: DWs cannot be created and each DW drifts to the closest DWuntil annihilation occurs. When temperature is switched on, two things happen: DW drift becomes anasymmetric DW diffusion and DW creation is permitted. Asymmetric diffusion makes dynamics muchmore complicated because the symmetric and the antisymmetric parts of DW hopping compete andtheir balance depends on the temperature, on how J ( r ) decreases with r , and on the typical distance L ( t ) between neighbouring DWs. DW creation at finite T makes possible to attain thermal equilibrium,with a distance L ( t ) between DWs which equals the equilibrium correlation length, ξ ( T ), similarly towhat happens in the nn case.Let us now move to the COP class, whose elementary processes at the basis of the dynamicsare shown in Fig. 1(b). We use again a particle language where now a particle represents n i = 1while an empty sites (or a hole) corresponds to n i = 0. The four elementary processes are particle and hole diffusion, particle attachment and particle detachment. If we limit to nn interaction it isobvious that diffusion keeps the energy constant, attachment decreases the energy and detachmentincreases the energy. Going beyond nn interaction makes impossible general statements about the signof ∆E = E F − E I , as in the NCOP case, but, again, if we focus on the simple configuration depictedin Fig. 2(b) we can make some statements: detachment costs energy, attachment gains energy anddiffusion is asymmetric with a drift towards the closest domain. It is clear that for COP, temperature has a more direct role in the dynamics, which can be easilyhighlighted in the nn model. At T = 0 the allowed processes are particle and hole diffusion, and particleattachment. This means that the dynamics stops as soon as a particle is attached to another particleand a hole is attached to another hole. Starting from a fully disordered configuration, T i = ∞ , weobtain an average domain size L nn ( T = 0) ≃ .
135 [15]. If J ( r ) extends beyond nearest neighbours,being attached to another particle/hole is not enought to avoid diffusion but for not so small domainsit is surely true that detachment costs energy and it is forbidden at T = 0. We can therefore affirm thatzero-temperature dynamics stops almost immediately, with a “small” average size of domains. Whenwe switch on temperature, particle detachment is allowed. Monomers perform an asymmetric diffusionand they can travel the whole empty space between two clusters and reach the other domain. Theprobability that this occurs will be evaluated in Sec. 5. This is the mechanism whereby neighbouringdomains exchange monomers and represents the basic process leading to coarsening since, due to theloss or gain of monomers, the length of each domain performs a sort of random walk.If a second particle is detached in the empty domain between two clusters while the first one is stilldiffusing (an extremely rare event at low T ) they can stick together forming a new cluster. This is themechanism arresting coarsening and producing equilibration when L ( t ) ≃ ξ ( T ). (ΔΕ) nn < 0(ΔΕ) nn > 0(ΔΕ) nn = 0(a) (b) Fig. 1 (a) NCOP (Glauber) dynamics, using spin language and domain wall (DW) language (a DW is repre-sented by a circle). The reversed spin is the thick spin and its flipping implies the hopping of a DW (top), theannihilation of two DWs (center), or the creation of two DWs (bottom). DWs are solid and open circles, beforeand after the flipping, respectively. (b) COP (Kawasaki) dynamics, using the lattice gas language. Full circlesare particles, empty circles are holes. Arrows indicate possible moves of a particle. NCOP and COP microscopicprocesses are classified according to the variation of energy for the nn Ising model, ( ∆E ) nn = E F − E I , betweenthe final and the initial state. More precisely, the drift points to the closest domain if the two domains are of equal length. This meansthat the direction of the drift changes when the diffusing particle reaches the middlepoint between the twodomains. In the general case of unequal domains, the drift changes direction when the diffusing particle movesfrom one domain to the other, but not in the middle point. N spins with periodic boundary conditions and implement standard MonteCarlo dynamics with Glauber transition rates, see Eq. (9). Since the interaction extends to all thespins, for each spin flip trial, the evaluation of the energy cost ∆E = E F − E I involves the sum overall spins in the system with a computation time N × N . This makes simulations much slower thanin the nn case and prevents one to obtain completely satisfactory results in some cases. An efficientalgorithm has been developed in Ref. [16] which has shown that, in the case of periodic boundaryconditions, it is possible to obtain an efficient diagonalization of the coupling matrix J ( r ) via FFT.By means of this method, one obtains an exact algorithm that scales with N × ln N . In our study weconsider two different approximated simulation schemes which scale with N and with the number ofdefects n , respectively. The two schemes are briefly introduced below and a more detailed discussionon simulations techniques is contained in Appendix A.S1 Simulations with truncated J ( r ) . In this case we consider the Hamiltonian (1) in terms of the spin variables and assume, for expo-nential couplings J exp ( r ), that J ( r ) ≡ r > M R , where M is a sufficiently large number. Wehave checked that with M of order 10 the results obtained with truncation are indistinguishablefrom exact simulations. With such large values of M , therefore, this kind of simulations is basicallyexact.S2 Simulations with a reduced number of interacting kinks.
This method implements the DW description illustrated in Fig.(1) where a spin flip or a pair ofspin flips is mapped in different moves of a particle . The key observation is that, indicating with n the number of DWs in the system at a given time, the energy difference ∆E after the flip of thespin in the i -th site can be always written in the form (see Appendix A) ∆E = ( ∆E ) nn + n/ X j = − n/ x j = i ( − j Q ( | x j − i | ) , (10)where x k is the position of the k -th DW and DWs are sorted according to their distance from thesite i . In the above equation ( ∆E ) nn is the quantity specified in Fig. 1, namely the nn contributionto the energy change for the different moves. Q ( r ) is a decreasing function of r , proportional to J ( r ) or to its integral for COP and NCOP dynamics, respectively. The numerical implementationof this simulation method is more complex but the evaluation of ∆E involves the sum of n terms,while method S1 involves the sum of M R terms, therefore making S2 particularly advantageous atlarge times, when n = N/L ( t ) ≪ M R . On the other hand, at short times n is comparable to thetotal number of spins in the system and S2 becomes less efficient.Within the exact S2 framework one can introduce an approximated simulation method correspond-ing to consider an interaction extending only to a finite number of kinks n K , by means of thesubstitution n → n K in Eq. (10), where n K is a parameter to be optimized. Clearly, the smaller is n K the worst, but the faster, the approximation is. This approximation is expected to provide exactresults for sufficiently large times when Q ( | x j − i | ) can be neglected for j > n K . In the followingwe will refer to S2 assuming a suitable n K .In the case of COP dynamics we always use the simulation method S2 whereas for NCOP we use S1for simulations at short times and S2 for longer simulations. In addition, for NCOP, to speed further upthe computations, we have considered simulations where activated processes (spin-flips in the bulk) areforbidden, preventing the equilibration of the system. This approximation becomes exact in the limit ofvery small temperature and at large t , when the average distance between domains is sufficiently largeso that the sum in Eq.(10) becomes much smaller than 2 J (1). For COP, instead, we adopt a rejectionfree algorithm of the type described in [17] where activated moves (such as monomers evaporations),which in the low T limit are very unlikely and delay the dynamics, are always accepted and time isincreased according to the likeliness of the accepted event (see Appendix A). This technique does notintroduce any error.In our simulations we have consider a chain of N = 10 − spins, depending on the various pa-rameters. These numbers are sufficiently large to avoid finite-size effects in the range of time considered. For each choice of the parameters, we take an average over 10 − realizations of the initial conditionsand of the thermal history. Additional details of the used algorithms can be found in Appendix A.We finally stress that for the models which are magnetized at finite temperature (power modelswith σ <
1) we always consider a quench to the ordered phase. In particular, for σ = 0 . T ≤ T c > After a quench from the fully disordered phase ( T i = ∞ ) to zero or low temperature the system relaxesto equilibrium, which is characterized by a local order on a length scale equal to the correlation length ξ ( T ). Upon increasing the interaction range, the correlation length increases as well and for the classof power law models we even have long range order at finite T , if σ ≤
1. In any case, even if T c = 0 wecan consider low enough T such that ξ ( T ) is arbitrarily large. (a) (b)S * X(t)X(t) x Fig. 2
Simple one dimensional configurations, with periodic boundary conditions. (a) A single domain ofdown spins and length X ( t ) (with X (0) = L ) and a neighbouring domain of initial length L (which scaleswith L ). The interaction of spin S ∗ with down spins is compensated by its interaction with dashed up spins. (b)Two clusters of particles, labelled as “1” and “2”, whose sizes change in time because they exchange monomers.The distance between the monomer and cluster 1 is x . Relaxation to equilibrium occurs through a coarsening kinetics where domains disappear leading tothe increase over time of their average size, L ( t ). Figure 2 represents the simplest configurations leadingto the disappearance of a single domain, for NCOP (a) and COP (b). The analysis of the dynamics ofsuch configurations not only explains the physics of the process, but it also allows to derive most ofthe coarsening laws. In fact, the evolution is a self-similar phenomenon characterized by a single lengthscale, L ( t ). According to the scaling hypothesis, this property means, e.g., that the correlation function C ( r, t ) = h s i ( t ) s i + r ( t ) i is actually a function of a single variable, C ( r, t ) = f ( r/L ( t )). Invoking scaling,we argue that the functional dependence L ( t ) can be found by determining the typical temporal scale t necessary to close a domain of initial size L and inverting the resulting function t ( L ). Let us now seehow this program can be implemented.Starting from the NCOP case, see Fig. 2(a), we consider a domain of the negative phase and initialsize X (0) = L and a neighbouring domain of the positive phase and initial size L ≥ L , which scaleswith L (periodic boundary conditions apply). Each DW performs an anisotropic random walk with adrift which favors the closing of the smallest domain, whose closure time is t ( L ). In the next Section wewill evaluate the drift in the configuration L = ∞ and the closure time in the configuration L = L .This choice is due to the fact that the drift vanishes for L = L and the closure time may diverge for L = ∞ . Any other choice of L would be equally arbitrary: our choice is justified a priori bysimplicity and a posteriori by the comparison of our analysis with numerical results.The COP case, Fig. 2(b), requires to consider two clusters of particles. This is because two clusterscan exchange matter until finally one of them disappears while a single cluster can never do that. Forsimplicity we assume that all domains (clusters of particles and clusters of holes) have an initial lengthequal to L . If T = 0 the configuration is frozen because the detachment of a particle requires energy.If T >
0, such process is permitted on the time scale τ det ≈ e J (1) /T . Once that a particle has detachedfrom cluster “1” it performs an asymmetric random walk ending its journey either reattaching to thesame cluster or attaching to cluster “2”. In the former case the net outcome is null. In the latter casethere is a net exchange of mass (1 →
2) between the first and the second cluster and since the reversedprocess (2 →
1) may occur in first approximation with the same probability, there is a symmetricexchange of matter between the two clusters leading to a diffusion process for the length X ( t ), seeFig. 2(b).In Sec. 5 we will determine the average time t cl needed by a domain of initial size X (0) = L toeither disappear ( X ( t cl ) = 0) or collect all the matter ( X ( t cl ) = 2 L ) due to the evaporation of theother domain. The key ingredient to evaluate t cl will be the effective diffusivity of X , D ( L ), which isinversely proportional to the probability p ( L ) that a monomer, detached from a cluster, attains theother one. L = L the domain walls feel a drift favoring the closure of thesmallest domain. As already said, we evaluate such drift for L → ∞ . The smaller domain has thetime-dependent length X ( t ), with X (0) = L . If we define the integrated quantity I ( x ) ≡ ∞ X r = x J ( r ) , (11)the process X → X + 1 requires the energy ( ∆E ) + = 4 I ( X ), while the process X → X − Therefore, using Eq. (9), the probabilities of such processes are p ± = 11 + e ± βI ( X ) , (12)and the drift is δ ( X ) = p + − p − = − tanh(2 βI ( X )) . (13)Next, we evaluate the closing time using a symmetric initial configuration, L = L (see discussionrelated to footnote 2). In terms of the random variable X ( t ) whose evolution is controlled by theprobabilities p ± , this amounts to have X (0) = L and absorbing barriers in X = 0 and X = 2 L .Upon mapping the discrete, asymmetric random walk onto a convection-diffusion equation, and in theapproximation of constant drift (because of scaling it is assumed to depend on X (0), not on X ( t )), theaverage exit time of the particle, given in [19], is t ( L ) = Lv tanh (cid:18) vLD (cid:19) . (14) In the absence of drift the length X of the domain performs a symmetric random walk with the initialcondition X (0) = L . The closing time is equivalent to the first passage time in the origin, X ( t cl ) = 0. It is wellknown that for symmetric hopping its average value diverges, h t cl ( L ) i = ∞ . At the beginning of the process the two clusters are of equal length and the two possible exchanges ofmatter, 1 → →
1, are perfectly symmetric. In the course of time the different lengths between the twoclusters creates an asymmetry which will be neglected in our calculations. More precisely, it is the process X + 1 → X to release the same energy, but for large X we can neglect thisdifference. This expression gives the correct limits t ( L ) = L /D for vanishing drift and t ( L ) = L/v for strong drift( v ≫ D/L ). According to the spirit of the above calculation and in view of Eq. (13), the appropriateexpression to be used for the drift is v ( L ) = v tanh(2 βI ( L )) , (15)where v is a constant. Therefore, Eqs. (14) and (15) give the closing time t of a domain as a functionof its initial size L .It must be stressed that the simplified model describes the true situation with many domains if thenext boundary is at a distance where J ( r ) is already very small. This implies that the interaction withthe subsequent boundaries can be neglected, which is basically what the present approximation does.Since the typical distance at which an interface is found is L ( t ) we ask the condition J ( L ) < e − M ,where M &
1, in order for the approximation to be valid. This provides the lower limit L a for L ( t )above which the results of the model with few domains holds, with L exp a = M R and L pow a = e M/ (1+ σ ) . (16)Notice that L a depends linearly on R for the exponential case while it is weakly dependent on σ forthe algebraic case.Before studying Eqs. (14) and (15), let us write in the continuum approximation the explicitexpression of the quantity I ( L ) for an exponentially decaying interaction, Eq. (2), and for a power-lawinteraction, Eq. (3): J exp ( r ) = e − r/R I exp ( L ) ≃ Re − L/R (17) J pow ( r ) = 1 r σ I pow ( L ) ≃ σ L σ . (18)In the following we demonstrate that Eqs. (14,15) imply the existence of three dynamical regimes,emerging as particular limits in which the arguments of the two hyperbolic tangents appearing inEqs. (14) and (15) are small or large. A summary of such regimes (plus others that are not captured bythis analytical method) and of the characteristic crossover lengths between them is provided in table 1. The ballistic regime —
In this regime L ( t ) grows linearly with time, which means a constant v (i.e.,not depending on L ). This is possible if both the hyperbolic tangents in Eqs. (14) and (15) can beapproximated to one, i.e. if both arguments are very large, v LD ≫ βI ( L ) ≫ . (19)The left inequality provides a lower limit for L , L ≫ ( D/v ), within the approximation with fewdomains. Notice however that this bound might be inadequate if D/v happens to be smaller than thetypical length L a above which the approximation holds true.On the other hand, the right inequality in (19), considering the decreasing behavior of I ( L ), providesun upper limit L b defined by the relation 2 βI ( L b ) = 1. More precisely, for the exponential and power-law models we find L exp b = R ln(2 βR ) and L pow b = (cid:18) σ β (cid:19) /σ , (20)which both diverge for T →
0. In this limit, therefore, the ballistic dynamics becomes the asymptoticone. For any finite T , instead, it is followed by the other regimes described below. The slow regime —
This regime corresponds to a small v , such that the argument of the trascen-dental function in Eq. (15) is small, and to a t ( L ), see Eq. (14), which is given by the relation t = L/v .This means that we must have 2 βI ( L ) ≪ vLD ≫ . (21)The left inequality gives L ≫ L b and the resulting velocity is v ( L ) = 2 v βI ( L ) . (22) Therefore, the right inequality of Eq. (21) gives2 v D LβI ( L ) ≫ , (23)which is consistent with the first of Eqs. (21) and it allows us to define the length L c through therelation 2 v D L c βI ( L c ) = 1 . (24)For the exponential case we have 2 v D βL exp c Re − L exp c /R = 1 , (25)which has the form Axe − x = 1, where x = L exp c /R and A = v D βR . An approximate solution for A ≫ x = ln A + ln x = ln A + ln ln A + ln (cid:18) x ln A (cid:19) ≃ ln( A ln A ) . (26)So, we obtain L exp c = R ln (cid:20) v D βR ln (cid:18) v D βR (cid:19)(cid:21) . (27)For the power-law case we have instead L pow c = (cid:18) v σD β (cid:19) / ( σ − = (cid:16) v D (cid:17) / ( σ − ( L pow b ) σ/ ( σ − . (28)In the range ( L b , L c ) the coarsening law is given by the relation t = Lv ( L ) = L v βI ( L ) , (29)which implies a logarithmically slow dynamics for the exponential case, L ( t ) ∼ R ln t, (30)and a non universal power-law growth for the power-law case, L ( t ) ≃ t / ( σ +1) . (31)It is obvious that L pow c , see Eq. (28), is meaningless for σ <
1. In fact, L a is a length of order oneand L b is a length which increases with decreasing the temperature or (for the exponential case) withincreasing R . These same properties apply to L c (and L c ≫ L b ) in the exponential case and in thepower-law case with σ ≥
1. Instead, if σ < L ≫ (cid:18) σD v β (cid:19) / (1 − σ ) , (32)which is automatically satisfied at low temperature, implying that the slow regime extends to infinityat any temperature T < T c for the power-law case with σ <
1. In all the other cases there will be athird regime, that we now discuss.
Diffusive regime —
For L ≫ L c it is vL/D ≪
1, so Eq. (14) gives t = Lv vLD = L D , (33)from which the usual growth law of the nn case, L ( t ) ≃ t / , (34)is recovered.We summarize all results for NCOP coarsening in Table 1. The regimes described by the approx-imation with few domains are those occurring for L ( t ) > L a , namely those to the right of the doublevertical line in the Table. The early regimes with L ( t ) < L a , which are observed only for J exp , will bediscussed later. L MF inter-mediate L a ballistic L b slow L c diffusive J exp e t R ??? MR t R ln(2 βR ) ln t R ln { (2 v /D ) βR · ln (cid:2) (2 v /D ) βR (cid:3)(cid:9) t J pow σ > e M σ t (cid:0) σ β (cid:1) σ t σ +1 (cid:0) v σD β (cid:1) σ − t J pow σ ≤ e M σ t (cid:0) σ β (cid:1) σ t σ +1 Table 1
Summary of the results for the NCOP class, for different models (see left column). Time ideally runsfrom left to right. For each model we give on the top line the crossover scales L MF , L a , L b , L c where the variousregimes start/end, and the corresponding regimes occuring between (or before or after) these lengths. In thelines below, the analytic expressions for the crossover lengths and the behavior of L ( t ) in the correspondingregimes is reported (for the intermediate regime, see the question marks, L ( t ) is not analytically known). Thequantities to the right of the double vertical line are those predicted by the approximation with few domains.Some regimes, and the corresponding crossover lengths, do not exist for certain models. For example, for thepower model with σ ≤ L c is not defined, the diffusive stage does not exists and the slow regimeis the asymptotic one. J exp ( r ) with varying R . In this figure we report results ofsimulations obtained with both methods of Sec. 2.2. Clearly, with the method S1 a much smaller timerange can be investigated. When using method S2, bulk flips are forbidden, and for any R we searchedfor a value of n K representing a good compromise between the accuracy of the simulations and theirefficiency (for the largest values of R , n K is of order 2 · ). For small values of R the two kind ofcalculations agree rather well at any time. This can be clearly seen in the case with R = 5. The twomethods also agree, for any R , at sufficiently long times, as expected. Besides that, for any R theyalso agree in a very early regime where a fast exponential growth of L ( t ) takes place. This regime isnot captured by our previous analytical arguments because, recalling the discussion below Eq. (19),it occurs when L ( t ) is still too small, L ( t ) < L a , for the model with a single domain to be adequate.Such regime, which is of a mean field character, will be worked out analytically in Sec. 4.2.1. On theother hand, for large R , simulations S1 and S2 disagree in an intermediate time interval after the meanfield regime. For instance, for R = 10 the curves obtained with methods S1 and S2 are quite differentin the time range between t ≃
10 and t ≃ . We remind that this discrepancy is partly due to thelimited number n K of kinks considered but, especially, to the fact that flips in the bulk cannot beneglected in this time domain.Regarding the behavior of L ( t ), the first observation is the impressive difference, particularly forlarge R , with respect to the behavior of the nn model. Looking at the curves obtained with thesimulation method S2 the three regimes discussed above, ballistic ( L ∼ t ), slow ( L ∼ ln t ), and diffusive( L ∼ t / ) are clearly visible, and the crossovers between them occur at values of L ( t ), L exp a , L exp b , L exp c ,which are consistent with their estimations given in Table 1 or equivalently in Eqs. (16,20,28). Theseestimations say that, with increasing R the ballistic regime ends later, the slow kinetics lasts longer,and the diffusive regime starts later. In the exact simulations performed with method S1 the ballisticregime is only barely observed for R = 10 , since the range of times where it shows up is small. Itshould be more clearly observed for larger R but the numerical effort to go to (reasonably) largervalues of R turns out to be much beyond the scope of this paper.It is interesting to display the analytical curves for L ( t ), obtained solving numerically the twocoupled equations (14) and (15), see Fig. 4. Here the three different regimes are very clearly observedand the figure has the merit of stressing that the behavior of L ( t ) in the ballistic and the diffusiveregimes does not depend on R (while, clearly, the duration of such regimes, namely the quantities L exp a , L exp b , L exp c , do depend on R ). This is true but less evident for simulations as well, see Fig. 3. Tobetter analyze this point one can have a look to the lower inset of Fig. 3 where the same data ofthe main panel are plotted on rescaled axes. Although the flat plateau occurring when the fast initialgrowth of L ( t ) is over is a spurious effect due to use of method S2 for simulating the system (indeed, asit can be seen in the upper panel this plateau is washed out in the exact simulations) it is instructiveto study its location. Its height lies around L ( t ) /R ≃
10, and after that the ballistic regime starts. Thisshows that, as already discussed, the convective regime starts at L ( t ) = L a ∼ M R , with M ≃
10. Sincethe ballistic regime is independent of R this implies that it starts at a time of order R , and indeed inthe figure it is seen that it begins at t/R ≃ t L ( t ) t / n n R = R = R = R = R = R=10 M F t × × L ( t ) l n ( t ) -2 t/R -2 L ( t ) / R t Fig. 3 (NCOP exp)
In the main panel the domain size L ( t ) for a quench from T i = ∞ to T = 10 − , for asystem of size N = 10 , is shown on a log-log plot. Continuous lines with symbols are obtained by means ofsimulations using method S2. Different curves correspond to the nn interaction and to the interaction J exp ( r )with different values of R , as indicated. For some values of R ( R = 5 , , ) we plot also the curve obtainedwith the simulation method S1, see dotted lines. In the simulation method S1 we set M = 100 and we haveverified that no significant change is observed for larger M values. The dashed green and violet lines are thealgebraic forms t / and t , respectively. The curve drawn with turquoise + symbols at short times representsthe exponential growth in the mean-field regime, obtained analytically in Eq. (35). In the upper inset the datafor R = 10 are plotted on a log-linear scale to appreciate the logarithmic behavior. The lower inset shows thesame data of the main panel (only simulations with method S2 for R = 10 , , ), but plotting the rescaledquantity L ( t ) /R against t/R .3 t L ( t ) R = 10R = 20R = 40R = 80R = 16010 t L ( t ) t L ( t ) Fig. 4 (NCOP exp)
Plot of L ( t ), obtained from reversing the function t ( L ) as given by Eqs. (14) and (15),on a double logarithmic scale. Upper left inset: we plot the same data in the short time regime with linearscales to show the ballistic regime. Lower right inset: the same data are plotted on a log-linear scale to showthe logarithmic regime. We want now to discuss the early regime, characterized by a fast (exponential) growth of order. Sincethis occurs for values of L ( t ) smaller than L a , the approximation with few domains fails. Luckily,in the limit L ≪ R a mean field approximation holds. We are therefore going to discuss such anapproximation.If any spin interacts with other spins within a distance R ≫ m increasesin time following the equation [6] dm/dt = [ − m + tanh(2 βRm )]. The initial value of m is the result ofthe imbalance between positive and negative spins on a scale of order R . Because of the central limittheorem, m (0) ≃ / √ R (we assume m (0) >
0) so that the minimal value of the argument of tanhis of order 2 β √ R ≫
1. Therefore, the approximation tanh(2 βRm ) ≃ dm/dt = (1 − m ), whose solution is m ( t ) = 1 − [1 − m (0)] exp( − t ).Because of the random character of the initial configuration, and due to the fact that the meanfield dynamics does not introduce any correlation among spins, the configuration corresponding toa magnetization density m can be obtained by choosing the i − th spin as s i = +1 with probability p + = (1 + m ) / s i = − p − = (1 − m ) /
2. In doing that, the probability ofhaving ℓ + consecutive positively aligned spins is P ( ℓ + ) = p − p ℓ + − and the average length of suchdomain is ¯ ℓ + = 1 /p − . Analogously, for negative spins one finds ¯ ℓ − = 1 /p + . Therefore the averagedomain size is L = (¯ ℓ + + ¯ ℓ − ) = 1 / (2 p − p + ). Using the explicit expression for p ± in terms of themagnetization, we obtain L ( t ) = 21 − m ( t ) = 21 + m ( t ) · e t − m (0) . (35) Since the quantity 1 + m ( t ) is very weakly dependent on t , it varies from (1 + m (0)) to 2, this equationclearly shows the exponential growth of L ( t ) in the mean field regime, L ( t ) ≃ e t . In Fig. 3 we cancheck that simulation results well reproduce the prediction of Eq. (35) (compare the dotted magentaline with the + turquoise symbols) at short times.The exponential growth ends when L ( t ) is of order R , which happens at a time of order ln R .Recalling that the ballistic regime starts at t/R ≃ L ( t ) = L a = M R , with M ≃
10, there is atime lag of order R − ln R ≃ R in which L ( t ) must fill the gap from the end of the mean-field regime with L ≃ R to the beginning of the ballistic one with L ≃ M R . In this intermediate regime both the mean-field approximation and the one with few domains fail, because one has neither many boundaries nor asingle boundary within the interaction distance. Also, this is a regime where simulations of kind S2 fail(basically for the same reason). We could not devise any scheme to derive quantitative informationsin this intermediate time range.4.3 Simulations: interactions decaying algebraicallyBefore starting our discussion of the case with algebraic interactions let us comment on the fact that,at variance with the exponential case, now the use of method S2 for the simulations can alway be madereliable, at any time, by tuning n K appropriately. Indeed, as shown in Fig. 5, the initial mean-fieldregime is never present in this case and the approximated method S2 always correspond to neglectterms of the order of ( n k L ( t )) − − σ which become sub-leading for a sufficienly large n k . In particular,we have compared S2 simulations with the exact ones for several choices of the parameters σ and T and we found excellent agreement in the whole time domain using values at most equal to n K = 200.In the following, therefore, we will present always data obtained with method S2.Let us remind that for J ( r ) = J pow ( r ) our theory predicts a ballistic regime ( L ∼ t ), then a slowregime ( L ∼ t / (1+ σ ) ) which is asymptotic for σ ≤ L ∼ t / )if σ >
1. In Fig. 5 we plot the NCOP results for different values of σ and for the nn case. Startingfrom the smallest value of σ , namely σ = 0 .
5, one observes that, after a short transient for t . L ( t ) . L pow a ≃ , the ballistic regime is entered which extends up to the longestsimulated time t max . For this choice of the parameters, indeed, the crossover to the slow regime isdelayed after t max , as we will prove in a while. The curves for increasing values of σ superimpose onthe σ = 0 . L pow b of L which gets smaller the larger σ is, according toEq. (16). For σ = 3, L pow b is so small that the ballistic regime is not even observed. The late stagediffusive regime that is expected for σ > σ = 1 . L ( t ) = L pow c but this quantity, according to Eq. (28), diverges for σ → σ = 3 and is incepientfor σ = 2 at late times. Notice also in this case, as for the exponential coupling, the profound differencewith respect to the nn case.In order to show the transition to the slow regime also for the case with σ = 0 . L ( t ) for quenches to different temperatures. Indeed, the slow regime starts at the length scale L powb of Eq. (20) which, although typically very large when σ <
1, can be reduced by increasing thetemperature. This figure shows the crossover to the slow regime as T is raised, as expected. t L ( t ) σ=0.5σ=1σ=1.5σ=2σ=3 nn t t Fig. 5 (NCOP pow) L ( t ) for NCOP quenched from T i = ∞ to T = 10 − on a double-logarithmic scale.Different symbols and colors correspond to different values of σ and to the nn case(see legend). The dashedorange line is the t / law and the dashed green one is the ballistic behavior. The color dotted lines (below thedata curves) are the power-laws t / ( σ +1) of the slow regime for each σ value.6 t L ( t ) t t T=0.0001T=0.01T=0.1T=1
Fig. 6 (NCOP pow) L ( t ) for NCOP quenched from T i = ∞ to different final T for σ = 0 .
5. The green dashedline indicate the linear, ballistic regime L ( t ) ∼ t . The magenta dashed line is the growth L ∼ t / ( σ +1) = t / in the slow regime. The system size is N = 8 × . X ( t ) is an integer positive random variable whichcan increase or decrease; (ii) each emitted particle performs a random walk between two clusters and X ( t ) actually increases or decreases only if an emitted particle is absorbed by a different cluster.The minimal model involves two clusters of particles in a ring geometry and, for simplicity, we aregoing to consider two clusters of initial length L , separated by a distance L . In spin language the initialconfiguration is composed by four domains of equal length. In this way the process is (at least initially)perfectly symmetric and we can limit to determine the probability p ( L ) that a particle emitted bycluster 1 is absorbed by cluster 2. Once we know p ( L ) the resulting closing time of one of the twodomains is simply t ( L ) = t L /p ( L ) , (36)where t = e βJ (1) is the characteristic time of particle emission and t /p ( L ) is the typical time becausea cluster varies its size by one.In the nn model a particle between two neighbouring clusters diffuses freely and it is straightforwardto derive that p ( L ) = 1 /L , as argued below [20]. For symmetry reasons p ( L ) = p ( L/
2) because oncethe particle has attained an equal distance to both clusters the probability to attach to the right clusteris equal to the probability to attach to the left cluster. Therefore, p ( L ) = aL and since p (2) = , weobtain a = 1. If the range of J ( r ) is not limited to nn, the random walk of the particle emitted by acluster is not symmetric. Indeed, when a monomer detaches from a cluster it feels the attraction fromthe same cluster, a fact that strongly reduces the probability p ( L ) with respect to the nn case. As wewill see, this produces an overall slow down of the kinetics upon increasing the interaction range. We are now going to determine the drift δ ( x ) felt by a particle at distance x from the closestdomain, see Fig. 2(b). The energy E ( x ) associated to the particle can be evinced from Eq. (6) bysingling out in all the sums the term with i = i x , i x being the site where the monomer is. Neglectingirrelevant additive constants which are independent of L one has E ( x ) = − x + L X r = x J ( r ) + L − x X r = L − x J ( r ) ! , (37)which is a function defined in the interval (0 , L ) and symmetric with respect to the midpoint, x = L/ x < L/ x > L/
2. For symmetry reasons it is always true that p ( L ) = ˜ p ( L/
2) where ˜ p ( L/
2) is the probabilityto attain the midpoint x = L/ δ ( x ) for x < L/
2. In this interval we can forget the effect of the farthest domain and we canassume a diverging length of the nearest domain, so that for x < L/ E ( x ) asfollows, E ( x ) ≃ − ∞ X r = x J ( r ) = − I ( x ) . (38)The probability p + ( x ) to hop from x to x + 1 is p + ( x ) = 11 + e β ( E ( x +1) − E ( x )) = 11 + e βJ ( x ) (39)while the probability to hop from x to x − p − ( x ) = 11 + e − βJ ( x ) , (40)so that δ ( x ) = p + ( x ) − p − ( x ) = − tanh(2 βJ ( x )) . (41)We now must determine ˜ p ( L/
2) from the knowledge of δ ( x ). Let us consider the following first-passage problem: a particle diffuses anisotropically on the integer sites of the interval [0 , N ] and wewonder what is the probability W N ( x ) that a particle currently at x reaches the site N before reachingthe site 0. We can write W N ( x ) = p + ( x ) W N ( x + 1) + p − ( x ) W N ( x − . (42)In a continuum approximation, writing W N ( x ± ≃ W N ( x ) ± W ′ N ( x ) + (1 / W ′′ N ( x ), using therelation p + + p − = 1 and Eq. (41), the above equation reads W ′′ N ( x ) = − δ ( x ) W ′ N ( x ), which shouldbe supplemented with the boundary conditions W N (0) = 0 and W N ( N ) = 1. The equation can beintegrated twice, giving W N ( x ) = R x dy e − R y dsδ ( s ) R N dy e − R y dsδ ( s ) . (43)Observing that ˜ p ( L/
2) = W N ( x ) for x = 1 (the site where a detached particle starts to diffuse) and N = L/
2, we can write ˜ p ( L/
2) = R dy e R y ds | δ ( s ) | R L/ dy e R y ds | δ ( s ) | ≡ AB ( L ) , (44)with | δ ( s ) | = tanh(2 βJ ( s )). With the knowledge of A and B , we have the following relation to determinethe coarsening law, t = 2 t A L B ( L ) . (45)Let us now evaluate the quantity B ( L ). Starting with the expression for | δ ( s ) | , the argument ofthe hyperbolic tangent, 2 βJ ( s ), varies from 2 βJ ( s ) ≫ s to 0 for diverging s , so there is acrossover between two regimes. If s ∗ is defined by the relation 2 βJ ( s ∗ ) = 1, we have | δ ( s ) | ≃ (cid:26) s ≪ s ∗ βJ ( s ) s ≫ s ∗ , (46) with a sharp transition between the two regimes. We can therefore approximate the integral appearingin the exponent as follows, C ( y ) ≡ Z y ds | δ ( s ) | ≃ (cid:26) y, y < s ∗ s ∗ + 2 β [ I ( s ∗ ) − I ( y )] , y > s ∗ . (47)We finally obtain B ( L ) = Z L/ dy e C ( y ) ≃ R L/ dy e y = ( e L − ≃ e L , L < s ∗ e s ∗ + e s ∗ R L/ s ∗ dy e β [ I ( s ∗ ) − I ( y )] , L > s ∗ . (48)As for the numerator, A = Z dy e C ( y ) ≃ Z dy e y = 12 ( e − ≡ c ≃ . . (49)We can now find the limiting behaviors for small L and diverging L . For L < s ∗ = L s we obtain t = t c L e L , L ≪ L s (50)which gives a logarithmically slow coarsening, L ( t ) ≃ ln t .For the opposite case, L > L s , we can observe that ( I ( s ∗ ) − I ( y )) is an increasing function from y = 0 (for s = s ∗ ) to the positive, constant value I ( s ∗ ) for y → ∞ . So, we expect the leading term fordiverging L to be B ( L ) ≃ e L s e βI ( L s / L , (51)and t ≃ t c e L s e βI ( L s / L , L → ∞ (52)thus obtaining L ≈ t / asymptotically. Notice that, quite interestingly, the logarithmic and the diffu-sive regime do not make explicit reference to the form of the interaction.However, we must observe that the transition of ( I ( s ∗ ) − I ( y )) from zero to I ( s ∗ ) depends on theexplicit form of the coupling: in the exponential case, J exp ( r ) rapidly decays to zero and such transitionis sharp; in the power law case, J pow ( r ) does not decay rapidly, the transition is not sharp and a thirdintermediate regime exists, as we are going to argue.For J ( r ) = 1 /r σ , I ( y ) = 1 / ( σy σ ) and for L > L s it is B ( L ) = e s ∗ e s ∗ e βσ s ∗ ) σ Z L/ s ∗ dye − βσ yσ (53)= e s ∗ e s ∗ e βσ s ∗ ) σ σ (cid:18) βσ (cid:19) /σ (cid:20) Γ (cid:18) − σ , βσ (cid:18) L (cid:19) σ (cid:19) − Γ (cid:18) − σ , βσ (cid:18) s ∗ (cid:19) σ (cid:19)(cid:21) , (54)where Γ ( α, x ) is the upper incomplete Gamma function.We define the length L cr such that the argument x of the Gamma function on the left betweensquare brackets, Eq. (54), is equal to one. We find L cr = 2 /σ ) ( β/σ ) /σ , (55)and L cr /L s = (2+ σ ) / [ σ (1+ σ )] σ /σ β / ( σ (1+ σ )) ≫ L ≫ L cr the incomplete Gamma function on the right between square brackets, Eq. (54), can beneglected with respect to the left Γ and using the asymptotic expansion for x ≪ L s ≪ L ≪ L cr we can use the expansion valid for x ≫ Γ ( − /σ, x ) ≃ x − − (1 /σ ) e − x , and weobtain t ≃ t c σ +2 e L s β L σ +3 exp (cid:20) β σ +2 σ (cid:18) L s ) σ − L σ (cid:19)(cid:21) ( L s ≪ L ≪ L cr ) . (56) L s intermediate L cr diffusive J exp ln t R ln(2 β ) t J pow σ > t β ) σ Eq. (56) 2 /σ ) ( β/σ ) /σ t J pow σ → + ln t β ) σ t β +3 Table 2
Summary of the results for the COP class. Time ideally runs from left to right. On the top line thescales L s , L cr where the various regimes start/end, and the corresponding regimes occuring between (or beforeor after) these lengths are indicated. In the lines below, the analytic expressions for the crossover lengths andthe behavior of L ( t ) in the corresponding regimes is reported, for the different forms of J ( r ). Some regimes, andthe corresponding crossover lengths, do not exist for certain models. For example, for the exponential modelthere is no intermediate regime because L s and L cr are not distinct. In the limiting case σ → L s = 4 β and L cr → ∞ , so that the intermediate regime is theasymptotic regime. In the same limit we find t ≃ t c (cid:18) e β (cid:19) β L β +3 β ( σ → + ) . (57)A summary of the regimes and of the crossover lengths separating them is given in Table 2.5.2 Simulations: interactions decaying exponentiallyFor an exponential coupling, J exp ( r ), our theory predicts a logarithmic coarsening, for L ( t ) < L s = 2 s ∗ ,followed by the asymptotic, power-law regime, L ∼ t / . Using the definition of s ∗ given above Eq. (46)we have L exp s = 2 R ln(2 β ) . (58)These two regimes and the sharp crossover between them are very neatly observed by plotting thetheoretical formulas in Fig. 7. Notice also that the crossover length scales as R , as expected accordingto Eq. (58). t L ( t ) R=1/2R=2/3R=1R=2R=5R=1010 t L ( t ) t Fig. 7 (COP exp) L ( t ) for COP quenched to T = 0 . J ( r ) = J exp ( r ) and various values of R , see key. The dashed violet lineis the asymptotic diffusive behavior L ( t ) ∼ t / . In the main figure the data are plotted on a double logarithmicscale, and the heavy circles correspond to the crossover lengths of Eq. (58) (for R = 10 it is beyond the largesttime in the plot). The inset shows the same data but using a logarithmic-linear scale in order to show theinitial logarithmic regime. Let us now discuss the results of our simulations performed according to the method S2 discussedin Sec. 2.2. We have checked that with COP this kind of simulations provide reliable results basically atany time, provided that n K is chosen appropriately (mostly, we used n K = 2 · ). This is true for bothkinds of interactions J ( r ) considered. Despite the speed-up provided by simulations with a reducednumber of kinks, calculations with COP are quite time demanding and, therefore, it is difficult to push R to rather large values. Nevertheless, even if data display the transition between the two regimes lessclearly than the solution of the model with few domains, there is still a rather clear evidence of a short-time logarithmic coarsening and of the asymptotic t / power law following it, see Fig. 8. Also, thecrossover between them is delayed by increasing R , as expected. Notice that in our analytical approachthe logarithmic regime turns out to be independent of R . Indeed curves for different R superimpose inFig. 7. This is not observed in the numerical simulations of Fig. 8. This could be possibly due to anoffset caused by a very early regime which is not captured by our analytical techniques, or/and to theimpossibility to reach sufficiently large values of R in simulations. t L ( t ) n.n.R=1/2R=2/3R=1R=2R=310 t L ( t ) t Fig. 8 (COP exp) L ( t ) for a quench from T i = ∞ to T = 0 . N = 10 and differentvalues of R , as detailed in the key. The plot in the main figure is on a double-logarithmic scale. The dashedgreen line is the the asymptotic diffusive behavior L ( t ) ∼ t / . The horizontal dotted line corresponds to L nn ( T = 0) ≃ . T = 0. In the inset the same data (onlyfor R = 2) are plotted with log-linear scales so as to show the initial, logarithmically slow coarsening. t / , regime, but inaddition we should observe an intermediate regime, as given by Eq. (56), for L s < L ( t ) < L cr , with(see Table 2): L pow s = 2(2 β ) σ and L cr = 2 /σ ) ( β/σ ) /σ . (59)These analytical predictions are corroborated by the explicit numerical solution of Eq. (45), see Fig. 9.First of all the asymptotic power law expected for σ → + is very well observed. For the largest valuesof σ we have considered, namely σ = 1 and σ = 2, after a relatively short preasymptotic regime, L ( t ) attains the asymptotic diffusive behavior L ( t ) ∼ t / . As σ is lowered the preasymptotic stageincreases, because both L s and L cr increase, and the asymptotic stage is pushed beyond the scaleof times presented in the figure. In the rightmost inset of Fig. 9 we show the early time behavior ofthe preasymptotic stage with linear-log scales, in order to detect the logarithmic law. Although thegrowth is definitely slower than an algebraic one, the resulting plot is not fully linear, not even forthe smaller values of σ , because there is a tiny upward curvature. This is perhaps due to the fact thatthe crossover to the next stages is broad. In addition, the leftmost inset shows that, after this slowregime, an intermediate regime where Eq. (56) holds is observed. Actually, according to this equation,by defining Λ − = β σ +2 σ L − σ , this quantity should behave as Λ − = const − τ , where τ = ln (cid:0) tL σ +3 (cid:1) .Indeed, a linear relation between Λ − and τ is observed in the inset of Fig. 9 after a certain τ . t L ( t ) σ =0 σ =0.05 σ =0.1 σ =0.2 σ =0.5 σ =1 σ =2 10 t L ( t ) t t β +3) τ Λ - ( τ ) Fig. 9 (COP pow) L ( t ) for a quench to T = 0 . J ( r ) = J pow ( r ) and with different values of σ , see key. In the main part of the figure, data are plotted ona log-log scale. The green dashed line is the behavior t β +3 expected for σ → + . The violet dashed line isthe diffusive behavior t / . In the rightmost inset the same data are plotted on a log-linear scale, to showthe logarithmic regime. In the leftmost inset the quantity Λ − = β σ +2 σ L − σ is plotted against τ = ln (cid:0) tL σ +3 (cid:1) .According to Eq. (56) one should have the intermediate regime Λ = cost − τ , which is indeed well observed. When we pass to simulations, the power law COP model is more elusive. Firstly, it is not possibleto clearly show both logarithmic and t / regimes for the same parameters, because simulations cannotaccess the whole range of time that would be needed. Indeed, if T is sufficiently high, given the form of L s and L cr , Eq. (59), one is able to enter the asymptotic diffusive regime but the preasymptotic onesare too compressed to be observable. On the contrary, lowering T one is able to see the preasymptoticregime (at least the slow logarithmic one, see below), particularly for small σ , but the asymptotic oneis so delayed to be unreachable. For this reason we present in the following two figures where we change σ and T separately in order to observe the early as well as the late regimes. Specifically, in Fig. 10we show a quench to a temperature T = 0 . L ( t ) ∼ t / is observed at late times for σ = 2 and σ = 3. For smaller values of σ thecrossover to the asymptotic stage is at most incipient (for σ = 1). Due to this incipient crossover, it isdifficult to identify a well defined preasymptotic regime. Then, in order to show its presence we showin Fig. 11, for a favorable case with small σ , i.e. σ = 0 .
1, how it emerges by lowering the temperature.What we see is that the curves for very small T tend to become straight lines in this log-linear plot,signaling a logarithmic growth of L ( t ). Regarding the intermediate regime, this is too elusive to beclearly recognized. However, as shown in the inset of Fig. 11, by plotting the quantity Λ − (actuallywe plot ( βΛ ) − to better compare curves at different T ) against τ , as already done in Fig. 9, one seesthat at large times (i.e. large τ ) the curves tend to have a linear behavior, which would signal thesetting in of the intermediate regime, Eq. (56). t L ( t ) σ=0.5σ=1σ=2σ=3 t Fig. 10 (COP pow) L ( t ) for a quench from T i = ∞ to T = 0 .
4, for N = 10 and different values of σ , seekey. The dashed green line is the t / law. t L ( t ) β =6 β =7 β =8 β =9 β =100 5 10 15 20 25 30 τ β - Λ - ( τ ) * * **** Fig. 11 (COP pow) L ( t ) for a quench from T i = ∞ to different values of β = 1 /T (see key) for σ = 0 . N = 10 . The plot is on a log-linear scale, hence a logarithmic regime is more and more visible with increasing β . In the inset the quantity Λ − is plotted against τ (see discussion around Fig. 9). Missing points in the regionof relatively short times is an effect due to the large time jumps introduced by the rejection free simulationscheme (see Appendix A).4 Although this is not the main focus of this paper, in this section we briefly discuss the growth laws foundin deterministic continuous models for growth kinetics. This will allow us to compare the behaviourof these models with the ones of the Ising system analysed insofar and to discuss to which extent thetwo approaches can be considered equivalent.For nn interactions with NCOP the kinetics can be described by means of the Time-DependentGinzburg-Landau equation, ∂ t φ ( x, t ) = 2 ∂ xx φ − φ + 4 φ. (60)This equation has time independent, single-kink solutions φ ( x ) = ± tanh( x ). In the presence of anexponential coupling J exp ( r ) = e − r/R , one can explicitly take into account the scale R of the interactionsobtaining φ ( x ) = ± tanh( x/R ). Combining two of such solutions to reproduce the single domainconfiguration plotted in Fig. 2(a) one has φ ( x, t ) = tanh (cid:18) x − X/ R (cid:19) − tanh (cid:18) x + X/ R (cid:19) + 1 . (61)The resulting time evolution of X ( t ) is given by [21] ˙ X ( t ) = − [ V ( φ ( x = 0)) − V ( φ ( x = ∞ ))], where V ( φ ) = ( φ − is the standard double well potential. Using Eq. (61) we obtain˙ X ( t ) = − [ V (1 − X/ R )) − V (1)] ≡ − v ( X/R ) , (62)where the drift v ( X/R ) is a positive function vanishing for small and large argument and with amaximum for
X/R = tanh − (1 / X (0) = L ≫ R , because of the exponential tails (withrespect to the asymptotic values ± v ( X/R ) ≃ e − X/R , which explains the logarithmic coarseningof the slow regime [22]. The opposite limit X (0) = L ≪ R (for which v ( X/R ) ≈ ( X/R ) ) is notphysically relevant because the two DWs model is not applicable. However, in the intermediate regime X ( t ) ≈ R the function v ( X/R ) is approximately constant because of the maximum, and such constantdrift originates a ballistic behavior. This shows that both the slow and the ballistic regimes observedin the Ising model have a counterpart in the continuum theory.For the algebraic coupling J pow ( r ), a continuum model has been extensively studied by Alan Brayand Andrew D. Rutenberg [2,3], finding L ( t ) ∼ t / (1+ σ ) for NCOP , which is the slow regime observedalso in the discrete Ising model. This regime has the same origin of the slow regime for the exponentialcoupling. In particular, the growth law in these regimes is a direct manifestation of the analytic formof the interaction: it is logarithmic for an exponential J ( r ) and power-law for an algebraic J ( r ). Inconclusion, for NCOP there is a rather general correspondence between the behavior of the discreteIsing model and continuum approaches.For COP there are not many available results for deterministic continuum theories, the only avail-able result concerning the short-range model [23], where a logarithmic coarsening is inferred. Despitethis, we argue on general grounds that the good correspondence between noiseless continuum ap-proaches and the discrete model found for NCOP does not apply to the conserved case. We say thisbecause any regime in the Ising model with COP is intrinsically stochastic and therefore cannot becaptured by a deterministic continuum theory. For this reason, even if with COP a logarithmic growthis found both in the continuum approach [23] and in the Ising model, these laws do not have a sim-ilar origin, and the corresponding regimes are physically different. In fact, the logarithmic stage inthe continuum model is due (similarly to what discussed above for NCOP) to an exponentially smallinteraction between DWs. Instead in the Ising model it stems from the exponential vanishing of theprobability that a particle detached from a domain reaches another cluster (because of the backwarddrift).This conclusion is corroborated by the fact that the short-time logarithmic regime in the Ising COPis completely general and independent of the details of J ( r ). On the contrary, the interaction between In this respect it is worth stressing that such slow regime is asymptotic for σ <
1, while it is replacedby the diffusive one ( L ( t ) ∼ t / ) for large t if σ ≥
1. This is simply due to the relevance of temperature. In d = 1, indeed, the model in equilibrium has a finite T c for σ < T c for σ ≥
1. Therefore, ina renormalization group language, T is an irrelevant parameter in the former case and a relevant one in thelatter. If it is irrelevant, the (continuum) result found at T = 0 is valid also switching on noise (temperature); ifit is relevant, the effect of T should be visible at large enough length scales (as it is for the short range model).5 DWs in the continuum theory depends on the form of the interaction, so that for an algebraicallydecaying coupling one does not expect such logarithmic coarsening.
Despite that phase ordering is an old problem, a thorough investigation of the kinetics of the Isingmodel with space decaying interactions was not pursued previously. In this paper we have consideredthis model in one dimension with a coupling J ( r ) between two spins at distance r which is a general,positive and decreasing function J ( r ) > , ∀ r .In the asymptotic regime, t → ∞ , coarsening dynamics is the same as for the short range model if J ( r ) decays faster than 1 /r , the condition to have a vanishing Curie temperature for the equilibriummodel. Hence it is L ( t ) ∼ t / for NCOP and L ( t ) ∼ t / for COP. However, our investigation ofthe dynamics at any time shows that the model with space decaying interactions displays a rich andunexpected variety of different regimes, with the asymptotic one being sometimes so delayed to behardly observable.Some features of coarsening dynamics are worth of note. The first one is that extending the rangeof the interactions produces an acceleration of the NCOP dynamics and a slowdown of the COP one.Surprisingly, these opposite effects originate from the same phenomenon: the diffusion of a DW in theNCOP models and the diffusion of a particle in the COP models are no more symmetric, because a driftappears. In the nonconserved case this drift tends to move a DW towards its closest neighbour, thereforefavoring the closure of domains and speeding up the dynamics. In the conserved case, instead, the driftis applied to a detached monomer and reduces its possibility to attain the other clusters, thereforehampering mass exchange between clusters and impeding the kinetics.A second feature is related to the comparison between the dynamics of the discrete Ising model andthe dynamics of continuum models, which has been discussed in the previous Section. We limit here tostress that we have noticed the existence of a ballistic regime (constant drift) in the continuum modelas well. Such regime appears when we pass from the nn to the exponential model, with a sufficientlylarge R . We have also pointed out that a deterministic continuum approach fails in reproducing thekinetics of the Ising model with COP in 1d, arguing that a stochastic model is needed to reproduceits behavior in any dynamical regime, from the early stage to the asymptotic one.In this manuscript we have provided a detailed study of the time behaviour of L ( t ), but thisquantity does not cover all features of coarsening dynamics, which is also characterized by correlationfunctions and by the full size distribution of domains. The general time dependent spin-correlationfunction h s i ( t ) s j ( t w ) i is usually studied at the same site, C ( t, t w ) = h s i ( t ) s i ( t w ) i , or at the same time, G ( r, t ) = h s i ( t ) s i + r ( t ) i . The autocorrelation function, whose scaling form is C ( t, t w ) = f ( L ( t ) /L ( t w )),has recently been studied for nonconserved dynamics by the same authors [24], discovering a new uni-versality class appearing in the power model when σ ≤ z the function f ( x ) and in particular the Fisher-Huse exponent λ , f ( x ) ≈ x − λ for x ≫
1, does notdepend on σ when σ ≤ G ( r, t ) = G ( r/L ( t ) and whose behavior is clearly related to the dis-tribution of domain lengths, n d ( ℓ, t ). It would be of interest to study both G and n d and finding outpossible connections between the two, within different models.Beyond the case of a quench from a disordered state addressed in this paper, many other topicsremains unexplored, as for instance the kinetics following a quench from a critical state [25,26], whichis present in the one-dimensional model with algebraic interactions when 0 < σ ≤ T c is finite.In addition, besides the determination of the growth law L ( t ), several other features of the Ising modelwith space decaying interactions are worth of further investigations. Let us mention here the agingproperties, i.e. the behavior of two-time quantities such as correlation and response functions, whoseunderstanding could provide useful hints for a general interpretation of aging systems [ ? ]. Furthermore,our studies can be extended to higher dimension d >
1, some results for d = 2 being contained in [6,27],and to σ <
0, where additivity is lost [8]. Another interesting point to be investigated is the robustnessof our results with respect to the presence of quenched disorder, which is often unavoidable in realsystems. It is well known, in fact, that even a tiny amount of such randomness may change radicallythe kinetics of coarsening systems [28], both in one dimension [29] and higher dimensions [30,31,32].The situation becomes even more complex if disorder introduces frustration [33,34]. In particular, in the case σ ≤
0, the one-dimensional Edwards-Anderson model with algebraically decaying couplingconstants, shows different behaviors depending on the exponent σ [35,36,37,38]. It would be thereforeinteresting to explore if and to which extent the formalism developed in this paper may provide somehints for the understanding disordered systems with or without frustration.A final comment concerns a recent preprint [7] whose focus is the study of the NCOP coarseningdynamics for a lattice model with a continuum local variable q i and falling in the equilibrium univer-sality class of the Ising model with power-law interactions. The system is a chain of oscillators withthe standard single site, double-well potential and an interaction potentials decaying as 1 /r σij , with0 ≤ σ ≤
1. Authors want to analyze if there is equivalence between canonical and microcanonicalensemble or not. The answer is negative: in the former case they obtain z = 1 + σ , in agreement withour results and with those in [2,3]. Instead, in the latter case they asymptotically find z = 2, showingan out-of-equilibrium ensemble inequivalence for σ -values where additivity (and therefore equilibriumadditivity) holds. A Algorithm S2 with a reduced number of interacting kinks
We indicate with n the total number of interfaces present in the system at a generic time t and with x k theposition of the k -th interface (with k = [ − n/ , ..., , ..., n/ i ∈ [ x k + 1 , x k +1 ] and j ∈ [ x m + 1 , x m +1 ], then s i ( t ) s j ( t ) = 1 for k − m even whereas s i ( t ) s j ( t ) = − k − m odd. As a consequence the Hamiltonina can be written as H = − n/ X k = − n/ n/ X m = − n/ ( − k − m x k +1 X i = x k +1 x m +1 X j = x m +1 J ( | i − j | ) , (63)where | i − j | indicate the spatial distance between the sites ( i, j ). Because of periodic boundary conditions,such distance is actually given by the minimum between | i − j | and N − | i − j | , but we use the expression J ( | i − j | ) for any pair ( i, j ) to avoid overloading the notation. Furthermore, we assume J (0) = 0.We next consider the energetic cost due to the flip of the i -th spin ∆E = 2 s i ( t ) X j J ( | i − j | ) s j ( t ) (64)and separate the discussion for the NCOP and COP dynamics. A.1 Fast NCOP dynamics
In our fast simulation protocol, flips of spins in the bulk are forbidden. Therefore only spins at the interface( i = x k or i = x k + 1) can flip and the dynamics is mapped to the displacement of a DW as in Fig.1a (upperpanel) ( x k → x k ± x k = x k +1 −
1, the displacement of the k -th interface towards the right leads to theannihilation of the two interfaces ( n → n − x k = x k − + 1. Without lack of generality we consider the dynamics of the defect x and a displacementtowards the right x → x + 1 corresponds to the flip of the spin i = x + 1. According to Eqs. (63,64) we find ∆E = 2 n/ X k = − n/ ( − k x k +1 X j = x k +1 J ( | x + 1 − j | ) . (65)Similarly, a displacement towards the left, x → x −
1, corresponds to the flip of the spin i = x and since, bydefinition s x = − s x +1 we immendiately obtain ∆E = − n/ X k = − n/ ( − k x k +1 X j = x k +1 J ( | x − j | ) . (66)Defining Q ( | x k − i | ) = x k +1 X j = x k +1 J ( | i − j | ) (67)we finally obtain Eq.(10).In the approximation scheme S2, we randomly choose one of the ( n + 1) interfaces present at time t andaccept its move towards the left or the right with a probability given in Eq.(9). A Monte Carlo step correspondsto n + 1 trials. In particular, the approximation scheme with a finite number of interacting kinks correspondsto the substitution n → n K in Eq. (66), where n K is a tunable parameter to be optimised.7 A.2 Fast COP dynamics
We consider the Kawasaki dynamics which corresponds to the exchange of two nearest-neighbor oppositespins. With this dynamics the only possible moves are the three listed in Fig. 1b. More precisely, diffusion(upper panel) and annihilation (central panel) correspond to the simultaneous motion of two consecutivedefects x k + 1 = x k +1 towards the right ( x k +1 → x k +1 + 1, x k → x k + 1) or towards the left ( x k → x k − x k +1 → x k +1 − x k +2 = x k +1 + 1 leading to the annihilation of two defects. The same situationoccurs for a displacement towards the left when x k − = x k −
1. Finally the detachment process (lower panel)corresponds to the nucleation of two new DWs close to an existing one.The energy difference can be still obtained from Eqs. (65,66) after taking into account that all moves involvethe simultaneous flip of two consecutive spins i = x and i + 1 = x = x + 1. As a consequence each move hasan energy cost ∆E = 2 n/ X k = − n/ ( − k x k +1 X j = x k +1 ( J ( | x − x k | ) − J ( | x − x k | )) (68)which leads to ∆E = 2 J (1) − n/ X k =2 ( − k J ( | x − x k | ) + 2 n/ X k =1 ( − k J ( | x − x − k | ) , (69)corresponding to Eq.(10) with Q ( | x j − i | ) = J ( r = | x j − i | ).Also in the case of COP dynamics we have considered the approximation scheme with a finite number n K of interacting kinks. Simulations have been performed according to a rejection free algorithm [17]. Ata given time t , a generic defect k can perform only one of the three moves in Fig. 1b, with a probability W k = 1 / (1 + exp( β∆E )) (see Eq. (9)). At this time t , we select one of the defects (say the k -th) amongthe n existing, with a probability W k / P n/ j = − n/ W j . The corresponding move is always accepted and timeis incremented by 1 /W k . Notice that, at sufficiently low temperature, after a transient the system will reachthe configuration where only detachment of the kind of Fig. 1b (lower panel) are possible. Since in this case( ∆E ) nn > /W k ≫
1) which are, for instance, clearly visible inFig. 11 by increasing β . Compliance with Ethical Standards
Funding: F.C. acknowledges financial support by MIUR project PRIN2015K7KK8L.Conflict of Interest: The authors declare that they have no conflict of interest.Research involving Human Participants and/or Animals: It does not apply.Informed consent: It does not apply.
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