Short-time critical dynamics of the three-dimensional systems with long-range correlated disorder
V. Prudnikov, P. Prudnikov, B. Zheng, S. Dorofeev, V. Kolesnikov
aa r X i v : . [ c ond - m a t . d i s - nn ] S e p Short-time critical dynamics of the three-dimensional systemswith long-range correlated disorder
Vladimir V.
Prudnikov , ∗ , Pavel V. Prudnikov , Bo Zheng ,Sergei V. Dorofeev and Vyacheslav Yu. Kolesnikov , Dept. of Theoretical Physics, Omsk State University, Omsk 644077, Russia Physics Dept., Zhejiang University, Hangzhou 310027, PR of China
Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional Ising and XY models with long-range correlated disorder at criticality, inthe case corresponding to linear defects. The static and dynamic critical exponentsare determined for systems starting separately from ordered and disordered initialstates. The obtained values of the exponents are in a good agreement with results ofthe field-theoretic description of the critical behavior of these models in the two-loopapproximation and with our results of Monte Carlo simulations of three-dimensionalIsing model in equilibrium state.
I. INTRODUCTION
The investigation of critical behavior of disordered systems remains one of the mainproblems in condensed matter physics and excites a great interest, because all real solidscontain structural defects. The structural disorder breaks the translational symmetry ofthe crystal and thus greatly complicates the theoretical description of the material. Theinfluence of disorder is particularly important near critical point where behavior of a systemis characterized by anomalous large response on any even weak perturbation. In mostinvestigations consideration has been restricted to the case of point-like uncorrelated defects[1]. However, the non-idealities of structure cannot be modeled by simple uncorrelateddefects only. Solids often contain defects of a more complex structure: linear dislocations,planar grain boundaries, clusters of point-like defects, and so on.Different models of structural disorder have arisen as an attempt to describe such com-plicated defects. In this paper we concentrate on model of Weinrib and Halperin (WH) [2] ∗ E-mail: [email protected] with the so-called long-range correlated disorder when pair correlation function for point-likedefects g ( x − y ) falls off with distance as a power law g ( x − y ) ∼ | x − y | − a . Weinrib andHalperin showed that for a ≥ d long-range correlations are irrelevant and the usual short-range Harris criterion [3] 2 − dν = α > d is the spatial dimension, and ν and α are the correlation-length and thespecific-heat exponents of the pure system. For a < d the extended criterion 2 − aν > a < d a new long-range (LR) disorder stable fixed point (FP) of the renormalization group recur-sion relations for systems with a number of components of the order parameter m ≥ ε = 4 − d ≪ δ = 4 − a ≪
1. The correlation-length exponent wasevaluated in this linear approximation as ν = 2 /a and it was argued that this scaling relationis exact and also holds in higher order approximation. In the case m = 1 the accidentaldegeneracy of the recursion relations in the one-loop approximation did not permit to findLR disorder stable FP. Korzhenevskii et al [4] proved the existence of the LR disorder stableFP for the one-component WH model and also found characteristics of this type of criticalbehavior.Ballesteros and Parisi [5] have studied by Monte Carlo means the critical behavior inequilibrium of the 3D site-diluted Ising model with LR spatially correlated disorder, in the a = 2 case corresponding to linear defects. They have computed the critical exponents ofthese systems with the use of the finite-size scaling techniques and found that a ν value iscompatible with the analytical predictions ν = 2 /a .However, numerous investigations of pure and disordered systems performed with the useof the field-theoretic approach show that the predictions made in the one-loop approxima-tion, especially based on the ε - expansion, can differ strongly from the real critical behavior[6, 7, 8, 9]. Therefore, the results for WH model with LR correlated defects received basedon the ε, δ - expansion [2, 4, 10, 11, 12] was questioned in our paper [13], where a renor-malization analysis of scaling functions was carried out directly for the 3D systems in thetwo-loop approximation with the values of a in the range 2 ≤ a ≤
3, and the FPs corre-sponding to stability of various types of critical behavior were identified. The static anddynamic critical exponents in the two-loop approximation were calculated with the use ofthe Pade-Borel summation technique. The results obtained in [13] essentially differ from theresults evaluated by a double ε, δ - expansion. The comparison of calculated the exponent ν values and ratio 2 /a showed the violation of the relation ν = 2 /a , supposed in [2] as exact.The models with LR-correlated quenched defects have both theoretical interest due tothe possibility of predicting new types of critical behavior in disordered systems and exper-imental interest due to the possibility of realizing LR-correlated defects in the orientationalglasses [14], polymers [15], and disordered solids containing fractal-like defects [4] or dislo-cations near the sample surface [16].To shed light on the reason of discrepancy between the results of Monte Carlo simulationof the 3D Ising model with LR-correlated disorder [5], in the a = 2 case, and the results ofour renormalization group description of this model [13], we have computed by the short-time dynamics method [17, 18] the static and dynamic critical exponents for site-diluted 3DIsing and XY models with the linear defects of random orientation in a sample.In the following section, we introduce a site-diluted 3D Ising model with the linear defectsand scaling relations for the short-time critical dynamics. In Section III, we give results ofcritical temperature determination for 3D Ising model with the linear defects for case withspin concentration p = 0 .
8. We analyze the critical short-time dynamics in Ising systemsstarting separately from ordered and disordered initial states. Critical exponents obtainedunder these two conditions with the use of the corrections to scaling are compared. Also,in Section III the results of measurements of the critical characteristics for 3D Ising modelin equilibrium state are presented in comparison with results of the short-time dynamicsmethod. In Section IV the results of Monte Carlo studies of critical behavior of 3D XY-model with linear defects for the same spin concentration p = 0 . II. MODEL AND METHODS
We have considered the following Ising model Hamiltonian defined in a cubic lattice oflinear size L with periodic boundary conditions: H = − J X h i,j i p i p j S i S j , (1)where the sum is extended to the nearest neighbors, S i = ± p i are quenched random variables ( p i = 1, when the site i is occupied by spin, and p i = 0, when the site is empty), with LR spatial correlation. An actual p i set will be calleda sample from now on. We have studied the next way to introduce the correlation betweenthe p i variables for WH model with a = 2, corresponding to linear defects. We start with afilled cubic lattice and remove lines of spins until we get the fixed spin concentration p in thesample. We remove lines along the coordinate axes only to preserve the lattice symmetriesand equalize the probability of removal for all the lattice points. This model was referredin [5] as the model with non-Gaussian distribution noise and characterized by the isotropicimpurity-impurity pair correlation function decays for large r as g ( r ) ∼ /r . In contrast to[5] we put a condition of linear defects disjointness on their distribution in a sample, whereasin [5] the possibility of linear defects intersection is not discarded. The physical grounds forthis condition are connected with fact that in real materials dislocations as linear defectsare distributed uniformly in macroscopic sample with probability of their intersection closeto zero. The condition of linear defects disjointness corresponds to WH model since theintersection of linear defects being taken into consideration results in additional vertices ofinteraction which are absent in the effective Hamiltonian of WH model.In this paper we have investigated systems with the spin concentration p = 0 .
8. We haveconsidered the cubic lattices with linear sizes L from 16 to 128. The Metropolis algorithmhas been used in simulations. We consider only the dynamic evolution of systems describedby the model A in the classification of Hohenberg and Halperin [19]. The Metropolis MonteCarlo scheme of simulation with the dynamics of a single-spin flips reflects the dynamicsof model A and enables us to compare the obtained dynamical critical exponent z withthe results of our renormalization group description of critical dynamics of this model [13]having LR-disorder.A lot of results have been recently obtained concerning the critical dynamical behavior ofstatistical models [17, 18] in the macroscopic short-time regime. This kind of investigationwas motivated by analytical and numerical results contained in the papers of Janssen etal [20] and Huse [21]. Important is that extra critical exponents should be introduced todescribe the dependence of the scaling behavior for thermodynamic and correlation functionson the initial conditions. According to the argument of Janssen, Schaub and Schmittman[20] obtained with the renormalization group method, one may expect a generalized scalingrelation for the k -th moment of the magnetization M ( k ) ( t, τ, L, m ) = b − kβ/ν M ( k ) (cid:0) b − z t, b /ν τ, b − L, b x m (cid:1) , (2)is realized after a time scale t mic which is large enough in microscopic sense but still verysmall in macroscopic sense. In (2) β , ν are the well-known static critical exponents and z isthe dynamic exponent, while the new independent exponent x is the scaling dimension ofthe initial magnetization m , τ = ( T − T c ) /T c is the reduced temperature.Since the system is in the early stage of the evolution the correlation length is still smalland finite size problems are nearly absent. Therefore we generally consider L large enoughand skip this argument. We now choose the scaling factor b = t /z so that the main t -dependence on the right is cancelled. Expanding the scaling form (2) for k = 1 with respectto the small quantity t x /z m , one obtains M ( t, τ, m ) ∼ m t θ F ( t /νz τ, t x /z m ) = m t θ (1 + at /νz τ ) + O ( τ , m ) , (3)where θ = ( x − β/ν ) /z has been introduced. For τ = 0 and small enough t and m the scalingdependence for magnetization (3) takes the form M ( t ) ∼ m t θ . For almost all statisticalsystems studied up to now [17, 18, 22], the exponent θ is positive, i.e., the magnetizationundergoes surprisingly a critical initial increase. The time scale of this increase is t ∼ m − z/x . However, in the limit of m the time scale goes to infinity. Hence the initialcondition can leave its trace even in the long-time regime.If τ = 0, the power law behavior is modified by the scaling function F ( t /νz τ ) withcorrections to the simple power law, which will be depended on the sign of τ . Therefore,simulation of the system for temperatures near the critical point allows to obtain the timedependent magnetization with non-perfect power behavior, and the critical temperature T c can be determined by interpolation.Other two interesting observables in short-time dynamics are the second moment of mag-netization M (2) ( t ) and the auto-correlation function A ( t ) = 1 L d *X i S i ( t ) S i (0) + . (4)As the spatial correlation length in the beginning of the time evolution is small, for a finitesystem of dimension d with lattice size L the second moment M (2) ( t, L ) ∼ L − d . Combiningthis with the result of the scaling form (2) for τ = 0 and b = t /z , one obtains M (2) ( t ) ∼ t − β/νz M (2) (cid:0) , t − /z L (cid:1) ∼ t c , c = (cid:18) d − βν (cid:19) z . (5)Furthermore, careful scaling analysis shows that the auto-correlation also decays by a powerlaw [23] A ( t ) ∼ t − c a , c a = dz − θ. (6)Thus, the investigation of the short-time evolution of system from a high-temperature initialstate with m = 0 allows to determine the dynamic exponent z , the ratio of static exponents β/ν and a new independent critical exponent θ .Till now a completely disordered initial state has been considered as starting point, i.e.,a state of very high temperature. The question arises how a completely ordered initial stateevolves, when heated up suddenly to the critical temperature. In the scaling form (2) onecan skip besides L , also the argument m = 1 M ( k ) ( t, τ ) = b − kβ/ν M ( k ) (cid:0) b − z t, b /ν τ (cid:1) . (7)The system is simulated numerically by starting with a completely ordered state, whoseevaluation is measured at or near the critical temperature. The quantities measured are M ( t ), M (2) ( t ). With b = t /z one avoids the main t -dependence in M ( k ) ( t ), and for k = 1one has M ( t, τ ) = t − β/νz M (1 , t /νz τ ) = t − β/νz (cid:0) at /νz τ + O ( τ ) (cid:1) . (8)For τ = 0 the magnetization decays by a power law M ( t ) ∼ t − β/νz . If τ = 0, the powerlaw behavior is modified by the scaling function M (1 , t /νz τ ). From this fact, the criticaltemperature T c and the critical exponent β/νz can be determined.We must note, that the short-time dynamic method in part of critical evolution de-scription of system starting from the ordered initial state is essentially the same as thenon-equilibrium relaxation method proposed by N. Ito in [24] for critical behavior study ofthree-dimensional pure Ising model. At present, this method was extended by N. Ito tonon-equilibrium relaxation study of Ising spin glass models [25], Kosterlitz-Thouless phasetransition [26] and fully frustrated XY models in two dimension [27].From scaling form (8) the power law of time dependence for the logarifmic derivative ofthe magnetization can be obtained in the next form ∂ τ lnM ( t, τ ) | τ =0 ∼ t /νz , (9)which allows to determine the ratio 1 /νz . On basis of the magnetization and its secondmoment the time dependent Binder cumulant U ( t ) = M (2) ( M ) − ∼ t d/z (10)is defined. From its slope one can directly measure the dynamic exponent z . Consequently,from an investigation of the system relaxation from ordered initial state with m = 1 thedynamic exponent z and the static exponents β and ν can be determined and their valuescan be compared with results of simulation of system behavior from disordered initial statewith m = 0. III. MEASUREMENTS OF THE CRITICAL TEMPERATURE AND CRITICALEXPONENTS FOR 3D ISING MODEL
We have performed simulations on three-dimensional cubic lattices with linear sizes L from 16 to 128, starting either from an ordered state or from a high-temperature state withzero or small initial magnetization. We would like to mention that measurements startingfrom from a completely ordered state with the spins oriented in the same direction ( m = 1)are more favorable, since they are much less affected by fluctuations, because the quantitiesmeasured are rather big in contrast to those from a random start with m = 1. Therefore, forcareful determination of the critical temperature and critical exponents for 3D Ising modelwith linear defects we begin to investigate the relaxation of this model from a completelyordered initial state. A. Evolution from an ordered state
Initial configurations for systems with the spin concentration p = 0 . t = 1000. We measured the time evolution of the magnetization M ( t ) = 1 N spin "*X i p i S i ( t ) + (11) FIG. 1: Time evolution of the magnetization M ( t ) for L = 128 and for different values of thetemperature T . and the second moment M (2) ( t ) = 1 N spin * X i p i S i ( t ) ! + , (12)which also allow to calculate the time dependent Binder cumulant U ( t ) (10). The anglebrackets in (11) and (12) denote the statistical averaging and the square brackets are foraveraging over the different impurity configurations.In Fig. 1 the magnetization M ( t ) for samples with linear size L = 128 at T = 3 . , T c = 3 . p = 0 . T c = 3 . U , defined as U = 12 (cid:18) − [ h M i ][ h M i ] (cid:19) , (13)and the correlation length [28] ξ = 12 sin ( π/L ) r χF − , (14) χ = 1 N spin [ h M i ] , (15) F = 1 N spin [ h Φ i ] , (16)Φ = 13 X n =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j p j S j exp (cid:18) πix n,j L (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (17)where ( x ,j , x ,j , x ,j ) are coordinates of j -th site of lattice.The cumulant U ( L, T ) has a scaling form U ( L, T ) = u (cid:0) L /ν ( T − T c ) (cid:1) . (18)The scaling dependence of the cumulant makes it possible to determine the critical tem-perature T c from the coordinate of the points of intersections of the curves specifying thetemperature dependence U ( L, T ) for different L . In Fig. 2a the computed curves of U ( L, T )are presented for lattices with sizes L from 16 to 128. As a result it was determined that thecritical temperature is T c = 3 . L = 16 , L = 64 , ξ/L was introduced as a convenient method for calculating of T c in[29]. In Fig. 2b the computed curves of temperature dependence of ratio ξ/L are presentedfor lattices with the same sizes, the coordinate of the points of intersections of which alsogives the critical temperature T c = 3 . T c we selected as the best forsubsequent investigations of the Ising model.0 FIG. 2: Binder cumulant U ( T, L ) (a) and ratio ξ/L (b) as a function of T for lattices with differentsizes L .FIG. 3: Time evolution of logarithmic derivative of the magnetization ∂ τ lnM ( t, τ ) | τ =0 (a) andthe cumulant U ( t ) (b) for L = 128 at the critical temperature T c = 3 . Also, we have determined the temperature of intersection of the curves specifying thetemperature dependence cumulants U ( L, T ) for L = 16 and L = 32 with the use of linear de-fects distribution in samples as in [5] with the possibility of their intersection. Computationgives T c ( L ) = 3 . T c ( L ) = 3 . /νz can be determined from relation (9) if we differentiate lnM ( t, τ ) with respect to τ .1 FIG. 4: Dependence of the mean squareerrors σ of the fits for the magnetization(a), logarifmic derivative of the magneti-zation (b), and the cumulant (c) as a func-tion of the exponents β/νz , 1 /νz , and d/z for ω = 0 . The dynamic exponent z can be determined fromanalysis of time dependent Binder cumulant U ( t )(10) for τ = 0. In Fig. 3 the logarithmic derivativeof the magnetization ∂ τ lnM ( t, τ ) | τ =0 with respectto τ (Fig. 3a) and the cumulant U ( t ) (Fig. 3b) forsamples with linear size L = 128 at T c = 3 . ∂ τ lnM ( t, τ ) | τ =0 have been obtained from a quadratic interpolationbetween the three curves of time evolution of themagnetization for the temperatures T = 3 . . . T c = 3 . U ( t ) and clarified that in the time in-terval [50,150] the U ( t ) is best fitted by powerlaw with the dynamic exponent z ≃ .
02, corre-sponding to the pure Ising model [30, 31], andthe linear defects are developed for t >
400 MCSonly. An analysis of the U ( t ) slope measuredin the interval [500,900] shows that the exponent d/z = 1 . z = 2 . β/νz = 0 . /νz = 0 . ν = 0 . β = 0 . TABLE I: Values of the exponents β/νz , 1 /νz , d/z , and minimal values of the mean square errors σ in fits for different values of the exponent ωω β/νz σ /νz σ d/z σ able X ( t ): X ( t ) ∼ t ∆ (cid:0) A x t − ω/z (cid:1) , (19)where ω is a well-known exponent of corrections to scaling. This expression reflects thescaling transformation in the critical range of time-dependent corrections to scaling in theform of t − ω/z to the usual form of corrections to scaling τ ων in equilibrium state for timet comparable with the order parameter relaxation time t r ∼ ξ z Ω( kξ ) [17]. Field-theoreticestimate of the ω value gives ω ≃ .
80 in the two-loop approximation [15]. Monte Carlostudy of Ballesteros and Parisi [5] shows that ω ≃ . Xt − ∆ )on t − ω/z with the changing values of the exponent ∆ and the exponent ω from the interval[0.7,1.0]. Then, we have investigated the dependence of the mean square errors σ of thisfitting procedure for the function Xt − ∆ ( t − ω/z ) on the changing ∆ and ω . In Fig. 4 we plotthe σ for the magnetization (Fig. 4a), logarifmic derivative of the magnetization (Fig. 4b),and the cumulant (Fig. 4c) as a function of the exponents β/νz , 1 /νz , and d/z for ω = 0 . σ determines the exponents z , ν , and β for every ω . In Table I we present thecomputed values of the exponents β/νz , 1 /νz , and d/z , and minimal values of the meansquare errors σ in these fits for values of the exponent ω = 0 . , . , . , .
0. We see thatthe values of β/νz , 1 /νz , and d/z are weakly dependent on the change of the exponent ω in the interval [0.7,1.0], but the ω = 0 . ω = 0 . z = 2 . ± . ,ν = 0 . ± . ,β = 0 . ± . . (20)It is interesting to compare these values of exponents with those obtained in [13] withthe use of the field-theoretic approach z = 2 . ,ν = 0 . ,β = 0 . , (21)which demonstrate a very good agreement with each other, but show an essential differencefrom Monte Carlo results of Ballesteros and Parisi [5] with ν = 1 . β = 0 . B. Evolution from a disordered state
We have also performed simulations of evolution of the system with linear defects on thelargest lattice with L = 128, starting from a disordered state with small initial magnetiza-tions m = 0 .
02 and m = 0 .
001 at the critical point. The initial magnetization has beenprepared by flipping in an ordered state a definite number of spins at randomly chosen sitesin order to get the desired small value of m . In accordance with Section II, a generalizeddynamic scaling predicts in this case a power law evolution for the magnetization M ( t ), thesecond moment M (2) ( t ) and the auto-correlation A ( t ) in the short-dynamic region.In Fig. 5, 6 and 7 we show the obtained curves for M ( t ) (Fig. 5), M (2) ( t ) (Fig. 6)and A ( t ) (Fig. 7), which are plotted in log-log scale up to t = 700. These curves wereresulted by averaging over 3000 different samples with 25 runs for each sample. FromFig. 5 we can see an initial increase of the magnetization, which is a very prominentphenomenon in the short-time critical dynamics [17, 18]. But in contrast to dynam-ics of the pure systems [17], we can observe the crossover from dynamics of the puresystem on early times of the magnetization evolution up to t = 70 to dynamics ofthe disordered system with the influence of linear defects in the time interval [100,650].4 FIG. 5: Time evolution of the magnetization M ( t ) for L = 128 with the initial magnetization m = 0 .
02 (a) and m = 0 .
001 (b) at the critical temperature T c = 3 . M (2) for L = 128 with the initial magnetization m = 0 .
02 (a) and m = 0 .
001 (b) at the critical temperature T c = 3 . A ( t ) for L = 128 with differentinitial magnetization values m at thecritical temperature T c = 3 . The same crossover phenomena were observed inevolution of the second moment M (2) ( t ) and the au-tocorrelation A ( t ). In result of linear approxima-tion of these curves in the both time intervals weobtained the values of the exponents θ , c and c a in accordance with relations in (3), (5) and (6) forinitial states with m = 0 .
02 and m = 0 .
001 (Ta-ble II). The final values of these exponents and alsothe critical exponents z , β/ν and x were obtainedby extrapolation to m = 0. In Table II we comparethe values of these exponents with values of corre-5 TABLE II: Values of the critical exponents obtained in present work for evolution from disorderedinitial states with different m and extrapolated to m = 0 and corresponding exponents for puresystems [17] and from [13] θ c c a z β/ν x t ∈ [10 , m = 0 ,
02 0.086(12) 0.964(28) 1.384(26) m = 0 ,
001 0.099(9) 0.973(19) 1.364(23) m = 0 0.101(10) 0.975(23) 1.363(26) 2.049(27) 0.501(27) 0.708(34) t ∈ [100 , m = 0 ,
02 0.152(12) 0.812(21) 1.103(16) m = 0 ,
001 0.149(10) 0.804(19) 1.047(12) m = 0 0.149(11) 0.801(20) 1.043(14) 2.517(32) 0.492(28) 0.867(37)TFD [13] 2.495 0.489pure [17] 0.108(2) 0.970(11) 1.362(19) 2.041(18) 0.510(14) 0.730(25) sponding exponents for the pure Ising model [18] and theoretical field description (TFD)results for system with linear defects [13]. The obtained values quite well agree with re-sults of simulation from an ordered state with m = 1 and with results from [13] and showthat LR-correlated defects lead to faster increasing of the magnetization in the short-timedynamic regime in compare with the pure system. C. Measurements of the critical characteristics in equilibrium state
With the aim to verify the short-time dynamics method and the results obtained we alsocarried out the study of the critical behavior of 3D Ising model with the linear defects ofrandom orientation by traditional Monte Carlo simulation methods in equilibrium state.For simulations we have used the Wolf single-cluster algorithm. We have computed for thecritical temperature T c = 3 . U for lattices with sizes L from 16 to 128 for thesame spin concentration p = 0 .
8. The use of well-known scaling critical dependences for6
FIG. 8: Dependence of the mean square errors σ of the fits for heat capacity (a), magnetization(b), susceptibility (c), and thermal derivative of cumulant (d) as a function of the exponents α/ν , β/ν , γ/ν , and ν for different values of ω . these thermodynamic and correlation functions with taking into consideration the finite sizescaling corrections C ( L ) ∼ L α/ν (cid:0) aL − ω (cid:1) (22) M ( L ) ∼ L − β/ν (cid:0) bL − ω (cid:1) (23) χ ( L ) ∼ L γ/ν (cid:0) cL − ω (cid:1) (24) dUdT ( L ) ∼ L /ν (cid:0) dL − ω (cid:1) (25)makes it possible to determine the critical exponents α , ν , β , γ , and ω by means of sta-tistical data processing of simulation results. To analyze simulation data we have used thelinear approximation of the ( XL − ∆ ) on L − ω and then investigated the dependence of themean square errors σ of this fitting procedure for the function XL − ∆ ( L − ω ) on the changing7 TABLE III: Values of the exponents α/ν , β/ν , γ/ν , and ν with values of the exponent ω , giv-ing the best fit in approximation procedure TABLE IV: Values of the critical exponents obtainedin present work for average value of exponent ω =0 .
76 and corresponding exponents from [13] α/ν β/ν γ/ν ν − . . . . ω .
90 0 .
65 0 .
70 0 . α β γ ν present − . . . . − . . . . exponent ∆ and ω values. In Fig. 8 we plot the σ for heat capacity (Fig. 8a), magnetiza-tion (Fig. 8b), susceptibility (Fig. 8c), and temperature derivative of cumulant (Fig. 8d) asa function of the exponents α/ν , β/ν , γ/ν , and ν for different values of ω . Minimum of σ determines the values of exponents. In Table III we present the obtained values of theexponents α/ν , β/ν , γ/ν , ν , and ω , which give minimal values of σ in these fits. Thenwe determine the average value of ω = 0 . IV. MEASUREMENTS OF THE CRITICAL TEMPERATURE AND CRITICALEXPONENTS FOR 3D XY-MODEL
FIG. 9: Binder cumulant U ( T, L ) for 3DXY-model as a function of T for latticeswith different sizes L . Also, we have carried out the Monte Carlostudy of the effect of LR-correlated quenched de-fects on the critical behavior of 3D XY-modelcharacterized by the two-component order param-eter. As is well-known [2, 13], renormalizationgroup analysis predicts the possibility of a newtype of the critical behavior for this model dif-ferent from the critical behavior of pure XY-likesystems or systems with point-like uncorrelated8defects. We considered the same site-diluted cubic lattices with linear defects of randomorientation in the samples with the spin concentration p = 0 .
8. The critical temperature T c = 1 . U ( L, T ) for latticeswith sizes L from 32 to 128 (Fig. 9). For simulations we have used the Wolff single-clusteralgorithm. A. Evolution from an ordered state
FIG. 10: Time evolution of the magne-tization M ( t ) (a), Binder cumulant U ( t )(b) and the logarithmic derivative of themagnetization ∂ τ lnM ( t, τ ) | τ =0 (c) for 3DXY-model with lattice size L = 128 at thecritical temperature T c = 1 . We have performed simulations of the criti-cal relaxation of the XY-model with linear de-fects starting from an ordered initial state. Asexample, in Fig. 10 we show the obtained curvesfor the magnetization M ( t ) (Fig. 10a), Bindercumulant U ( t ) (Fig. 10b) and the logarithmicderivative of the magnetization ∂ τ lnM ( t, τ ) | τ =0 (Fig. 10c), which are plotted in log-log scale upto t = 1000 for lattices with L = 128. Thesecurves were resulted by averaging over 3000 dif-ferent samples. On these figures we can observethe crossover from dynamics, which is similar tothat in the pure system on early times of the evo-lution up to t = 150, to dynamics of the dis-ordered system with the influence of linear de-fects in the time interval [350,800]. The slopeof M ( t ), U ( t ) and ∂ τ lnM ( t, τ ) | τ =0 over the in-terval [350,800] provides the exponents β/νz =0 . d/z = 1 . /νz = 0 . β/νz = 0 . d/z = 1 .
268 and 1 /νz = 0 . z = 2 . ± . ,ν = 0 . ± . ,β = 0 . ± . ,ω = 1 . ± . . (26)The comparison of these values of exponents with those obtained in [13] with the use ofthe field-theoretic approach z = 2 . ν = 0 . β = 0 . ω = 1 .
15 in [15] showstheir good agreement within the limits of statistical errors of simulation and numericalapproximations.The obtained results confirm the strong effect of LR-correlated quenched defects onboth the critical behavior of 3D Ising model and the systems characterized by the many-component order parameter.
B. Evolution from a disordered state
We have also performed simulations of evolution of the system with linear defects onthe largest lattice with L = 128, starting from a disordered state with small initial mag-netizations m = 0 . m = 0 . m = 0 .
005 at the critical point. The initialmagnetization has been prepared by flipping in an ordered state a definite number of spinsat randomly chosen sites in order to get the desired small value of m . In accordance withSection II, a generalized dynamic scaling predicts in this case a power law evolution forthe magnetization M ( t ), the second moment M (2) ( t ) and the auto-correlation A ( t ) in theshort-dynamic region.In Fig. 11 and 12 we show the obtained curves for M ( t ) (Fig. 11), M (2) ( t ) and A ( t )(Fig. 12), which are plotted in log-log scale up to t = 700. These curves were resulted byaveraging over 3000 different samples with 25 runs for each sample. From Fig. 11 we cansee also an initial increase of the magnetization, which is a very prominent phenomenonin the short-time critical dynamics. But in contrast to dynamics of the pure systems, wecan observe, as previously for Ising model, the crossover from dynamics of the pure systemon early times of the magnetization evolution up to t = 100 to dynamics of the disorderedsystem with the influence of linear defects in the time interval [200,650]. In this time interval0 FIG. 11: Time evolution of the magnetization M ( t ) for 3D XY-model with the initial magnetization m = 0 .
01 (a), m = 0 . m = 0 .
005 (c) at the critical temperature T c = 1 . θ in the limit m → we determined the values of the dynamic critical exponent θ , which are equal θ = 0 . m = 0 . θ = 0 . m = 0 . θ = 0 . m = 0 . θ = 0 . m → M (2) ( t ) (Fig. 12a) and theautocorrelation A ( t ) (Fig. 12b), obtained for simulation of systems with the initial magne-tization m = 0 (in reality for m = 10 − such as for XY-model the spin configuration with m = 0 is impossible to prepare), gives directly the values of exponents c = 0 . c a = 0 . M (2) ( t ) and A ( t ) from dynamics of the pure system on early times to dynamicsof the disordered system with the influence of linear defects.On basis of these values of exponents θ , c and c a we obtained the exponents z = 2 . FIG. 12: Time evolution of the second moment M (2) (a) and the auto-correlation A ( t ) (b) for 3DXY-model with the initial magnetization m = 0 at the critical temperature T c = 1 . and β/ν = 0 . m = 1 and with results of theoretical field description for XY-like systemswith linear defects [13] within the limits of statistical errors of simulation and numericalapproximations. V. CONCLUSION REMARKS
The present results of Monte Carlo investigations allow us to recognize that the short-time dynamics method is reliable for the study of the critical behavior of the systems withquenched disorder and is the alternative to traditional Monte Carlo methods. But in contrastto studies of the critical behavior of the pure systems by the short-time dynamics method,in case of the systems with quenched disorder corresponding to randomly distributed lineardefects after the microscopic time t mic ≃
10 there exist three stages of dynamic evolution.For systems starting from the ordered initial states ( m = 1) in the time interval of 50-200MCS, the power-law dependences are observed in the critical point for the magnetization M ( t ), the logarithmic derivative of the magnetization ∂ τ lnM ( t, τ ) | τ =0 and Binder cumulant U ( t ), which are similar to that in the pure system. In the time interval [450,900], the power-law dependences are observed in the critical point which are determined by the influenceof disorder. However, careful analysis of the slopes for M ( t ), ∂ τ lnM ( t, τ ) | τ =0 and U ( t )reveals that a correction to scaling should be considered in order to obtain accurate results.The dynamic and static critical exponents were computed with the use of the corrections2to scaling for the Ising and XY models with linear defects, which demonstrate their goodagreement with results of the field-theoretic description of the critical behavior of thesemodels with long-range correlated disorder. In intermediate time interval of 200-400 MCSthe dynamic crossover behavior is observed from the critical behavior typical for the puresystems to behavior determined by the influence of disorder.The investigation of the critical behavior of the Ising model with extended defects startingfrom the disordered initial states with m ≃ M ( t ), thesecond moment M (2) ( t ) and the autocorrelation A ( t ) are observed in the critical point, whichare typical for the pure system in the interval [10,70] and for the disordered system in theinterval [100,650]. In intermediate time interval the crossover behavior is observed in thedynamic evolution of the system. The obtained values of exponents demonstrate a goodagreement within the limits of statistical errors of simulation and numerical approximationswith results of simulation of the pure Ising model by the short-time dynamics method [13]for the first time interval and with our results of simulation of the critical relaxation of thismodel from the ordered initial states.Also, we would like to note that over complicated critical dynamics of the systems withquenched disorder the accurate determination of the critical temperature is better to carryout in equilibrium state from the coordinate of the points of intersections of the curves spec-ifying the temperature dependence of Binder cumulants U ( L, T ) or ratio ξ/L for differentlinear sizes L of lattices.The obtained results for 3D XY-model confirm the strong influence of LR-correlatedquenched defects on the critical behavior of the systems described by the many-componentorder parameter. As a result, a wider class of disordered systems, not only the three-dimensional Ising model, can be characterized by a new type of critical behavior induced byquenched disorder.We are planning to continue the Monte Carlo study of critical behavior of the model withLR-disorder for different spin concentrations p and investigate the universality of critical be-havior of diluted systems with LR-disorder focusing on the problem of disorder independenceof asymptotic characteristics.3 Acknowledgements
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