Ordering Kinetics in the Random Bond XY Model
OOrdering Kinetics in the Random Bond XY Model
Manoj Kumar , Swarnajit Chatterjee , Raja Paul ∗ and Sanjay Puri † School of Physical Sciences, Jawaharlal Nehru University, New Delhi – 110067, India. and Indian Association for the Cultivation of Science, Kolkata – 700032, India.
We present a comprehensive Monte Carlo study of domain growth in the random-bond XY modelwith non-conserved kinetics. The presence of quenched disorder slows down domain growth in d = 2 ,
3. In d = 2, we observe power-law growth with a disorder-dependent exponent on the time-scales of our simulation. In d = 3, we see the signature of an asymptotically logarithmic growthregime. The scaling functions for the real-space correlation function are seen to be independent of thedisorder. However, the same does not apply for the two-time autocorrelation function, demonstratingthe breakdown of superuniversality. I. INTRODUCTION
The XY model has been widely studied in the litera-ture. Experimentally, a large number of physical systemshave been described by the XY model. For dimensional-ity d = 2, typical realizations of the XY model includemagnetic films with planar anisotropy [1], thin-film su-perfluids or superconductors [2], Josephson junction ar-rays [3, 4], hexatic liquid crystals [5], melting of two-dimensional solids [6], etc. In d = 3, physical systemssuch as superfluid He [7, 8] and planar spin magnetshave been described by the XY model.The XY model in d = 2 exhibits the well-known Berezinskii-Kosterlitz-Thouless (BKT) transition at tem-perature T BKT [9–11]. This system shows long-range or-der (LRO) only at temperature T = 0. However, for0 < T < T BKT , the system shows quasi-long-range or-der (QLRO), where the correlation-function decays as apower-law with temperature-dependent exponent η ( T ).In this state, the morphology consists of bound statesof vortex-antivortex pairs. For T > T
BKT , the vortex-antivortex pairs unbind and the correlation function de-cays exponentially. In this paper, we are interested inthe disordered XY model. The presence of quenched dis-order has a strong effect on the BKT phase transition.For instance, various numerical studies [12–17] of the XYmodel with dilution (e.g., site-vacancies, bond-vacancies)have shown that T BKT decreases with increasing disorder.It becomes zero at a critical value of dilution, which isreferred to as the percolation threshold. In d = 3, thedisorder-free XY model exhibits true long-range order(LRO) for T < T c [18].In this paper, we are interested in the nonequilibriumordering kinetics of the disordered XY model in d = 2 , T . When a pure XYsystem is quenched to T < T
BKT or T < T c , the coarsen-ing process is characterized by the annihilation of vortex-antivortex pairs [19, 20]. The characteristic length scale R ( t ) ∼ t / for non-conserved vector fields in d ≥ ∗ author for correspondence: [email protected] † author for correspondence: [email protected] For d = 2, Yurke et al. [22] predicted a logarithmic cor-rection to the diffusive growth as R ( t ) ∼ ( t/ ln t ) / .In recent years, there has been intense interest in thesubject of domain growth in disordered systems [23–25].In general, the domain boundaries become trapped atlate times by energy barriers introduced by the disor-der, thereby slowing down the asymptotic domain growthlaw [26]. There has been some debate about the precisenature of the asymptotic growth law in the case withscalar order parameter, e.g., random-field Ising model (RFIM) or random-bond Ising model (RBIM). The earlyMonte Carlo (MC) simulations of the RBIM by Paul etal. [25] reported a power law growth with a disorder-dependent exponent. However, the recent works of Cor-beri et al. [27–29] have demonstrated that there is a slowcrossover to a logarithmic growth regime, which is nu-merically very difficult to access.It would be fair to say that we now have a good under-standing of the case with scalar order parameter. How-ever, to the best of our knowledge, there have been nostudies of the case with vector order parameter – eventhough this is experimentally very important. This pa-per is a first step in that direction, i.e., we present com-prehensive numerical results for domain growth in the random-bond XY model (RBXYM) with nonconservedkinetics in d = 2 ,
3. Our study covers two importantaspects. First, we study the effect of disorder on thetransition temperature by using the Wolff single-clusterupdating algorithm [30]. Second, we study nonconservedordering kinetics via the Metropolis algorithm [31] byquenching the system below the transition temperature.The main results of our study are as follows:(a) The critical temperature is found to decrease withincreasing disorder.(b) For domain growth, the correlation function showsdynamical scaling in d = 2 ,
3. Also, the scaling functionis independent of disorder, and therefore shows a uni-versal behavior. However, the two-time autocorrelationfunction is not universal.(c) In d = 2, the growth law over our simulation time-scales is algebraic with a disorder-dependent exponent.(d) In d = 3, the domain growth law is asymptoticallylogarithmic, as in the scalar case.This paper is organized as follows. In Sec. II, we dis- a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t cuss the model and present details of our numerical simu-lations. In Sec. III, we present detailed numerical resultsfrom our simulations of the d = 2 , II. MODELING AND SIMULATION DETAILSA. Random Bond XY model
Consider a lattice of size L (in d = 2) or L (in d = 3).Each lattice site labeled by i has a two-component vectorspin S i . The Hamiltonian for the RBXYM is defined as H = − (cid:88) (cid:104) ij (cid:105) J ij S i · S j = − (cid:88) (cid:104) ij (cid:105) J ij cos( θ i − θ j ) , (1)where J ij is the exchange coupling between thenearest-neighbor pair denoted by (cid:104) ij (cid:105) . Each spin S i =(cos θ i , sin θ i ) is a unit vector and is described by an angle θ i ∈ ( − π, π ). The quenched random-bond variables { J ij } are distributed uniformly on the interval [1 − (cid:15)/ , (cid:15)/ (cid:15) quantifies the degree of disorder. The limit (cid:15) = 0corresponds to the pure case ( J ij = 1). Here, we fo-cus on the ferromagnetic case where J ij >
0. Therefore, (cid:15) = 2 corresponds to the maximum value of disorder inour simulations.
B. Simulation Details for Study of TransitionTemperature
Before we discuss nonequilibrium studies, it is im-portant to understand the equilibrium properties of theRBXYM. In this context, let us discuss the simulationdetails for determining the transition temperature T BKT (in d = 2) or T c (in d = 3) in the presence of disorder( (cid:15) ). For the pure XY model, the transition temperatureis T BKT (cid:39) .
89 in d = 2 [32, 33], and T c (cid:39) .
203 in d = 3[7, 34].A standard tool to determine the transition tem-perature is the fourth-order Binder cumulant U ( T, L )[35, 36], defined as U ( T, L ) = 1 − [ (cid:104) m (cid:105) ]3[ (cid:104) m (cid:105) ] . (2)Here, m is the magnetization, and (cid:104)· · · (cid:105) and [ · · · ] de-note the thermal and disorder averages, respectively. TheBinder cumulants U ( T, L ) are plotted against tempera-ture T for different lattice sizes L . Then, in the scalingregion near T BKT or T c , the U vs. T curves for different L have a unique intersection point [35], which is identifiedas the transition temperature. The magnetization m for the XY model is defined as m = 1 N (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) N (cid:88) i =1 cos θ i (cid:33) + (cid:32) N (cid:88) i =1 sin θ i (cid:33) , (3)where N is the total number of sites, i.e., N = L d .The magnetization m is measured when the system hasreached thermal equilibrium. To equilibrate the systemat temperature T , we use the canonical sampling MCmethod with the Wolff single-cluster updating algorithm[30].A single MC update in the Wolff single-cluster algo-rithm can be described as follows:(a) Choose a random reflection r = (cos φ, sin φ ) and arandom spin S i = (cos θ i , sin θ i ) as the starting point fora cluster C to be built.(b) Flip S i → R ( r ) S i = S i − S i · r ) r , i.e., θ i → θ (cid:48) i = π − θ i + 2 φ .(c) Visit all neighboring spins S j of S i , and add them tothe cluster C with the probability [30] P ( S i , S j ) = 1 − exp { min[0 , βJ ij ( r · S i )( r · S j )] } , (4)where β = ( k B T ) − . In terms of the angle variables, thecorresponding expression for the probability is P ( θ i , θ j ) = 1 − exp { min[0 , βJ ij cos( θ i − φ ) cos( θ j − φ )] } . (5)(d) Keep visiting all nearest neighbors of newly-addedspins, and add them to the cluster C with probability P . Continue this process until no spin is left to add to C . One Monte Carlo step (MCS) corresponds to N suchupdates. C. Simulation Details for Study of OrderingKinetics
We study ordering kinetics in the RBXYM by assign-ing a random initial orientation to each spin θ i ∈ ( − π, π ),mimicking the high-temperature disordered state. Attime t = 0, the system is rapidly quenched to T < T
BKT (in d = 2) or T < T c (in d = 3), and evolved via noncon-served kinetics. We let the system evolve upto 10 MCSwith the help of the Metropolis algorithm [31]. For theXY system, our algorithm can be described as follows:(a) Select a random spin S i and give θ i a small rotation δ ∈ ( − . , . θ i → θ (cid:48) i = θ i + δ .(b) The new spin θ (cid:48) i is accepted with the Metropolis tran-sition probability P = min[1 , exp ( − β∆ H )] . (6)Here, ∆ H is the change in energy resulting from the anglechange θ i → θ (cid:48) i :∆ H = (cid:88) k J ik [cos( θ i − θ k ) − cos( θ (cid:48) i − θ k )] , (7)where k refers to the nearest neighbors of site i .A useful quantity in studying phase ordering kineticsis the spatial correlation function , which is defined as [37] C ( r , t ) = 1 N N (cid:88) i =1 { [ (cid:104) S i ( t ) · S i + r ( t ) (cid:105) ] − [ (cid:104) S i ( t ) (cid:105) ] · [ (cid:104) S i + r ( t ) (cid:105) ] } , (8)where [ (cid:104)· · · (cid:105) ] indicates an averaging over independentruns, and different realizations of bond randomness.The quantity C ( r , t ) characterizes the morphology of thecoarsening system. If the system is isotropic and charac-terized by a single length scale R ( t ), then the correlationfunction has a dynamical scaling form [37, 38] : C ( r , t ) = f (cid:18) rR ( t ) (cid:19) , (9)where f ( x ) is the scaling function. The characteristiclength scale R ( t ) is defined as the distance over whichthe correlation function decays to (say) 0.2 of its max-imum value. In the XY model, the typical length-scalecan also be determined from the density of topologicaldefects, e.g., ρ def ( t ) ∼ /R v ( t ) in d = 2, where R v is thediameter of a vortex. In the scaling regime, this defini-tion will differ from the former one only by a prefactor.In this paper, we use the R ( t ) determined from the decayof C ( r, t ) [39, 40].Another nonequilibrium quantity of interest is the two-time autocorrelation function : A ( t, t w ) = 1 N N (cid:88) i =1 { [ (cid:104) S i ( t w ) · S i ( t ) (cid:105) ] − [ (cid:104) S i ( t w ) (cid:105) ] · [ (cid:104) S i ( t ) (cid:105) ] } , (10)which is important in studies of aging [41]. The quan-tity t w is referred to as the waiting time . In the scalingregime, we expect A ( t, t w ) = h (cid:20) R ( t ) R ( t w ) (cid:21) ∼ (cid:20) R ( t ) R ( t w ) (cid:21) − λ for t (cid:29) t w . (11)Here, h ( y ) is a scaling function, and the exponent λ wasfirst introduced in the context of spin glasses [42].The morphology of the ordering system is usually stud-ied by scattering experiments which measure the struc-ture factor S ( k , t ), defined as the Fourier transform ofthe correlation function C ( r , t ). It has the correspondingdynamical scaling form: S ( k, t ) = R ( t ) d g ( kR ( t )) . (12)Bray and Puri (BP) [21] and Toyoki (T) [43] have inde-pendently obtained the scaling functions f ( x ) and g ( p )for domain growth with an n -component vector fieldwithout disorder. They found that the scaling function g ( p ) has the large- p behavior: g ( p ) ∼ p − ( d + n ) for p → ∞ . (13)This is referred to as the generalized Porod tail , asPorod [44, 45] emphasized that scattering off sharp inter-faces in a scalar field ( n = 1 case) yields the structure-factor tail g ( p ) ∼ p − ( d +1) . III. NUMERICAL RESULTS
Now, we will present numerical results from our simu-lations of the d = 2 , (cid:15) = 0, 0.5, 1.0, 1.5, 2.0; (cid:15) = 0 is the pure casefor reference, and (cid:15) = 2 corresponds to the case withmaximum disorder. First, we determine the transitiontemperatures as a function of (cid:15) . Then, we study order-ing kinetics by quenching the system below T BKT ( (cid:15) ) or T c ( (cid:15) ). A. RBXYM in d = 2
1. Estimation of T BKT
Let us first consider the d = 2 RBXYM. We study thissystem on a square lattice ( L ) of linear sizes L = 96, 128,and 256. Starting from a random initial configuration, welet the system equilibrate using the Wolff single-clusterupdate algorithm (see Sec. II B). After equilibration, weaverage m and m upto 6 × MCS. Further, we per-form a disorder average over 200 independent runs ofrandom-bond configurations. Then, one can determinethe Binder cumulant U ( T, L ) from Eq. (2). U T ε = 1.0 L = 96L = 128L = 256 T K T ε FIG. 1: Plot of fourth-order Binder cumulant U ( T, L ) vs.temperature T for the d = 2 RBXYM. The data is shownfor (cid:15) = 1.0, and square lattices with L = 96 , , T BKT ( (cid:15) ) is determined from the in-tersection of different L -curves. The inset shows the plot of T BKT ( (cid:15) ) with disorder (cid:15) . The numerical values of T BKT ( (cid:15) ) areprovided in Table I. Figure 1 is a plot of U vs. T for (cid:15) = 1.0. (We havezoomed the plot in the vicinity of T BKT .) The transitiontemperature ( T BKT ) can be accurately estimated fromthe intersection of Binder cumulant curves for different L . (cid:15) T BKT z ± ± ± ± ± ± ± ± ± T BKT ( (cid:15) ) and growth ex-ponents z ( (cid:15) ) for the d = 2 RBXYM. The T BKT -values for various (cid:15) are plotted in the inset ofFig. 1, and tabulated in Table I. For the pure case ( (cid:15) = 0),we found T BKT = 0 . ± . T BKT (cid:39) .
893 in the literature [10, 11, 32, 33].With increasing (cid:15) , T BKT decreases from T BKT (cid:39) .
902 to T BKT ( (cid:15) = 2) = 0 . ± . Coarsening dynamics
Next, we present numerical results for coarsening dy-namics in the d = 2 RBXYM. The simulations are per-formed on a square lattice of size 1024 with periodicboundary conditions applied on both sides. As describedin Sec. II C, the initially disordered system is quenched to T = 0 . . < T BKT , see Table I) at t = 0 MCS. Thesystem is evolved upto t = 10 MCS using the Metropolisalgorithm. All statistical results presented here are av-eraged over 20 runs (sometimes more) with independent { J ij } -configurations. FIG. 2: Evolution snapshots of the d = 2 RBXYM at t = 10 MCS, after a quench from T = ∞ to T = 0 . (cid:15) = 0, (b) (cid:15) = 1, and (c) (cid:15) = 2. The lattice size is 512 . In these plots, { θ i } are marked in the interval [ θ − . , θ + 0 .
1] with thefollowing color coding: θ = 0 (black), θ = 2 π/ θ = − π/ In Fig. 2, we show the typical evolution snapshots for T = 0 .
2. The snapshots correspond to t = 10 MCS,and disorder amplitudes (cid:15) = 0 , ,
2. The three colors de-note small angle-windows, as specified in the caption. Ajunction point of the three colors corresponds to a vortexor anti-vortex, depending on the direction of rotation.These plots show an increase in defect density with dis-order, corresponding to slowing down of domain growth.In Fig. 3, we show vector plots corresponding to Fig. 2,in which we draw a unit vector for each spin S i = (b) ε = 1 (c) ε = 2 xy (a) ε = 0 FIG. 3: Vector plots for { θ i } -configurations in Fig. 2. Ateach lattice site i , we draw a vector corresponding to S i =(cos θ i , sin θ i ). For a better view, we show only a 32 cornerof the 512 lattice. Solid circles denote vortices, and solidtriangles denote antivortices. (cos θ i , sin θ i ). For a better visualization of vectors, wehave shown only a small portion of the lattice in Fig. 2.Vortices and anti-vortices are marked by solid circles andtriangles, respectively. These are characterized by cal-culating the net change in spin direction while movingclockwise on a square plaquette. A vortex is identified ifa spin rotates through 2 π , and an anti-vortex is identifiedif a spin rotates through − π . r/R C ( r , t ) ε = 0ε = 1ε = 2 -1 kR -2 -1 S ( k , t ) R - ε = 0ε = 1ε = 2 t = 10 MCS t = 10 MCS(a) (b) -4 FIG. 4: (a) Scaled correlation functions, C ( r, t ) vs. r/R ( t ),for the evolution of the d = 2 RBXYM after a quench to T = 0 .
5. We show data for t = 10 MCS, and (cid:15) = 0 , , S ( k, t ) R ( t ) − vs. kR , for thedata sets in (a). The solid curves in (a) and (b) denote theBray-Puri-Toyoki (BPT) function in Eq. (14) for n = 2, andits Fourier transform, respectively. The line of slope − S ( k, t ) ∼ k − ( d + n ) for d = n = 2. In Fig. 4, we plot the scaled forms of (a) the correlationfunction, C ( r, t ) vs. r/R ; and (b) the structure factor, S ( k, t ) R − vs. kR . The data sets correspond to differentvalues of (cid:15) . We have confirmed (not shown here) that thedata sets for a fixed value of (cid:15) and different times showa good data collapse. Thus, dynamical scaling holds foreach value of (cid:15) . In Fig. 4, we check for the robustness ofthis scaling function by plotting scaled data at t = 10 MCS for (cid:15) = 0 , ,
2. The scaling functions are seen tobe independent of the disorder amplitude. This featurewas first seen in simulations of ordering kinetics in theRBIM [23, 24], and was referred to as super-universality(SU). We will shortly see that the SU property does notapply to the two-time function A ( t, t w ).The solid curve in Fig. 4(a) is a plot of the Bray-Puri-Toyoki (BPT) function [21, 43] for n = 2. As mentionedearlier, BPT obtained the scaling function f ( x ) for or-dering dynamics of the O ( n ) model: f BPT ( r/R ) = nγ π (cid:20) B (cid:18) n + 12 , (cid:19)(cid:21) F (cid:18) ,
12 ; n + 22 ; γ (cid:19) , (14)where γ = exp( − r /R ). In Eq. (14), B ( x, y ) ≡ Γ( x )Γ( y ) / Γ( x + y ) is the beta function, and F ( a, b ; c ; z )is the hypergeometric function. In Fig. 4(b), the solidcurve is the Fourier transform of the BPT function, andthe line of slope − S ( k, t ) ∼ k − ( d + n ) for d = 2 , n = 2. R(t)/R(t w ) A ( t , t w ) ε = 0ε = 1ε = 2 FIG. 5: Plot of the autocorrelation function, A ( t, t w ) vs. R ( t ) /R ( t w ), for waiting time t w = 10 MCS, and (cid:15) = 0 , , In Fig. 5, we examine the scaling properties of A ( t, t w ).First, for a fixed value of (cid:15) , we superpose data for A ( t, t w )vs. R ( t ) /R ( t w ) from different times (not shown here).These data sets show a good scaling collapse. In Fig. 5,we plot the corresponding data sets for t w = 10 MCSand (cid:15) = 0 , , A ( t, t w ). The decay exponent λ is determinedfrom the asymptotically linear portion of the log-log plotin Fig. 5. Clearly, λ is larger for higher values of (cid:15) . Asimilar observation was made earlier in the context of theRBIM with nonconserved kinetics [27, 28].The most important characteristic of a coarsening pro-cess is the growth law of R ( t ). In Figs. 2 and 3, we haveseen that the evolution occurs via the annihilation of vor- tices and anti-vortices with typical size R ( t ). It is usefulto review the arguments for the growth law in the pureXY model before examining data for the disordered case.Consider a single vortex-antivortex pair separated bya distance R , with a as the dimension of the vortex-core. For a n -component vector model in d dimen-sions, the topological defects lie on a surface of dimension d − n . Therefore, the volume of the defect-core scales as R d − n [38, 46]. For the XY model ( n = 2) in d = 2, thecore volume is a dimensionless constant. The defect pairenergy E p ( R ) ∼ ln( R/a ) [38, 46, 47]. Thus, the drivingforce [ F ( R ) = − dE p /dR ] per unit core-volume, which isresponsible for the annihilation of the vortex-antivortexpair, scales as F ( R ) ∼ − /R .The annihilation time of the defect pair is governed bythe vortex mobility µ , which depends logarithmically onthe pair separation, i.e., µ ∼ [ln( R/a )] − [38, 46, 47]. Asthe mobility is related to the velocity via v = µF , wehave dRdt ∼ − R ln( R/a ) . (15)For a pair separated by a large distance R (cid:29) a , in-tegrating this equation gives the annihilation time t ∼ R ln( R/a ). This can be inverted to obtain R ∼ (cid:20) t ln( t/a ) (cid:21) / . (16)Eq. (16) yields the growth law R ( t ) ∼ ( t/ ln t ) / for thepure XY model in d = 2.What is the effect of random-bond disorder on thisgrowth law? As a reference point, it is useful to recallthe scenario for the RBIM [25, 27, 28]. In that case,coarsening interfaces are trapped by disorder sites withenergy barriers which have a power-law dependence onthe length scale R : E B ( R ) ∼ R ϕ , where ϕ is the bar-rier exponent. This yields an asymptotically logarithmicgrowth regime: R ( t ) ∼ (ln t ) /ϕ , which is preceded bya power-law regime where the exponent depends on thedisorder amplitude. For the d = 2 RBXYM, Fig. 6(a)shows the plot of R ( t ) vs. t/ ln t on a log-log scale fordifferent (cid:15) -values. This plot is motivated by the loga-rithmic correction in the domain growth law for the purecase. The dashed line denotes the power-law growth forthe pure-case: R ( t ) ∼ ( t/ ln t ) / [39, 46, 48]. Figure 6(a)shows that the presence of disorder slows down domaingrowth. The data sets in Fig. 6(a) suggest a power-law(over three decades of t ) with a disorder-dependent ex-ponent: R ( t ) ∼ (cid:18) t ln t (cid:19) φ ( (cid:15) ) (cid:39) (cid:18) t ln t (cid:19) / ¯ z ( (cid:15) ) . (17)Before proceeding, we should stress that the log-log plotof R ( t ) vs. t on the same time-window is also consistentwith a power-law behavior. The only difference fromFig. 6(a) is that the effective exponent φ ( (cid:15) ) is reduceddue to the logarithmic correction. We need at least fivedecades of data to differentiate between t ¯ φ and ( t/ ln t ) φ on a log-log scale. t/ln t R ( t ) ε= 0ε= 1.0ε= 1.5ε= 2.0 t z e ff ε= 0 ε= 0.5ε= 1.0ε= 1.5ε= 2.0 (b)(a) (t/ln t) FIG. 6: (a) Plot of R ( t ) vs. t/ ln t on a log-log scalefor the specified values of (cid:15) . The dashed line is of slope0.5, which indicates the growth law for the pure case: R ( t ) ∼ ( t/ ln t ) / . (b) Plot of the effective exponent, z eff = [ d (ln R ) /d (ln[ t/ ln t ])] − vs. t , for (cid:15) = 0 , . , . , . , . z ( (cid:15) ). For a quantitative study of the growth law, we deter-mine the effective growth exponent, defined as1 z eff = d [ln R ( t )] d [ln ( t/ ln t )] . (18)In Fig. 6(b), we plot z eff vs. t for the data in Fig. 6(a).This plot clearly shows an extended flat regime upto thetime-scale (10 MCS) of our simulation. The dashed linesdenote the corresponding values of the effective exponent z ( (cid:15) ), which are specified in Table I. This power-law be-havior is consistent with the RBIM results at intermedi-ate times [25, 27, 28]. In the RBIM studies of Lippielloet al. [27, 28], there is an upward curvature of the z eff vs. R plot at late times, signaling the onset of the log-arithmic regime. We do not clearly see this signature inFig. 6(b) for the d = 2 RBXYM. Clearly, even longer sim-ulations are needed to determine whether the RBXYMshows a crossover to an asymptotic logarithmic regime.This may be conjectured, as we expect the trapping en-ergy of a vortex in the RBXYM to scale with the vortexsize.In the early papers of Paul et al. [25] on the RBIM,it was argued that ¯ z eff scales linearly with the disorderamplitude (cid:15) . This is a consequence of energy barrierswhich scale logarithmically with the length scale R ( t ).In Fig. 7, we plot z vs. (cid:15) . We see that z increases some-what faster than linearly [49, 50] – a best fit to the datasuggests z = 2 . . (cid:15) . . B. RBXYM in d = 3 Next, we briefly present results for the d = 3 RBXYM.As in the d = 2 case, we first determine the transitiontemperature T c ( (cid:15) ). Then, we study phase ordering kinet-ics after quenching the system below T c ( (cid:15) ). ε z FIG. 7: Plot of the disorder-dependent growth exponent z ( (cid:15) )vs. (cid:15) . The solid line is the best power-law fit: z = 2 . . (cid:15) . . Estimation of T c To determine T c ( (cid:15) ), we perform simulations on a simplecubic lattice of linear sizes L = 16, 24 and 32. Afterequilibration, data for m and m are thermally averagedover 10 MCS. We further average over 100 independent { J ij } -configurations. Figure 8 shows the plot of U vs. T for (cid:15) = 1.0. As before, T c is determined from theintersection of Binder-cumulant curves for different L .The T c -values for various (cid:15) are given in Table II, andplotted in the inset of Fig. 8. For the pure case ( (cid:15) =0), we obtain T c (cid:39) . (cid:15) , T c decreases to T c ( (cid:15) = 2) (cid:39) . d = 3, T c does notchange significantly with (cid:15) , in contrast to the d = 2 case. (cid:15) T c ± ± ± ± ± T c ( (cid:15) ) for the d = 3 RBXYM. ε = 1.0 U T L = 16L = 24L = 32 T c ε FIG. 8: Plot of U ( T, L ) vs. T for the d = 3 RBXYM on cubiclattices of size L = 16 , ,
32. We show data for (cid:15) = 1 .
0. Inthe inset, we plot T c ( (cid:15) ) vs. (cid:15) . The numerical values of T c ( (cid:15) )are provided in Table II. Coarsening dynamics
Let us now discuss numerical results for domain growthin the d = 3 RBXYM. The simulations are performed oncubic lattices of size 128 . The system is quenched to T = 1 . < T c ( (cid:15) ) at t = 0 MCS, and evolved upto t =10 MCS. The statistical data presented here is averagedover 10 independent realizations of disorder. (c) ε = 2(b) ε = 1(a) ε = 0 X YZ
FIG. 9: Evolution snapshots of the d = 3 RBXYM for atemperature quench to T = 0 .
5. The lattice size is 128 . Thedefects (vortex and anti-vortex strings) are shown at t = 10 MCS, and for (cid:15) = 0 , , Figure 9 shows the typical defect configurations for theevolution of the d = 3 RBXYM system. The relevantdefects in this case are vortex and anti-vortex strings.Here, we show the string configurations at t = 10 MCSfor (cid:15) = 0 , ,
2. The defect density reduces as the sys-tem evolves, due to the annihilation of vortices and anti-vortices. In Fig. 9, we see that the defect density is higherfor larger (cid:15) , i.e., domain growth is slower for larger values of (cid:15) . -1 kR -2 -1 S ( k , t ) R - ε = 0ε = 1ε = 2 r/R C ( r , t ) ε= 0ε= 1ε= 2 t = 10 MCS t = 10 MCS(a) (b) -5 FIG. 10: (a) Scaled correlation functions, C ( r, t ) vs. r/R ( t ),for the d = 3 RBXYM after a quench to T = 1 .
0. We showdata sets for t = 10 MCS, and (cid:15) = 0 , ,
2. (b) Scaled struc-ture factors, S ( k, t ) R ( t ) − vs. kR , corresponding to the datasets in (a). The solid curves in (a) and (b) denote the BPTfunction [Eq. (14)] for n = 2 and its Fourier transform, re-spectively. The dashed line of slope − S ( k, t ) ∼ k − ( d + n ) for d = 3 and n = 2. Figure 10 shows the scaled correlation function [ C ( r, t )vs. r/R ( t )], and the scaled structure factor [ S ( k, t ) R ( t ) − vs. kR ], for the evolution of the d = 3 RBXYM. The datais shown for t = 10 MCS, and for (cid:15) = 0 , ,
2. The solidlines in Fig. 10(a) and (b) denote the BPT function for n = 2, and the corresponding Fourier transform, respec-tively. In Fig. 10(b), a dashed line of slope − S ( k, t ) ∼ k − ( d + n ) for d = 3and n = 2. The data collapse in Fig. 10 is excellent, con-firming the SU behavior of the scaling function in d = 3,similar to the d = 2 case. However, this SU does notextend to the autocorrelation function, again as in the d = 2 case. For the sake of brevity, we do not show thisdata here.The most important feature of the growth process isthe domain growth law. Let us first understand thegrowth law for the pure XY model. In d = 3, the de-fects are of dimension 1, i.e., strings, as seen in Fig. 9.Therefore, the defect pair energy E p ( R ) ∼ R ln( R/a ),where we have included a factor of R for the defect-corevolume. Then, the driving force per unit defect-core vol-ume is F ( R ) ∼ − ln( R/a ) /R . The relation v = µF yields dRdt ∼ R , (19)so that R ( t ) ∼ t / .In Fig. 11, we plot R ( t ) vs. t for (cid:15) = 0 , . , . , . R ( t ) ∼ t / . We see that the growth law for the disor-dered cases follows the pure case for a while, and thenbecomes slower at late times. To understand the natureof the asymptotic growth law, we define the effective ex-ponent as 1 z eff = d [ln R ( t )] d [ln t ] . (20) t R (t) ε= 0ε= 0.5ε= 1.5ε= 2.0 t FIG. 11: Plot of R ( t ) vs. t (on a log-log scale) for the d = 3RBXYM. The dashed line denotes the t / -growth for thepure case. In Fig. 12(a), we plot z eff vs. t for the data sets inFig. 11. At intermediate times, we have z eff (cid:39)
2, whichis consistent with the pure case. At late times, the datafor the disordered case shows an upward curvature, whichhas been understood by Corberi et al. [27–29] as a signalof logarithmic growth. For each disordered data set, weidentify z as the exponent value in the “flat regime”. t z e ff ε= 0ε = 0.5ε = 2.0 R/ λ z e ff - z ε = 0.5ε = 1.0ε = 2.0 (b)(a) FIG. 12: (a) Plot of the effective exponent, z eff vs. t . (b)Scaling collapse of z eff − z vs. R/λ ( (cid:15) ). The solid line is thebest power-law fit: z eff − z (cid:39) . R/λ ) . . In Fig. 12(b), we plot z eff − z vs. R/λ ( (cid:15) ), where λ ( (cid:15) )is a scaling variable. The late-stage data for different (cid:15) -values shows a reasonable collapse. The scaling functionis z eff − z (cid:39) b ( R/λ ) ϕ with b (cid:39) .
022 and φ (cid:39) .
16. Asshown by Corberi et al. [29], this suggests the logarithmic growth law Rλ (cid:39) (cid:20) ϕb ln (cid:18) tλ z (cid:19)(cid:21) /ϕ , (21)with 1 /ϕ (cid:39) . IV. SUMMARY AND DISCUSSION
Let us conclude this paper with a summary and dis-cussion of our results. We have undertaken a compre-hensive Monte Carlo (MC) study of domain growth inthe random-bond XY model (RBXYM) in d = 2 ,
3. Re-call that the pure XY model exhibits the
Berezinskii-Kosterlitz-Thouless (BKT) transition in d = 2, and showsa regular phase transition for d = 3. To the best ofour knowledge, this is the first study of the effects ofquenched disorder on coarsening in systems with a vec-tor order parameter. We find that the coarsening sce-nario observed in scalar systems by Corberi et al. [27, 28]applies in the present case also.We first summarize our d = 2 study. We used theBinder-cumulant method to determine T BKT as a func-tion of the disorder amplitude (cid:15) , where the random bonds J ij ∈ [1 − (cid:15)/ , (cid:15)/ (cid:15) ≤
2, so thatthere is no frustration in the system. We found that T BKT decreased substantially as (cid:15) was increased. We then un-dertook a coarsening study by quenching an initially dis-ordered system to
T < T
BKT ( (cid:15) ), where the equilibriumstate is characterized by quasi-long-range-order (QLRO).Domain growth is characterized by the annealing of vor-tices and antivortices, which are point defects in d = 2.The disorder sites trap these point defects and slow downcoarsening. The evolution morphology is characterizedby the spatial correlation function C ( r, t ) and the auto-correlation function A ( t, t w ). The scaling form of C ( r, t )[or its Fourier transform, the structure factor S ( k, t )] isnot affected by the presence of disorder. However, thisdoes not hold good for A ( t, t w ). On the time-scales ofour simulation, the growth law exhibits a power-law be-havior with an exponent which depends on the disorderamplitude. This is analogous to the intermediate-timebehavior for coarsening in the RBIM [25, 27]. We donot see a logarithmic growth regime in the RBXYM, butexpect it to arise on even later time-scales than thosestudied here ( t = 10 MCS).Next, we summarize our d = 3 study. In this case,the low-temperature [ T < T c ( (cid:15) )] state is characterizedby LRO. Again, we use the Binder cumulant techniqueto determine T c ( (cid:15) ). In d = 3, the critical temperaturedoes not show a strong dependence on (cid:15) . Our results fordomain growth in the d = 3 RBXYM are analogous tothose for d = 2, with one major difference. In d = 3, wedo not see an extended intermediate regime of power-lawgrowth with an (cid:15) -dependent exponent. Further, we see aclear signature of logarithmic growth in the asymptoticregime.Clearly, experimental systems always contain bothquenched and mobile impurities. Therefore, it is impor-tant to have a good theoretical understanding of domaingrowth with quenched disorder. This now exists, as a re-sult of our work and that of several other groups. How-ever, experimental studies have not kept pace with thesedevelopments. We urge experimentalists to undertakecareful experiments to confirm (or contradict) the theo-retical scenario. Acknowledgments
SC thanks grant no. 09/080(0897)/2013-EMR-I,CSIR, India. RP thanks grant no. 03(1414)/17/EMR-II,CSIR, India. [1] D.J. Bishop and J.D. Reppy, Phys. Rev. Lett. , 1727(1978).[2] K. Epstein, A.M. Goldman and A.M. Kadin, Phys. Rev.Lett. , 534 (1981).[3] A.F. Hebard and A.T. Fiory, Phys. Rev. Lett. , 291(1980).[4] M.R. Beasley, J.E. Mooij and T.P. Orlando, Phys. Rev.Lett. , 1165 (1979).[5] A.N. Pargellis, S. Green and B. Yurke, Phys. Rev. E ,4250 (1994).[6] K.J. Strandburg, Rev. Mod. Phys. , 161 (1988).[7] M. Hasenbusch and S. Meyer, Phys. Lett. B , 238(1990).[8] A.P. Gottlob and M. Hasenbusch, Physica A , 593(1993).[9] V.L. Berezinskii, Sov. Phys. JETP , 493 (1970).[10] J.M. Kosterlitz and D.J. Thouless, J. Phys. C , 1181(1973).[11] J.M. Kosterlitz, J. Phys. C , 1046 (1974).[12] T. Surungan and Y. Okabe, Phys. Rev. B , 184438(2005).[13] S.A. Leonel, P.Z. Coura, A.R. Pereira, L.A.S. Ml andB.V. Costa, Phys. Rev. B , 104426 (2003).[14] J.J. Alonso, J. Magn. Magn. Mater. , 1330 (2010).[15] A.K. Murtazaev and A.B. Babaev, J. Magn. Magn.Mater. , 2630 (2009).[16] A.B. Harris, J. Phys. C , 1671 (1974).[17] G.M. Wysin, A.R. Pereira, I.A. Marques, S.A. Leoneland P.Z. Coura, Phys. Rev. B , 094418 (2005).[18] G. Kohring, R.E. Shrock and P. Willst, Phys. Rev. Lett. , 1358 (1996).[19] S. Puri and C. Roland, Phys. Lett. A , 500 (1990).[20] S. Puri, Phys. Lett. A , 211 (1992).[21] A.J. Bray and S. Puri, Phys. Rev. Lett. , 2670 (1991).[22] B. Yurke, A.N. Pargellis, T. Kovacs and D.A. Huse, Phys.Rev. E , 1525 (1993).[23] S. Puri, D. Chowdhury and N. Parekh, J. Phys. A ,L1087 (1991).[24] S. Puri and N. Parekh, J. Phys. A , 4127 (1992); J.Phys. A , 2777 (1993).[25] R. Paul, S. Puri and H. Rieger, Europhys. Lett. , 881(2004); Phys Rev. E , 061109 (2005).[26] S. Puri, Phase Transitions , 469 (2004).[27] E. Lippiello, A. Mukherjee, S. Puri and M. Zannetti, Eu-rophys. Lett. , 46006 (2010).[28] F. Corberi, E. Lippiello, A. Mukherjee, S. Puri and M.Zannetti, J. Stat. Mech., P03016 (2011).[29] F. Corberi, E. Lippiello, A. Mukherjee, S. Puri and M.Zannetti, Phys. Rev. E , 021141 (2011).[30] U. Wolff, Phys. Rev. Lett. , 361 (1989). [31] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H.Teller and E. Teller, J. Chem. Phys. , 1087 (1953).[32] J. Tobochnik and G.V. Chester, Phys. Rev. B , 3761(1979).[33] J.F. Fernandez, M.F. Ferreira and J. Stankiewicz, Phys.Rev. B , 292 (1986).[34] M. Ferer, M.A. Moore and M. Wortis, Phys. Rev. B ,5205 (1973).[35] K. Binder, Z. Phys. B , 119 (1981).[36] D. Loison, J. Phys. Condens. Matter , L401 (1999).[37] S. Puri and V. Wadhawan (eds.), Kinetics of Phase Tran-sitions (CRC Press, Boca Raton, 2009).[38] A.J. Bray, Adv. Phys. , 357 (1994).[39] F. Rojas and A.D. Rutenberg, Phys. Rev. E , 212(1999).[40] R.E. Blundell and A.J. Bray, Phys. Rev. E , 4925(1994).[41] M. Zannetti, in Ref. [37].[42] D.S. Fisher and D.A. Huse, Phys. Rev. B , 373 (1988).[43] H. Toyoki, Phys. Rev. B , 1965 (1992).[44] G. Porod, in Small-Angle X-Ray Scattering , O. Glatterand O. Kratky (eds.) (Academic Press, New York, 1982).[45] Y. Oono and S. Puri, Mod. Phys. Lett. B , 861 (1988).[46] A.D. Rutenberg and A.J. Bray, Phys. Rev. E , 5499(1995).[47] A.J. Bray, Phys. Rev. E , 103 (2000).[48] A.J. Bray and A.D. Rutenberg, Phys. Rev. E , R27(1994).[49] M. Henkel and M. Pleimling, Europhys. Lett. , 561(2006).[50] M. Henkel and M. Pleimling, Phys. Rev. B78