Static and dynamic simulation in the classical two-dimensional anisotropic Heisenberg model
SStatic and dynamic simulation in the classical two-dimensional anisotropic Heisenberg model
J. E. R. Costa and B. V. Costa
Departamento de Fı´sica, Instituto de Cieˆncias Exatas, Universidade Federal de Minas Gerais, Caixa Postal 702,30161-970 Belo Horizonte, Brazil (cid:126)
Received 21 December 1995 (cid:33)
By using a simulated annealing approach, Monte Carlo and molecular-dynamics techniques we have studiedstatic and dynamic behavior of the classical two-dimensional anisotropic Heisenberg model. We have obtainednumerically that the vortex developed in such a model exhibit two different behaviors depending if the valueof the anisotropy (cid:108) lies below or above a critical value (cid:108) c . The in-plane and out-of-plane correlation functions( S xx and S zz ) were obtained numerically for (cid:108) (cid:44) (cid:108) c and (cid:108) (cid:46) (cid:108) c . We found that the out-of-plane dynamicalcorrelation function exhibits a central peak for (cid:108) (cid:46) (cid:108) c but not for (cid:108) (cid:44) (cid:108) c at temperatures above T KT . (cid:64) S0163-1829 (cid:126) (cid:33) (cid:35) I. INTRODUCTION
In the last ten years much attention has been dedicated tothe study of the classical two-dimensional anisotropicHeisenberg model (cid:126)
CTDAHM (cid:33) . Such attention is groundedin the fact that a large variety of models may be mapped inthe CTDAHM. Some examples are superfluid, superconduct-ing films and roughening transitions.
The CTDAHM canbe described by the Hamiltonian H (cid:53)(cid:50) J (cid:40) (cid:94) i , j (cid:38) (cid:126) S ix S jx (cid:49) S iy S jy (cid:49) (cid:108) S iz S jz (cid:33) , (cid:126) (cid:33) where J (cid:46) (cid:108) is an anisotropy, S (cid:97) are clas-sical spin components defined on the surface of a unit sphere,and (cid:94) i , j (cid:38) are to be understood as first neighbor indices in asquare lattice. For (cid:108) (cid:53) XY model (cid:126) that should not be confused with the planar model, that hasonly two spin components (cid:33) and for (cid:108) (cid:53) (cid:126) (cid:33) are well understood in the limit (cid:108) (cid:53) and Kosterlitz and Thouless. It has a phasetransition of infinite order at temperature T KT , namedKosterlitz-Thouless phase transition, with no long-range or-der, which is characterized by a vortex-anti-vortexunbinding. A vortex (cid:126) antivortex (cid:33) is a topological excitationwhere spins on a closed path around the excitation precessby 2 (cid:112) ( (cid:50) (cid:112) ) in the same direction. When T KT is reachedfrom above the correlation length (cid:106) and magnetic suscepti-bility (cid:120) behave as (cid:106)(cid:59) e b (cid:106) t (cid:50) (cid:110) , (cid:120)(cid:59) e b (cid:120) t (cid:50) (cid:104) ,where t (cid:53) ( T (cid:50) T KT )/ T KT , with b (cid:106) , b (cid:120) (cid:59) (cid:110) (cid:53) and (cid:104) (cid:53) .When the system goes through T KT a vortex-anti-vortex un-binding process occurs increasing the entropy in the system.For (cid:108) (cid:44) XY model.The variation of T KT with (cid:108) is experimentally important.Both analytical as well simulational results show that T KT depends weakly on (cid:108) , except for (cid:108)(cid:59) T KT !
0. The dynamical behavior of the XY model was studiedtheoretically with different predictions for the nature of theneutron-scattering function. Villain analyzed the model inthe low- T limit in the harmonic approximation. He foundthat the in-plane S xx correlation function behaves as S xx (cid:126) q , (cid:118) (cid:33) (cid:59) (cid:117) (cid:118) (cid:50) (cid:118) q (cid:117) (cid:50) (cid:49) (cid:104) /2 with (cid:104) (cid:53) T KT , (cid:118) q is the magnon frequency, and S xx ( q , (cid:118) ) is obtained by Fourier transforming the space-timecorrelation function.In a hydrodynamic description, without vortex contribu-tion, Nelson and Fisher found the in-plane correlation func-tion S xx (cid:126) q , (cid:118) (cid:33) (cid:59) q (cid:50) (cid:104) (cid:67) (cid:83) (cid:118) q (cid:68) ,where (cid:67) (cid:126) y (cid:33) (cid:59) (cid:117) (cid:50) y (cid:117) (cid:50) (cid:104) around the spin-wave peak and S xx (cid:126) q , (cid:118) (cid:33) (cid:59)(cid:118) (cid:104) (cid:50) for large values of (cid:118) / q .Both, Villain and Nelson and Fisher predicted a narrowspin-wave peak to the out-of-plane correlation function S zz .By performing a low-temperature calculation which includesout-of-plane contributions, Menezes et al. found a spin-wave peak similar to that of Nelson and Fisher. In addition tothe spin-wave peak they found a central logarithmically di-vergent peak.At T KT vortex-anti-vortex pairs start to unbind, and vorti-ces may diffuse through the system leading to a strong cen-tral peak, at the same time the stiffness jumps to zero mean-ing that the spin-wave peaks disappear. Mertens et al. have proposed a phenomenologicalmodel to calculate the correlation function above T KT . Theirapproach was based on a well succeeded ballistic approachto the one-dimensional soliton dynamics in magnetic spin PHYSICAL REVIEW B 1 JULY 1996-IIVOLUME 54, NUMBER 2540163-1829/96/54 (cid:126) (cid:33) /994 (cid:126) (cid:33) /$10.00 994 © 1996 The American Physical Society hains. They found a Lorentzian central peak for S xx and aGaussian central peak for S zz .In a recent work, Evertz and Landau using spin-dynamics techniques have performed a large-scale computersimulation of the dynamical behavior of the XY model. Theyfound an unexpected central peak in the S xx correlation func-tion for temperatures well below T KT , and their results arenot adequately described by above theories.From the experimental point of view, Wiesler et al. studied a very anisotropic material that is expected to have a XY behavior. For T (cid:44) T KT they found spin-wave peaks but itis not clear if a central peak is present. Above T KT theyfound the expected central peak in the in-plane correlationfunction and the out-of-plane function exhibits damped spinwaves. More recently Song performed Y NMR experi-ments on a powder sample of type-II superconductorYBa Cu O (cid:50) (cid:100) around the Kosterlitz-Thouless temperaturein a magnetic field. In their experiment was observed onlylocal vortex motion and diffusive behavior seems to be ab-sent.Since recent experimental works and numerical onespresent results that are not in accordance with existent theo-ries, mainly in aspects related to the central peak and vortexmotion, more work to investigate this subject is justifiable. Infact, we found some results which are in agreement withSong’s observation about vortex motion, that we present atthe conclusion.In this work we present some Monte Carlo and molecular-dynamics simulation in the CTDAHM defined by the Hamil-tonian (cid:126) (cid:33) for (cid:108)(cid:222)
0. In Sec. II we discuss the effect of finiteanisotropy to both vortex components, in-plane and out-of-plane. A limiting value (cid:108) c for the anisotropy is numericallyobtained. For (cid:108) (cid:44) (cid:108) c the most stable spin configuration is a planar one. For (cid:108) (cid:46) (cid:108) c it develops a large out-of-plane S z component near the center of the vortex. The behavior of S z as a function of (cid:108) is obtained. In Sec. III we calculate, byusing Monte Carlo (cid:126) MC (cid:33) and molecular-dynamics simulationthe in-plane correlation function S xx and out-of-plane S zz , fortwo values of (cid:108) , (cid:108) (cid:44) (cid:108) c and (cid:108) (cid:46) (cid:108) c . Finally in Sec. IV wepresent our conclusions pointing out the relevant aspects in-troduced by a finite anisotropy. II. STATIC VORTEX SOLUTIONS
In this section we discuss the static vortex solutions to theHamiltonian (cid:126) (cid:33) for arbitrary 0 (cid:60)(cid:108) (cid:44)
1, firstly in the con-tinuum limit and then we obtain numerical solutions to thediscrete case.The classical spin vector may be parametrized by thespherical angles (cid:81) n and (cid:70) n as S (cid:87) n (cid:53) (cid:126) cos (cid:81) n cos (cid:70) n ,cos (cid:81) n sin (cid:70) n ,sin (cid:81) n (cid:33) . (cid:126) (cid:33) In the continuum approximation for the Hamiltonian (cid:126) (cid:33) , (cid:70) n and S nz (cid:53) sin (cid:81) n constitute a pair of canonically conjugatevariables, which allow us to write the equations of motion (cid:81) ˙ n (cid:53) (cid:93) H / (cid:93)(cid:70) n cos (cid:81) n , (cid:70) ˙ n (cid:53) (cid:93) H / (cid:93)(cid:81) n cos (cid:81) n . (cid:126) (cid:33) If S (cid:97) has an expansion like S (cid:97) (cid:126) x n (cid:54) a , y n (cid:33) (cid:53) (cid:40) k (cid:53) (cid:96) (cid:83) (cid:54) a ddx n (cid:68) k S (cid:97) (cid:126) x n , y n (cid:33) , (cid:126) (cid:33) where a is the lattice constant, we may rewrite (cid:126) (cid:33) in a con-tinuum version as (cid:81) ˙ (cid:53) J (cid:64) cos (cid:70)(cid:185) (cid:126) cos (cid:81) sin (cid:70) (cid:33) (cid:50) sin (cid:70)(cid:185) (cid:126) cos (cid:81) cos (cid:70) (cid:33)(cid:35) (cid:126) (cid:33) (cid:50) cos (cid:81)(cid:70) ˙ (cid:53) J (cid:36) (cid:108) cos (cid:81) (cid:64) sin (cid:70)(cid:185) sin (cid:81) (cid:49) cos (cid:70)(cid:185) (cid:126) sin (cid:81) cos (cid:70) (cid:33)(cid:35) (cid:50) sin (cid:81) (cid:64) sin (cid:70)(cid:185) (cid:126) cos (cid:81) sin (cid:70) (cid:33) (cid:49) cos (cid:70)(cid:185) (cid:126) cos (cid:81) cos (cid:70) (cid:33)(cid:35) (cid:37) , (cid:126) (cid:33) where we kept term up to order a in the expansion.Single static vortex solution may be obtained from Eqs. (cid:126) (cid:33) and (cid:126) (cid:33) with the appropriate boundary conditions S (cid:97) (cid:126) x , y (cid:33) (cid:53) S (cid:97) (cid:126) (cid:50) x , y (cid:33) , lim y ! (cid:54) (cid:96) , S (cid:97) (cid:126) x , y (cid:33) (cid:53) S (cid:97) (cid:126) x , (cid:50) y (cid:33) , lim x ! (cid:54) (cid:96) , (cid:126) (cid:33) where (cid:97) (cid:53) x , y . One solution can be readily seen as (cid:81) (cid:53) (cid:70) (cid:53) arctan yx , (cid:126) (cid:33) which describes an in-plane vortex (cid:126) KT (cid:33) .We expect that the expression given by (cid:126) (cid:33) should be astable solution for moderate values of the anisotropy (cid:108) sinceas long as (cid:108) grows a smaller energy configuration may beachieved if the S z component develops a nonzero value nearthe vortex center. Unfortunately a complete analytical solu-tion with (cid:81)(cid:222) r ! r ! (cid:96) , where r is the distance fromthe center of the vortex, we may write approximate solu-tions: (cid:70) (cid:53) (cid:70) , (cid:126) (cid:33) sin (cid:81) (cid:53) (cid:72) (cid:50) Ar , if r ! B exp (cid:72) (cid:50) (cid:83) (cid:50) (cid:108)(cid:108) (cid:68) r (cid:74) , if r ! (cid:96) We do not really expect that the expressions given by (cid:126) (cid:33) and (cid:126) (cid:33) are good solutions for the discrete case in the limit r ! (cid:81) should be stronger there. It isstraightforward to calculate the contributions to the energy inthe continuum limit due to both configurations, Eqs. (cid:126) (cid:33) and (cid:126) (cid:33) , they are dominated by a ln r term. We will see below thatthis behavior persists up to values of (cid:108) quite near (cid:108) (cid:53)
1. Atthe Heisenberg limit the energy has a completely different
54 995STATIC AND DYNAMIC SIMULATION IN THE CLASSICAL . . . ehavior, the relevant excitations become instantons ratherthan vortices which energy is a constant. In order to solve the discrete equations of motion given by (cid:126) (cid:33) in the static case, we use a simulated annealing approach,which was shown to be quite powerful in determining abso-lute minimum in spin-glass models. We minimize theHamiltonian (cid:126) (cid:33) using diagonally antiperiodic boundary con-ditions to the x and y spin components and diagonally peri-odic one to the z component S i ,0 (cid:97) (cid:53) kS L (cid:50) i , L (cid:97) , (cid:126) (cid:33) S j (cid:97) (cid:53) kS L , L (cid:50) j (cid:97) , FIG. 1. In-plane spin components in a square lattice of linearlength L (cid:53) T min (cid:53) (cid:50) . The values ofanisotropy are (cid:126) a (cid:33) (cid:108) (cid:53) (cid:126) b (cid:33) (cid:108) (cid:53) (cid:126) c (cid:33) (cid:108) (cid:53) L (cid:53) here k (cid:53)(cid:50) (cid:97) (cid:53) x , y and k (cid:53) (cid:97) (cid:53) z and 0 (cid:60) i , j (cid:60) L .These boundary conditions are enough to create an odd num-ber of vortex (cid:126) antivortex (cid:33) in the system and the ground statehas only one vortex (cid:126) antivortex (cid:33) , so that we can find numeri-cally the stable vortex solution (cid:126) in-plane or out-of-plane (cid:33) foreach value of (cid:108) anisotropy.We started the iteration by using the exact continuum vor-tex solution given by (cid:126) (cid:33) in a square lattice of linear size L (cid:53) L (cid:53)
400 with no signifi-cant change in the final results. The iterative simulated an-nealing process is implemented starting at temperature T init (cid:53) T min (cid:53) (cid:50) . Steps in tempera-ture, (cid:68) T , are chosen so that the acceptation rate is main-tained in 50%.Results for S x , y and S z are shown in Figs. 1 and 2 forsome values of the anisotropy (cid:108) . Two types of behavior arequite clear. There is a region of (cid:108) where the stable solution is S z (cid:53) S z (cid:222) (cid:108)(cid:39)(cid:108) c the S z component is appre-ciable only inside a small region near the vortex core. Aslong as (cid:108) increases, S z becomes larger and the vortex coregrows. Because of the S z symmetry (cid:49) S z and (cid:50) S z areequivalent solutions. In order to determine the critical valueof (cid:108) we have obtained a series of solutions for S z and S x , y for different values of (cid:108) . By measuring the S z component at x (cid:53) y (cid:53) a (cid:126) where a is a lattice constant (cid:33) we determined whereit goes to zero. A plot of such results is shown in Fig. 3. The S z component goes to zero at (cid:108) c (cid:46) (cid:54) S z as a function of (cid:108) is well described by a function( S z ) (cid:59) ( (cid:108) (cid:50) (cid:108) c ) (cid:110) with (cid:110) (cid:53) (cid:54) (cid:108) c does not mean that the system undergoes aphase transition, but just that the vortex develops an out-of-plane component from this value of (cid:108) . We have also calcu-lated the energy as a function of the distance to the center ofthe vortex for some values of (cid:108) . Energy curves obtained bysimulation are shown in Fig. 4 as circles, diamonds, andsquares for (cid:108) (cid:53) (cid:126) (cid:33) . The inset shows a log-linear plot of energy as afunction of ln2 r . The deviation from the logarithmic behav-ior is clear. FIG. 3. Out-of-plane squared component ( S z ) as a function of (cid:108) . Circles are simulation points (cid:126) from numerical vortex solution at T (cid:39)
0) and the solid line is the fit using ( S z ) (cid:59) ( (cid:108) (cid:50) (cid:108) c ) (cid:110) .FIG. 4. Vortex energy (cid:126) measured in units of J ) as a function ofthe vortex diameter 2 r . (cid:126) r is measured in units of lattice spacing. (cid:33) Energy curves for (cid:108) (cid:53) (cid:108) (cid:53) (cid:108) (cid:53) (cid:126) (cid:33) . The inset shows a log-linear plot of energy as afunction of ln2 r . FIG. 5. In-plane correlation function S xx ( q , (cid:118) ) as a function of (cid:118) for (cid:108) (cid:53) (cid:126) a (cid:33) T (cid:53) J / k ; (cid:126) b (cid:33) T (cid:53) J / k . Values for q are shown in the inset.54 997STATIC AND DYNAMIC SIMULATION IN THE CLASSICAL . . . ow a natural question arises, how a finite anisotropychanges the dynamical correlation functions S xx and S zz ?Because the in-plane symmetry is not changed we do notexpect any drastic change in S xx . However, since for (cid:108) (cid:46) (cid:108) c the most stable vortex solution is for S z (cid:222) S zz will not be surprising. In thenext section we numerically calculate both S xx and S zz . III. DYNAMICAL CORRELATION FUNCTIONS
In the last section we observed a drastic change in the S z spin component when the anisotropy (cid:108) exceeds (cid:108) c . Be-cause the in-plane symmetry is not changed when (cid:108) goesthrough (cid:108) c we do not expect any drastic change in S xx . How-ever, since the most stable configuration changes suddenlyfrom S z (cid:53) S zz correlation function will not be surprising.In this section we present Monte Carlo–molecular-dynamic simulation results we carried out to obtain the cor-relation function S xx and S zz for two values of (cid:108) , below( (cid:108) (cid:53) (cid:108) (cid:53) (cid:51)
64 square lattice with peri-odic boundary conditions at temperature T (cid:53) T (cid:53) J / k which are below and above the Kosterlitz-Thouless temperature T KT .Equilibrium configurations were created at each tempera-ture using a Monte Carlo method which combines clusterupdates of in-plane spin components with Metropolis reori-entation. After each single cluster update, two Metropolissweeps were performed. The cluster update is essential at thelow-temperature region, since the critical slowing down issevere and it should not be possible to achieve thermody-namic equilibrium in a reasonable computer time using onlythe Metropolis algorithm. We have used in our simulation200 independent configurations discarding the first 5000 hy-brid sweeps for equilibration.Starting with each equilibrated configuration, the timespin evolution was determined from the coupled equations ofmotion for each spin ddtS (cid:87) i , j (cid:53) S (cid:87) i , j (cid:51) V (cid:87) i , j , (cid:126) (cid:33) where V (cid:87) (cid:53) J (cid:40) (cid:97) (cid:126) S i (cid:50) j (cid:97) (cid:49) S i , j (cid:50) (cid:97) (cid:49) S i (cid:49) j (cid:97) (cid:49) S i , j (cid:49) (cid:97) (cid:33) eˆ (cid:97) FIG. 6. Out-of-plane correlation function S zz ( q , (cid:118) ) as a functionof (cid:118) for (cid:108) (cid:53) (cid:126) a (cid:33) T (cid:53) J / k ; (cid:126) b (cid:33) T (cid:53) J / k . Values for q are shown in the inset. FIG. 7. In-plane correlation function S xx ( q , (cid:118) ) as a function of (cid:118) for (cid:108) (cid:53) (cid:126) a (cid:33) T (cid:53) J / k ; (cid:126) b (cid:33) T (cid:53) J / k . Values for q are shown in the inset.998 54J. E. R. COSTA AND B. V. COSTA nd (cid:97) (cid:53) x , y , z , eˆ x and eˆ y are unit vectors in the x and y direction, respectively. Equation (cid:126) (cid:33) was numerically inte-grated by using a fourth-order predictor-corrector method with a time step of (cid:100) t (cid:53) J (cid:50) . The maximum integrationtime was t max (cid:53) J (cid:50) . A few runs with t max (cid:53) J (cid:50) weredone with the same results for our purpose, giving the samephysical results. The numerical integration stability ischecked out verifying that the constants of motion (cid:126) energyand z magnetization (cid:33) remain constants with a relative varia-tion of less than 10 (cid:50) after 1200 time steps. To obtain S (cid:97)(cid:97) ( q , (cid:118) ) we first calculated the space-time correlation func-tions, S (cid:97)(cid:97) ( i (cid:50) j , t ) as (cid:94) S i (cid:97) (cid:126) (cid:33) S j (cid:97) (cid:126) r , t (cid:33) (cid:38) (cid:53) N (cid:40) i (cid:53) N (cid:40) j (cid:53) N S i (cid:97) (cid:126) (cid:33) S j (cid:97) (cid:126) t (cid:33) (cid:126) (cid:33) for time steps of size (cid:68) t (cid:53) J (cid:50) up to 0.9 t max and finallyaveraging over all configuration.By Fourier transformation in space and time we have ob-tained the neutron-scattering function S (cid:97)(cid:97) ( q , (cid:118) ). We restrictourselves to momenta q (cid:87) (cid:53) ( q ,0) and (0, q ) with q given by q (cid:53) n (cid:112) L , n (cid:53) L . Since these two directions are equivalent we averaged themtogether to get better statistical accuracy. The frequencyresolution of our results is determined by the time integrationcutoff ( (cid:53) t max ) which introduces oscillations into S (cid:97)(cid:97) ( q , (cid:118) ). To reduce the cutoff effects we introducedGaussian spatial and temporal functions replacing S (cid:97)(cid:97) ( r , t ) by S (cid:97)(cid:97) (cid:126) r , t (cid:33) e (cid:50) (cid:126) (cid:33)(cid:126) t (cid:100)(cid:118) (cid:33) e (cid:50) (cid:126) (cid:33) (cid:126) r (cid:100) q (cid:33) to compute S (cid:97)(cid:97) ( q , (cid:118) ). Cutoff parameters are (cid:68)(cid:118) (cid:53) (cid:68) q (cid:53) (cid:126) a (cid:33) and 5 (cid:126) b (cid:33) show S xx ( q , (cid:118) ) for T (cid:53) (cid:108) (cid:53) (cid:108) (cid:53)
0. At low T we have only spin-wave peaks and T (cid:46) T KT only central peaks are displaced. The out-of-plane S zz correlation function is shown in Fig. 6 to the same pa-rameters and only spin-wave peaks are observed. The inter-esting behavior comes up when we go through (cid:108) c . In Figs. 7and 8 we show the in-plane and out-of-plane correlationfunctions, respectively. To S xx we observed the same quali-tative behavior for (cid:108) (cid:44) (cid:108) c , however for S zz a very clear cen-tral peak is developed for T (cid:46) T KT . As commented before thesource of such a central peak seems to lie on the vortexstructure developed for (cid:108) (cid:46) (cid:108) c . IV. CONCLUSIONS
We have obtained numerically that the vortex developedin the CTDAHM exhibit very different behavior dependingif the value of the anisotropy (cid:108) lies below or above thecritical value (cid:108) c . For (cid:108) (cid:44) (cid:108) c the spin components lie prefer-entially in the XY plane, while for (cid:108) (cid:46) (cid:108) c the most stableconfiguration develops an out-of-plane component thatgrows with (cid:108) . We have shown that the out-of-plane dynami-cal correlation function has a central peak for (cid:108) (cid:46) (cid:108) c but notfor (cid:108) (cid:44) (cid:108) c . Theories developed so far did not describe cor-rectly the correlation function as discussed in Refs. 19, 21,and 22. In an earlier work (cid:126) Costa et. al (cid:33) suggested thatcentral peak might be due to a vortex-anti-vortex creation FIG. 8. Out-of-plane correlation function S zz ( q , (cid:118) ) as a functionof (cid:118) for (cid:108) (cid:53) (cid:126) a (cid:33) T (cid:53) J / k ; (cid:126) b (cid:33) T (cid:53) J / k . Values for q are shown in the inset. FIG. 9. Number of vortices ( N v ) as a function of time (cid:64) t (0.1 J (cid:50) ) (cid:35) .54 999STATIC AND DYNAMIC SIMULATION IN THE CLASSICAL . . . nnihilation process. We have calculated in our simulationsthe fluctuation of the number of vortex with time for allconfigurations, anisotropies and temperatures. Figure 9 is atypical plot of the number of vortex as a function of time.Below and above T KT the fluctuation of the number is verystrong. Pairs may annihilate at the position r (cid:87) on time t , re-appearing at r (cid:87) (cid:56) on t (cid:56) . This process may introduce the dy-namics to give the central peaks. This is in accordance withthe NMR results of Song in Ref. 22 who found only localvortex motion in his measurements and with the central peak for T (cid:44) T KT found by Evertz and Landau in the in-planecorrelation function. An analytical calculation using a Masterequation approach in order to incorporate the creation-annihilation process is now in progress. ACKNOWLEDGMENTS
We thank FAPEMIG and CNPq for financial support. Partof our computer simulations were carried out on the CrayYMP at CESUP (cid:126)
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