A Monte Carlo Study of the Specific Heat of Diluted Antiferromagnets

Size: px

Start display at page:

Download "A Monte Carlo Study of the Specific Heat of Diluted Antiferromagnets"

Rodger Edwards
6 years ago
Views:

1 A Monte Carlo Study of the Specific Heat of Diluted Antiferromagnets M. Staats and K. D. Usadel Theoretische Physik and SFB 166 Gerhard-Mercator-Universität Gesamthochschule Duisburg Duisburg, Germany Abstract The -dimensional diluted antiferromagnet in a magnetic field has a longe-range ordered state for sufficiently low temperatures and external fields. We study the irreversibilities at the phase transition from this state into the paramagnetic state at higher temperatures. Performing field cooling and zero field cooling simulation procedures we find a small irreversibility in the internal energy and the specific heat, although the behavior of the order parameter is irreversible and the phase transition is suppressed in the FC case. The specific heat shows a non-critical broad maximum above the transition temperature Tc. The results are compatible with recent experimental results obtained by Satooka et. al., whereas our interpretation ofthedataisdifferent. Keywords: Critical-point effects, specific heats, short-range order, Numerical simulation studies, Spin-glass and other random models

2 ii 1 Introduction The behavior of diluted antiferromagnets in a field (DAFF) is of both theoretical and experimental interest. The DAFF is expected to be in the same universality class as the Random Field Ising Model (RFIM) [1, ], or at least as the RFIM with a Gaussian distribution of the random fields [, 4]. For sufficiently low temperatures below a critical temperature T c (B) and external fields lower than a critical B c the system is in a long-range ordered antiferromagnetic state [5, 6]. When increasing the temperature, a phase transition into a paramagnetic state occurs at T c (B). Up to now, the nature of this phase transition is not completely understood, and the question of the universality class of the DAFF is not finally settled. Also, the value and even the sign of the critical exponent for the specific heat is not known. Experiments usually yield a positive or 0 [7, 8], whereas in simulations and other numerical works is negative [9, 10]. While, as shown in this work, it is possible to analyze simulation data of the order parameter with finite size scaling as in ordinary second order phase transitions, the experimental situation is not clear, e. g. concentration gradients of the dilution in the measured sample may influence the critical behavior and leave room for a wide range of interpretations. It is well known that because of the strong disorder in such systems equilibrium is not reached when cooling the system in an external field from the paramagnetic state. This is called the field cooling (FC) procedure. On the other hand, if the system is in the ground state at low temperatures, one can expect to measure equilibrium behavior when heating the system. In experiments this is done by first cooling the system in zero field (ZFC), so that the ordered antiferromagnetic state is reached before the external field is switched on. In simulations it is easy to prepare the system in such an antiferromagnetically ordered state. Since FC and ZFC procedures are well distinguished, it is no surprise that in general there are differences in physical observables between both cases, i. e. that there are irreversibilities. This is found in experiments [11] as well in simulations [1, 1]. In contrast to quantities like the magnetization or the staggered magnetization (antiferromagnetic order parameter), recent experiments [14, 15] find, that surprisingly there are no irreversibilities for the specific heat, or at least, that they are very small [8]. These experimental findings are the motivation of this work, in which we investigate the equilibrium and non-equilibrium (hysteresis) behavior of the specific heat in FC and ZFC procedures with Monte Carlo techniques.

3 iii Model and Simulation To study the thermal phase transition in the DAFF, we simulate a threedimensional diluted antiferromagnet in an external field using Monte Carlo methods. The Ising-Hamiltonian is given by H =,J X hiji " i " j i j, B X i " i i ; (1) where the summation hiji runs over the nearest neighbor interactions in a simple cubic lattice. The coupling constant J is set to,1. The dilution is introduced through the quenched variables " i, which are 0 with a probability p and 1 otherwise. For all simulations in this work p is set to 0:5, which is well above the percolation threshold. We set the external field fixed to a value of B = 0:8, which is large enough to show all the irreversibilities which are observed in experiments and other simulations, but which is not too large so that the ground state is still long range ordered. Periodic boundary conditions are used. We study the phase transition by simulating heating-cooling cycles of the system, corresponding to ZFC and FC measurements. At T = 0 the system is prepared in the antiferromagnetically ordered ground state. Then the temperature is increased in temperature dependent steps. For each temperature we performed Monte Carlo Steps, if not mentioned otherwise. When the maximum temperature of our simulation is reached, we initialize the system with a random spin configuration, corresponding to infinite temperature. This is done to destroy any remaining correlations in the system. After that, the system is cooled down the same way it was heated. To study the equilibrium behavior we used the Swendsen-Wang cluster algorithm [16]. The coupling to the external field was taken into account via applying the heat-bath switching probability computed from the magnetization of the clusters. For a realistic simulation of the irreversibilities at the phase transition, we used Glauber dynamics (heat-bath algorithm), which is believed to reproduce the dynamics of the experimental system. We simulated systems with of size 5 ; 50 ; 100 and 160 with the single spin flip algorithm and 0 ; 0 ; 40 and 56 with the cluster algorithm.

4 iv Results.1 Equilibrium behavior First we are interested in the equilibrium behavior at the phase transition. Our aim is to determine the critical temperature T c for our given set of parameters using finite size scaling techniques. Therefore, we only analyzed the ZFC data, as it is widely accepted, that they show the phase transition from the ordered state into the paramagnetic state, which is generally doubted for the FC data. We used the Swendsen-Wang algorithm to obtain the data, since non-equilibrium effects are much smaller with this algorithm compared to a single spin flip algorithm. This can be seen from a comparison of the specific heat obtained either from the calculated internal energy c = [@U=@T] av, or by calculating the energy fluctuations c = N=T [hu i,hui ] av,wheren is the number of spins in the system, hi denotes the thermal, and [] av the configurational average. Only very small differences in both quantities are obtained when using the Swendsen-Wang algorithm. M st L = L = 0 L = 0 L = 40 L = (T, T c ) L 1= Figure 1 Figure 1: Scaling Plot of the ZFC data for the antiferromagnetic order parameter M st To determine the critical temperature T c we use finite size scaling anal-

5 v ysis for both the staggered magnetization M st and the disconnected susceptibility dis = N P 1 N ih i i. The finite size scaling relations (see av e. g. [17]) for the staggered magnetization M st;l and the disconnected susceptibility dis;l of finite systems with size L are given by M st;l = L,= ~ M st (T, T c )L 1= () dis;l = L = ~ dis (T, T c )L 1= : () From a scaling analysis of both M st and dis we obtain the transition temperature T c 1:5 0:05 and the critical exponents 1:09 0:05 (correlation length), 0:15 0:0 (order parameter), and :9 0:1. These exponents fulfill the modified hyper-scaling equation d =. Because of the limited range of system sizes we do not claim to improve the values of critical exponents for the DAFF. Nevertheless, our values of and are in agreement with other results [18, 9, 4], whereas our result for turns out to be larger than previous results. The scaling plot for M st is shown in Figure 1. The scaling plot for dis is of similar quality and is therefore not shown. Figure shows the equilibrium behavior of the specific heat, the inset shows the specific heat for several system sizes. The specific heat show a broad maximum, which occurs at temperatures well above the critical temperature T c.. Irreversibilities Irreversible behavior as observed in experiments is also found in simulations, but it is important to note that this may depend on the algorithm used. To compare with experiments one therefore has to use an algorithm which reflects the dynamics of the experimental system. It is widely agreed that this is achieved by using the single spin flip algorithm with random updates (Glauber dynamics). With this algorithm we analyzed the order parameter M st, the internal energy U and the specific heat c, to compare our data with experiments. Figure shows the order parameter for various system sizes in ZFC- FC loops. In agreement with other works (e. g. [1]) we find that the order parameter will not increase at T c when cooling the system in an external field. This indicates that the phase transition is suppressed and the system is frozen in a domain state. The non-zero value for M st below T c in our

6 vi 0.7 c pl (hu i,huihu T Figure Figure : The ZFC data for the specific heat for a system size L = 0 calculated with two different methods. The vertical line is drawn at T c.the inset shows the specific heat for several system sizes (L =0; 0; 40; 56). simulations can be clearly identified as a finite size effect, since the larger the systems are, the smaller the order parameter after the FC. The internal energy for a ZFC-FC cycle is shown in figure 4a. In contrast to the order parameter and the magnetization (see [11, 1]), there is only a small split between the ZFC an FC curves of the internal energy. This behavior is reflected in the specific heat c. Apart from a small region around T c there is no difference between the ZFC and FC curves (figure 4b). 4 Discussion We have shown that the specific heat has a broad maximum and shows only very little hysteresis as compared for instance to the order parameter. This surprising result has also been seen in experiments [14, 15]. The critical temperature T c clearly lies below the maximum of the specific heat. This is not a finite size effect, since the position of the maximum is practically independent of system size. We conclude that the broad maximum of the specific heat has nothing to do with critical behavior. Note that the

7 vii M st ZFC L = FC L = ZFC L = FC L = 50 ZFC L = FC L = 100? 44 ZFC L = FC L = ??????????? 0.1???????????????????????? ??????? 4? T Figure Figure : The order parameter M st loop for several systems sizes in a ZFC-FC U a) ZFC FC c (ZFC) (FC) T Figure 4 Figure 4: ZFC-FC cycles: a) The internal energy b) The specific heat

8 viii true T c may be lower than the one we determined from finite size scaling. A more elaborate analysis only would yield an even lower T c, because we are heating the system from the long range ordered phase and the order parameter has to go to zero at T c. Thus, longer simulations can only result in a decay of the order parameter at lower temperatures. An still open question is the critical behavior of the specific heat of the DAFF and the RFIM. The existence of irreversibilities in experiments is still unclear and experimental and theoretical values for the critical exponent do not agree [7, 8, 9, 10]. We have shown that the broad maximum does not correspond to critical behavior, by showing that the critical temperature lies on the left of the maximum of the specific heat and that there is no size dependence of its position. This opens new questions on how to interpret experimental data of the DAFF. Although the DAFF is in the same universality class as the RFIM with Gaussian random fields, the critical behavior of the DAFF is covered by the broad, non-critical peak, whereas in the RFIM the critical behavior of the specific heat can be measured [9]. It could be possible that the same effect occurs in experiments which measure the specific heat directly. The experimental data can be reproduced in the sense, that, although there has to be a small hysteresis in ZFC-FC loops, as indicated by the split in the internal energy, this hysteresis will be very small and difficult to measure. Our results are in contradiction with the interpretation of experimental data in [15], where it was claimed that the critical behavior is also present in the FC procedure. Our measurements of the order parameter clearly indicate that the phase transition is suppressed when cooling the system in an external field. It is expected that detailed analysis in the region around T c will show, that the equilibrium specific heat develops a singularity at T c in the ZFC case, i. e. in equilibrium. The resolution obtained by our simulations does not allow a final answer to this question. We believe that a detailed analysis, which will of course be quite expensive in computing time, will reveal the same behavior as it has been found recently in two-dimensional diluted ferromagnets: A small critical peak at T c followed by a broad, noncritical maximum at higher temperatures [19]. Since the phase transition does not occur in the FC case, irreversibilities in the specific heat must exist, but obviously they are very small in both experiments and in simulations. Acknowledgment: We would like to thank U. Nowak and A. Hucht for helpful discussions. This work was in part supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 166.

9 ix References [1] S. Fishman and A. Aharony, J. Phys. C1(1979) L79. [] J.L.Cardy,Phys.Rev.B9 (1984) 505. [] J.-C. A. d Auriac and N. Sourlas, Europhys. Lett. 9 (1997) 47. [4] A. K. Hartmann and U. Nowak, Eur. Phys. J. B (1998) (accepted for publication). [5] J. Z. Imbrie, Phys. Rev. Lett. 5 (1984) [6] J. Bricmont and A. Kupiainen, Phys. Rev. Lett. 59 (1987) 189. [7] D.P.Belanger,A.R.King,V.Jaccarino,andJ.L.Cardy, Phys.Rev.B 8 (198) 5. [8] Z. Zlani c and D. P. Belanger, J. Magn. Magn. Mater. 186 (1998) 65. [9] H. Rieger, Phys. Rev. B 5 (1995) [10] U. Nowak, K. D. Usadel, and J. Esser, Physica A 50 (1998) 1. [11] F. C. Montenegro, A. R. King, V. Jaccarino, S.-J. Han, and D. P. Belanger, Phys. Rev. B 44 (1991) 155. [1] C. Ro, G. S. Grest, C. M. Soukulis, and K. Levin, Phys. Rev. B 1 (1985) 168. [1] U. Nowak and K. Usadel, Phys. Rev. B 9 (1989) 516. [14] R. J. Birgenau, J. Magn. Magn. Mater (1998) 1. [15] J. Satooka, H. A. Katori, A. Tobo, and K. Katsumata, Phys. Rev. Lett. 81 (1998) 709. [16] R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58 (1987) 86. [17] K. Binder and D. W. Heermann, Monte Carlo Simulation in Statistical Physics, Springer Series in Solid State Sciences, Springer, [18] A. T. Ogielski and I. Morgenstern, Phys. Rev. Lett. 54 (1985) 98. [19] W. Selke, L. N. Shchur, and O. A. Vasilyev, Physica A 59 (1998) 88.

Monte Carlo studies of slow relaxation in diluted antiferromagnets

Monte Carlo studies of slow relaxation in diluted antiferromagnets U. Nowak and K. D. Usadel Theoretische Physik and Sonderforschungsbereich No. 166, Universitiit-GH-Duisburg, Lotharstrasse 1, 4100 Duisburg,