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Rounding of phase transition in the 2D 8-state random bond Potts model
Computer Physics Communications 121–122 (1999) 194–196 www.elsevier.nl/locate/cpc
Rounding of phase transition in the 2D 8-state random bond Potts model Tarık Çelik 1 , Fatih Ya¸sar, Yi˘git Gündüç Department of Physics, Hacettepe University, Beytepe 06532, Ankara, Turkey
Abstract The two-dimensional q = 8 state Potts model with varying degree of bond randomness has been simulated by using the cluster algorithm. We have shown that there exists a finite size dependent threshold value of the introduced quenched bond randomness for rounding the first-order phase transition and this threshold becomes smaller as one moves towards infinite systems. 1999 Elsevier Science B.V. All rights reserved.
Although the effect of adding quenched bond randomness to classical systems whose pure version displays a continuous phase transition is well understood, it is not so clear what happens when randomness is introduced to systems undergoing a first-order phase transition and this problem has recently attracted much interest. Phenomenological renormalization group arguments by Hui and Berker [1] and rigorous proof of the vanishing of the latent heat by Aizenman and Wehr [2] showed that bond randomness will induce a second-order phase transition in a system that would otherwise undergo a symmetry-breaking firstorder phase transition. The renormalization group arguments state that any infinitesimal amount of bond randomness will drive the system to exhibit a secondorder phase transition for d 6 2 where d is the dimension of the space. The numerical evidence for this prediction is provided by Monte Carlo simulations of the two-dimensional 8-state Potts model at different chosen values of the strength of randomness [3,4]. While the pure model is known to exhibit a strong first-order phase transition, simulations gave clear evidence of 1 E-mail: [email protected].
a second-order transition, but a discrepancy appeared concerning to whether the critical exponents fall into the 2D Ising model universality class or not. It was reported that the discrepancies between several works might be due to the choice of the strength of disorder [5]. In these simulation works, for the sake of convenience of establishing the histograms, the strong to weak bond ratio was chosen to have fixed values such as r = 0.5 or 0.1 and the aspect of varying randomness is not addressed. Specifically, the conversion in two-dimensions of a first-order phase transition into a second-order one even for even infinitesimal randomness is a result with important consequences. This claim is not completely established yet. In this context, it would be quite relevant to study the changes in the characteristic behaviors of the system with respect to the introduction of a gradually increasing degree of bond randomness as well as its finite size dependence. The main objective of this work is to understand the effect of quenched bond randomness on the transition and the correlation between the amount of randomness and the finite size of the system by consecutive increases of quenched randomness and deduce
0010-4655/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 9 9 ) 0 0 3 1 1 - 2
T. Çelik et al. / Computer Physics Communications 121–122 (1999) 194–196
Fig. 1. Energy histograms (a) for several values of r on a 64 × 64 lattice and (b) for r = 0.53 on several lattices.
the interplay of finite size and strength of randomness crossover [6]. ThePHamiltonian of the Potts model is given by H = hi,j i Kij δσi ,σj . In a pure system, Kij is constant and in the random bond Potts model, the couplings are selected from two positive values with a weak to strong ratio r = Kw /Ks chosen randomly from the distribution P (K) = pδ(K − Kw ) + (1 − p)δ(K − Ks ). We have simulated the 8-state Potts model on lattices of sizes 322 , 642 and 1282 . Measurements are done around the transition temperature; at each temperature 5×105 iterations are performed following the thermalization runs of 5 × 104 to 105 iterations. For all simulation works, the cluster update algorithm is employed. We have studied the phase structure of the 8-state Potts model with respect to the quenched randomness in the range starting from r = 1 (pure case) to
195
r = 0.1 (weak bonds are ten times weaker). For a fixed value of r, twenty replicas are created by randomly distributing the strong and weak bonds over the lattice. In Fig. 1(a), we show the energy histogram on a 64 × 64 lattice for several values of r. Starting from the pure case which exhibits a rather strong first-order phase transition, the 8-state Potts model with quenched bond randomness displays a double-peak structure in the energy histogram down to r = 0.5, the latter value corresponds to the case studied in Ref. [3]. By further tuning, one observes a single Gaussian in the energy histogram for r = 0.46. This Gaussian gets sharper as the value of r chosen is smaller. The observed threshold value for the randomness is specific to the lattice size used. Therefore the next step is to investigate how this threshold value changes with the size of the lattice. Fig. 1(b) displays the energy histograms for a fixed value of r, namely r = 0.53, obtained for different size lattices. According to finite size scaling arguments, for a first-order phase transition, the valley between the double-peaks in the energy histogram should become deeper and scale as the volume with increasing lattice size. From the distributions shown in Fig. 1(b), one observes a first-order phase transition on a 64 × 64 lattice while a second-order one on the 128 × 128 lattice. We will then need to introduce less amount of randomness (r smaller than 0.53) to regain the double-peak structure in the energy distribution on the 128 × 128 lattice. Apparently, the observed threshold value for the amount of quenched impurities for the characteristic behavior of the system to change from having a first-order phase transition to a second-order one depends on the finite size of the system under consideration. This threshold gets smaller and smaller as one moves towards infinite systems. In order to demonstrate that randomness is the crucial aspect, we also considered totally periodic distributions of Kw and Ks in both directions on square lattices. We gradually destroyed the periodicity by randomly picking up two lattice sites, exchange their positions and then continue to pick up another pair of sites. When this way of destroying the periodicity reaches a certain (lattice size dependent) threshold value, we observe that the phase transition becomes of second order. Our conclusion is that the threshold value for the randomness which induces a change of the order of
196
T. Çelik et al. / Computer Physics Communications 121–122 (1999) 194–196
the phase transition on a lattice is inversely related to the lattice size. That is, increasing the lattice size, the system requires less randomness for the softening of a first-order phase transition. This observation is in favor of the renormalization group arguments. For an infinite system, an infinitesimal amount of randomness changes the order of phase transition. Acknowledgements This research was supported by Hacettepe University Research Fund under the project 95.01.010.003 and TÜB˙ITAK project TBAG-1699.
References [1] K. Hui, A.N. Berker, Phys. Rev. Lett. 62 (1989) 2507. [2] M. Aizenman, J. Wehr, Phys. Rev. Lett. 62 (1989) 2503. [3] S. Chen, A.M. Ferrenberg, D.P. Landau, Phys. Rev. E 52 (1995) 1377. [4] C. Chatelain, B. Berche, Phys. Rev. Lett. 80 (1998) 1670. [5] J. Cardy, J.L. Jacobsen, Phys. Rev. Lett. 79 (1997) 4063. [6] F. Ya¸sar, Y. Gündüç, T. Çelik, Phys. Rev. E 58 (1998), in print.
Rounding of phase transition in the 2D 8-state random bond Potts model Tarık Çelik 1 , Fatih Ya¸sar, Yi˘git Gündüç Department of Physics, Hacettepe University, Beytepe 06532, Ankara, Turkey
Abstract The two-dimensional q = 8 state Potts model with varying degree of bond randomness has been simulated by using the cluster algorithm. We have shown that there exists a finite size dependent threshold value of the introduced quenched bond randomness for rounding the first-order phase transition and this threshold becomes smaller as one moves towards infinite systems. 1999 Elsevier Science B.V. All rights reserved.
Although the effect of adding quenched bond randomness to classical systems whose pure version displays a continuous phase transition is well understood, it is not so clear what happens when randomness is introduced to systems undergoing a first-order phase transition and this problem has recently attracted much interest. Phenomenological renormalization group arguments by Hui and Berker [1] and rigorous proof of the vanishing of the latent heat by Aizenman and Wehr [2] showed that bond randomness will induce a second-order phase transition in a system that would otherwise undergo a symmetry-breaking firstorder phase transition. The renormalization group arguments state that any infinitesimal amount of bond randomness will drive the system to exhibit a secondorder phase transition for d 6 2 where d is the dimension of the space. The numerical evidence for this prediction is provided by Monte Carlo simulations of the two-dimensional 8-state Potts model at different chosen values of the strength of randomness [3,4]. While the pure model is known to exhibit a strong first-order phase transition, simulations gave clear evidence of 1 E-mail: [email protected].
a second-order transition, but a discrepancy appeared concerning to whether the critical exponents fall into the 2D Ising model universality class or not. It was reported that the discrepancies between several works might be due to the choice of the strength of disorder [5]. In these simulation works, for the sake of convenience of establishing the histograms, the strong to weak bond ratio was chosen to have fixed values such as r = 0.5 or 0.1 and the aspect of varying randomness is not addressed. Specifically, the conversion in two-dimensions of a first-order phase transition into a second-order one even for even infinitesimal randomness is a result with important consequences. This claim is not completely established yet. In this context, it would be quite relevant to study the changes in the characteristic behaviors of the system with respect to the introduction of a gradually increasing degree of bond randomness as well as its finite size dependence. The main objective of this work is to understand the effect of quenched bond randomness on the transition and the correlation between the amount of randomness and the finite size of the system by consecutive increases of quenched randomness and deduce
0010-4655/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 9 9 ) 0 0 3 1 1 - 2
T. Çelik et al. / Computer Physics Communications 121–122 (1999) 194–196
Fig. 1. Energy histograms (a) for several values of r on a 64 × 64 lattice and (b) for r = 0.53 on several lattices.
the interplay of finite size and strength of randomness crossover [6]. ThePHamiltonian of the Potts model is given by H = hi,j i Kij δσi ,σj . In a pure system, Kij is constant and in the random bond Potts model, the couplings are selected from two positive values with a weak to strong ratio r = Kw /Ks chosen randomly from the distribution P (K) = pδ(K − Kw ) + (1 − p)δ(K − Ks ). We have simulated the 8-state Potts model on lattices of sizes 322 , 642 and 1282 . Measurements are done around the transition temperature; at each temperature 5×105 iterations are performed following the thermalization runs of 5 × 104 to 105 iterations. For all simulation works, the cluster update algorithm is employed. We have studied the phase structure of the 8-state Potts model with respect to the quenched randomness in the range starting from r = 1 (pure case) to
195
r = 0.1 (weak bonds are ten times weaker). For a fixed value of r, twenty replicas are created by randomly distributing the strong and weak bonds over the lattice. In Fig. 1(a), we show the energy histogram on a 64 × 64 lattice for several values of r. Starting from the pure case which exhibits a rather strong first-order phase transition, the 8-state Potts model with quenched bond randomness displays a double-peak structure in the energy histogram down to r = 0.5, the latter value corresponds to the case studied in Ref. [3]. By further tuning, one observes a single Gaussian in the energy histogram for r = 0.46. This Gaussian gets sharper as the value of r chosen is smaller. The observed threshold value for the randomness is specific to the lattice size used. Therefore the next step is to investigate how this threshold value changes with the size of the lattice. Fig. 1(b) displays the energy histograms for a fixed value of r, namely r = 0.53, obtained for different size lattices. According to finite size scaling arguments, for a first-order phase transition, the valley between the double-peaks in the energy histogram should become deeper and scale as the volume with increasing lattice size. From the distributions shown in Fig. 1(b), one observes a first-order phase transition on a 64 × 64 lattice while a second-order one on the 128 × 128 lattice. We will then need to introduce less amount of randomness (r smaller than 0.53) to regain the double-peak structure in the energy distribution on the 128 × 128 lattice. Apparently, the observed threshold value for the amount of quenched impurities for the characteristic behavior of the system to change from having a first-order phase transition to a second-order one depends on the finite size of the system under consideration. This threshold gets smaller and smaller as one moves towards infinite systems. In order to demonstrate that randomness is the crucial aspect, we also considered totally periodic distributions of Kw and Ks in both directions on square lattices. We gradually destroyed the periodicity by randomly picking up two lattice sites, exchange their positions and then continue to pick up another pair of sites. When this way of destroying the periodicity reaches a certain (lattice size dependent) threshold value, we observe that the phase transition becomes of second order. Our conclusion is that the threshold value for the randomness which induces a change of the order of
196
T. Çelik et al. / Computer Physics Communications 121–122 (1999) 194–196
the phase transition on a lattice is inversely related to the lattice size. That is, increasing the lattice size, the system requires less randomness for the softening of a first-order phase transition. This observation is in favor of the renormalization group arguments. For an infinite system, an infinitesimal amount of randomness changes the order of phase transition. Acknowledgements This research was supported by Hacettepe University Research Fund under the project 95.01.010.003 and TÜB˙ITAK project TBAG-1699.
References [1] K. Hui, A.N. Berker, Phys. Rev. Lett. 62 (1989) 2507. [2] M. Aizenman, J. Wehr, Phys. Rev. Lett. 62 (1989) 2503. [3] S. Chen, A.M. Ferrenberg, D.P. Landau, Phys. Rev. E 52 (1995) 1377. [4] C. Chatelain, B. Berche, Phys. Rev. Lett. 80 (1998) 1670. [5] J. Cardy, J.L. Jacobsen, Phys. Rev. Lett. 79 (1997) 4063. [6] F. Ya¸sar, Y. Gündüç, T. Çelik, Phys. Rev. E 58 (1998), in print.