arXiv:1012.5780v1 [cond-mat.stat-mech] 28 Dec 2010 Phase diagram for a zero-temperature Glauber dynamics under partially synchronous updates. Bartosz Skorupa and Katarzyna Sznajd–Weron Institute of Theoretical Physics, University of Wroc law, pl. Maxa Borna 9, 50-204 Wroclaw, Poland (Dated: December 30, 2010) We consider generalized zero-temperature Glauber dynamics under partially synchronous updat- ing mode for a one dimensional system. Using Monte Carlo simulations, we calculate the phase diagram and show that the system exhibits phase transition between ferromagnetic and active an- tiferromagnetic phase. Moreover, we provide analytical calculations that allow to understand the origin of the phase transition and confirm simulation results obtained earlier for synchronous up- dates. PACS numbers: 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) I. INTRODUCTION In the last decade renewed interest in Glauber dynam- ics [1] is observed, especially at zero temperature [2–14]. This is partially caused by recent experiments with so called single-chain magnets (for a recent review see [15]), but also due to the development of the nonequilibrium statistical physics. From this point of view especially one-dimensional systems at zero-temperature are inter- esting [9]. The dynamical rules of stochastic models, such as Glauber dynamics, can be defined in terms of various update schemes, the most important ones being paral- lel (synchronous) and random-sequential (asynchronous) updates [16]. Although, Glauber dynamics was originally introduced as a sequential updating process, it occurs that interesting theoretical results can be obtained also using synchronous updating mode [4, 8, 10, 12, 13, 17]. Moreover, clear evidence of a relaxation mechanism which involves the simultaneous reversal of spins have been shown experimentally for magnetic chains at low temperatures [18]. In computer simulations under syn- chronous updating mode all units of the system are up- dated at the same time. However, in real systems one can expect that simultaneous reversal of spins concerns only a part of the system. From this point of view partially synchronous updates are the most realistic. We have introduced such a partially synchronous up- dating scheme in 2006 [12] to investigate the differences between Glauber and Sznajd dynamics for a chain of L Ising spins. Within such an update in each elementary time step we visit all sites and select each of them with probability c as a candidate to get flipped, i.e. in average cL randomly chosen spins are considered in a single time step [12]. Of course c = 1 corresponds to the synchronous updating scheme and c =1/L to random sequential up- dates. Partially synchronous updates has been used also in 2007 by Radicchi et al [13] to investigate Ising spin chain at zero-temperature for the Metropolis algorithm [19]. They have observed, as a function of c, a criti- cal phase transition between two phases - ferromagnetic and so called active phase. Similar phase transition has been already observed earlier for the generalized zero- temperature Glauber dynamics by Menyhard and Odor in a case of synchronous updating scheme [4]. It should be noticed that Metropolis algorithm at zero temperature is a special case of a broader class of zero- temperature Glauber dynamics. Within the Glauber dy- namics for Ising spins with a spin s =1/2, in a broad sense, each spin is flipped S i (t) →-S i (t + 1) with a rate W (δE) per unit time and this rate is assumed to depend only on the energy difference implied in the flip. At zero temperature it can be defined as [9]: W (δE)=    1 if δE < 0, W 0 if δE =0, 0 if δE > 0, (1) The zero-temperature limits of the original Glauber dynamics [1] and Metropolis rates [19] (two the most pop- ular choices) are respectively W G 0 =1/2 and W M 0 = 1. Very recently generalized Glauber dynamics defined by (1) under synchronous updating mode have been stud- ied [17]. It has been shown that the system exhibits a well-defined phase transition for W 0 =1/2 between ferro- and antiferromagnetic phase. As an order parameter, the density ρ of active bonds has been used: ρ = 1 2L L  i=1 σ i σ i+1 , (2) where L has been the number of spins and σ i = ±1 has been Ising spin variable at the i-th site on the one- dimensional chain with the periodic boundary condition. Results obtained in [17] has been very recently confirmed and corrected by Yi and Kim [20]. They have shown, us- ing finite-size scaling technique, that the system exhibits continuous phase transition at W 0 =1/2. In this paper we consider generalized Glauber dynam- ics defined by (1) under partially synchronous updating mode and show that both parameters W 0 and c are re- sponsible for the phase transition between ferromagnetic and antiferromagnetic phase. We construct the phase diagram in (c, W 0 ) space based on the Monte Carlo sim- ulations. Moreover, we provide exact analytical calcula- tions for a simple case with only 3 active bonds. Such