Nonequilibrium critical relaxation at a first-order phase transition point

We study numerically the nonequilibrium dynamical behavior of an Ising model with mixed two-spin and four-spin interactions after a sudden quench from the high-temperature phase to the first-order phase transition point. The autocorrelation function is shown to approach its limiting value, given by the magnetization in the ordered phase at the transition point, mc, through a stretched exponential decay. On the other hand relaxation of the magnetization starting with an uncorrelated initial state with magnetization, mi, approaches either mc, for mi > 0.5, or zero, for mi < 0.5. For small mi and for mi slightly larger than 0.5 the relaxation of the magnetization shows an asymptotic power-law time dependence, thus from a nonequilibrium point of view the transition is continuous. Introduction. – Most of our knowledge about the properties of phase transitions has accumulated for continuous or second-order transitions for which the concepts of scaling and universality, as well as the application of the method of renormalization group, have provided a deep understanding [1]. Considerably less is known about singularities at first-order phase transitions, although in nature this type of transition is very common [2]. At a first-order phase transition point one observes phase coexistence where several thermodynamical quantities, such as the internal energy and the order parameter (magnetization), have a discontinuity, whereas the correlation length is generally finite. In spite of this finite correlation length some response functions, such as the magnetic susceptibility, have a scaling behavior in a finite system [3] that involves the discontinuity fixed-point exponent yd = d where d denotes the dimension of the system [4]. From a dynamical point of view the relaxation time is finite at a first-order transition point, yielding equilibrium autocorrelations and relaxation functions that decay exponentially. Many studies have been devoted to the investigation of the formation of the ordered phase when the temperature is lowered below the phase transition point, Tc. Thus, the nonequilibrium dynamics of the nucleation process [5] and the morphology of the solidification [6] are key problems in material science. From a more theoretical perspective one is interested in the coarsening process [7] that takes place when the system is suddenly quenched from the high-temperature phase at or below the phase transition point. If the quench is performed below Tc, then the competition between the stable ordered phases leads to the same type of qualitative picture independently of the order of the transition at Tc [8]. If, however, we quench the system at the very transition point, important differences are expected in the two cases due to the different behavior of the correlation length. In the past this type of phenomena, i.e. nonequilibrium critical relaxation, has been thoroughly studied in the case of a second-order transition point [9,10], whose investigations were also motivated by the appearence of ageing. On the contrary only little attention has been paid to the case when the transition is of first order. It has been demonstrated that nonequilibrium relaxation starting from an ordered or a mixed-phase initial state can be used to locate accurately the phase transition point and to decide about the order of the transition [11,12]. For models with quenched disorder the tricritical value of the dilution has