Fourth-Order Cumulants to Characterize the Phase Transitions of a Spin-1 Ising Model

Abstract

Fourth-order cumulants of physical quantities have been used to characterize the nature of a phase transition. In this paper we report some Monte Carlo simulations to illustrate the behavior of fourth-order cumulants of magnetization and energy across second and rstorder transitions in the phase diagram of a well known spin-1 Ising model. Simple ideas from the theory of thermodynamic uctuations are used to account for the behavior of these cumulants. I IntroductionThere are many attempts to characterize the order of a phase transition on the basis of the analysis of numerical data obtained from simulations of nite spin systems. One of the approaches to this problem consists in the analysis of the behavior of fourth-order cumulants of physical quantities as the order parameter and the energy associated with the systems under consideration 1, 2 . Properties of the fourth-order cumulants of magnetization and energy have been investigated in the context of nite-size e ects in magnetically and thermally driven rst-order transitions in Ising and Potts models 3, 4 , 5 , 6 as well as in the case of some other systems 7, 8 .In this paper, we perform Monte Carlo simulations for the well known Blume-Capel model 9, 10 to illustrate the behavior of the fourth-order cumulants of magnetization and energy across rst and second-order transitions in the phase diagram of this system. We show that it is possible to draw some conclusions from the study of relatively small lattices. The general features of the cumulants can be accounted for by simple arguments from the theory of thermodynamic uctuations. In particular, we emphasize the di erences between the two t ypes of cumulants, and the alternative de nitions of the cumulant of energy which has not been fully appreciated in previous investigations.The layout of this paper is as follows. In Section 2 we de ne the cumulants of a physical quantity. In Section 3 we i n troduce some further de nitions, and discuss some properties of the Blume-Capel model. Simulations for the fourth-order cumulants of magnetization and energy are reported in Sections 4 and 5, respectively. W e h o p e t o h a ve provided another example of the use of these cumulants to characterize the order of a phase transition. II De nition of the cumulantsThe cumulants of a quantity x can be obtained from an expansion of the form

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