New Method of Monte Carlo Simulations and Phenomenological Theory of Phase Transition in the Two-Dimensional XY-Model

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Simulations of the two-dimensional XYmodel showing the appearance of vortices have been reported in previous papers. 1) •2) It seems, however, necessary for studying the feature of phase transition to find out a more rapid method of simulation. The purposes of this short note are (i) to describe such a new method of simulation together with explicit results, and (ii) to represent briefly an idea to explain a new phase transition with vortices and without long-range order. 3) In contrast to the ordinary method!) our method has the following two new points: (1) The 2 d-classical XY-model is described approximately except at very low temperatures by the discrete vector model which can assume u discrete angles e j = 2rrj/n with j=1, 2, ···, n, and (2) each step of simulation is determined by generating a uniform random number p in the range O:c;:ps;1 and by making each local spin jump to a state e j> if p comes into the region pj as shown in Fig. 1, in which the probability pj rs given by Pj= exp ([3Ej) /~k exp ([3Ek) with a local energy Ej in the state e j· This procedure assures the detailed balance and equilibrium state. Clearly this gives a very rapid simulation for our problem. In fact, our preliminary calculations have already shown the system-size dependence of vortexsize and 'time' -dependence of disclina tion formation, as shown in Figs. 2 and 3. It is seen from these results together with P, P,

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