9 tion becomes the basis of new algorithms as the original Fourtuin-Kasteleyn representation lead to the Swendsen- Wang cluster algorithm [24] in the Ising ferromagnetic model. Acknowledgments This work is supported by a Grant-in-Aid for Scientific Research (#21740279) from MEXT in Japan. In this appendix, we consider the Ising ferromagnetic model whose Hamiltonian is described as Appendix A: Derivation of the Fourtuin-Kasteleyn representation of the partition function in the Ising ferromagnetic model from eq. (11) purely dipolar ferromagnetic dipolar case (a) 1 . 9 × 10 4 3 . 3 × 10 3 case (b) 4 . 4 × 10 4 3 . 5 × 10 3 case (c) 6 . 2 × 10 4 4 . 5 × 10 3 TABLE II: Ergod time in the purely dipolar system and the ferromagnetic dipolar system. Measurements are done in the three cases (see text). the replica exchange probability P$_{ex}$ ( T$_{i}$ ). Figure 6 shows the result. We see that the reduction of the exchange probability in case (c) is not as large as that of the purely dipolar system (Fig. 5). This result is reasonable because the SCO method is used only for the long range interac- tions and its contribution ( X L SPS in eq. (43d)) is not large. Recall that the ratio D/J is 0 . 1. We also measure the ergod time defined by the average MC step for a specific replica to move from the lowest to the highest tempera- ture and return to the lowest one. The result is shown in Table II. In both the systems, the ergod time in the cases (b) and (c) is larger than than that in the case (a), meaning that the use of the SCO method increases the ergod time. However, the increase of the ergod time in the ferromagnetic dipolar system is not as large as that in the purely dipolar system. These results show that the SCO method is particularly efficient for systems with strong short-range interactions and weak long-range in- teractions. This feature of the SCO method has already been pointed out in the previous work [11]. H = − J ∑ 〈 ij 〉 σ$_{i}$σ$_{j}$, (A1) P$_{ij}$ = { exp( − 2 βJ ) ( σ$_{i}$ = σ$_{j}$ ) , 1 ( σ$_{i}$ = − σ$_{j}$ ) , (A2) where σ$_{i}$ = ± 1 and J > 0. We set V$_{ij}$ = − Jσ$_{i}$σ$_{j}$ and consider a special case that ˜ V$_{ij}$ = 0 and V ∗ ij = J . Then, we find and ¯ V$_{ij}$ = { − J − β − $^{1}$log[1 − exp( − 2 βJ )] , ( σ$_{i}$ = σ$_{j}$ ) , ∞ ( σ$_{i}$ = − σ$_{j}$ ) . (A3) By substituting this equation into eq. (7), we obtain VII. SUMMARY In the present work, we have derived useful formulae for the SCO method to estimate internal energy, heat capacity, and replica exchange probability in the replica exchange MC method. We can reduce the computational time of these quantities greatly by using the formulae be- cause they only contain terms which are not cut off by the SCO method. On the other hand, we have found that the use of the formulae could cause a decline in the efficiency of the measurement and that in the exchange probability. When the new methods do not work well, the analyses done in the present paper, such as the estimations of the statistical error, the replica exchange probability, and the ergod time, might be helpful to figure out the reason and to get rid of it. Anyway, we hope that these formulae make the SCO method more useful and attractive. The other achievement of the present work is the derivation of the new Fourtuin-Kasteleyn representation of the partition function, i.e. , eq. (11). This representa- tion is more comprehensive than the original one because our representation includes arbitrariness in the choice of pseudointeractions { ˜ V$_{ij}$ } . Furthermore, this representa- tion can be used no matter whether the variables { S$_{i}$ } are discrete or continuous. We hope that this representa- As it has been pointed out in ref. [19], we can eas- ily obtain the Fourtuin-Kasteleyn representation of the partition function from this joint distribution. We first rewrite ∏ 〈 ij 〉 ω$_{ij}$ as It is important to note that the product ∏ 〈 ij 〉 ω$_{ij}$ is pro- portional to the joint distribution of { S$_{i}$ } and { g$_{ij}$ } in- troduced by Edwards and Sokal [19]. In this sense, we can regard the product of ω$_{ij}$ defined by eq. (7) as a generalization of Edwards and Sokal’s joint distribution. ∏ 〈 ij 〉 ω$_{ij}$ = e N$_{int}$$^{βJ}${ exp( − 2 βJ ) } N$_{int}$ − N$_{b}$ ×{ 1 − exp( − 2 βJ ) } N$_{b}$ ∏ (1) 〈 ij 〉 δ$_{σ}$$_{i}$$_{,σ}$$_{j}$ , (A5) where N$_{int}$ = ∑ 〈 ij 〉 and N$_{b}$ = ∑ 〈 ij 〉 δ$_{g}$$_{ij}$$_{,}$$_{1}$ . The product ∏ (1) 〈 ij 〉 runs over all the pairs with g$_{ij}$ = 1. By using ω$_{ij}$ ( σ$_{i}$, σ$_{j}$ ; g$_{ij}$ ) = { exp( − βJ ) ( g$_{ij}$ = 0) , exp( βJ )[1 − exp( − 2 βJ )] δ$_{σ}$$_{i}$$_{,σ}$$_{j}$ ( g$_{ij}$ = 1) . (A4)