content_image/399957.jpg content_image/399958.jpg content_image/399959.jpg content_image/399963.jpg content_image/399964.jpg content_image/399965.jpg content_image/399966.jpg content_image/399945.jpg content_image/399951.jpg content_image/399952.jpg
Since X$_{SPS}$ in this formula is calculated only from the pairs whose graph variables are one, it can be calculated with very short computational time as ˜ E ′ and ∂ ˜ E ′ ∂β in eqs. (32) and (33) are. 

## V. THE CASE WHEN LONG-RANGE INTERACTIONS AND SHORT-RANGE INTERACTIONS COEXIST 

When long-range interactions { V (L) ij } and short-range interactions { V (S) ij } coexist, we do not need to use the SCO method for short-range interactions because it does not reduce the computational time significantly. In other words, we can switch V (L) ij to 0 or ¯ V (L) ij with V (S) ij un- changed. This can be realized by setting ˜ V (L) ij = 0 and ˜ V (S) ij = V (S) ij . In this case, we can decompose E$_{const}$ , ˜ E , ∂ ˜ E $_{∂β}$, and X$_{SPS}$ in eqs. (27), (28), (29), and (40) as 

where the first terms and the second terms in the right hand sides denote contributions from long-range inter- actions and those from short-range interactions, respec- tively. As we have already mentioned, the first terms are given by eqs. (31), (32), (33), and (42), respectively. On the other hand, by substituting ˜ V$_{kl}$ = V (S) kl and Δ V ∗ kl = 0 into eqs. (27), (28), (29), and (40), we readily obtain 

<img src='content_image/399957.jpg'>

<img src='content_image/399958.jpg'>

<img src='content_image/399959.jpg'>

where we have used the fact that all the potentials are switched to ˜ V$_{kl}$ . Note that P$_{kl}$ = 1 when ˜ V$_{kl}$ = V$_{kl}$ and Δ V ∗ ij = 0. These results are quite natural because they coincide with the results in the usual MC procedure. 

In order to check the validity of the formulae for inter- nal energy and heat capacity measurements, we perform 

<img src='content_image/399963.jpg'>

<img src='content_image/399964.jpg'>

<img src='content_image/399965.jpg'>

<img src='content_image/399966.jpg'>

## VI. NUMERICAL TESTS 

## A. Internal energy and heat capacity measurements 

<img src='content_image/399945.jpg'>

MC simulations of a three dimensional magnetic dipolar system on a L 3 simple cubic lattice. The boundary condi- tion is open. The Hamiltonian of the system is described as 

<img src='content_image/399951.jpg'>

where S$_{i}$ is a classical Heisenberg spin of | S$_{i}$ | = 1, r$_{ij}$ is the vector spanned from a site i to j in the unit of the lattice constant a , r$_{ij}$ = | r$_{ij}$ | , and D = ( gµ$_{B}$S ) /a $^{3}$. We hereafter call the system purely dipolar system . 

In simulations with the SCO method, we regard each term in the right hand side of eq. (45) as V$_{ij}$ . Since ˜ V$_{ij}$ = 0 in the SCO method, Δ V$_{ij}$ defined by eq. (3) is equal to V$_{ij}$ . Interaction V$_{ij}$ has the maximum value 2 D/r 3 ij when S$_{i}$ and S$_{j}$ are anti-parallel along r$_{ij}$ . We therefore set Δ V ∗ $_{ij}$, which should be equal to or greater than Δ V$_{ij}$ over all S$_{i}$ and S$_{j}$ , to 2 D/r 3 $_{ij}$. By substituting these into eq. (2), we obtain 

<img src='content_image/399952.jpg'>

