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84931502401
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To be precise the bonds now must be considered fictitious like in the CK clusters. We label the bond configurations by the same symbol α used to label the interaction configurations (∞ interactions are mapped into present bonds).
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20
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84931502407
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When l corresponds to a single pair (ij), Eq. ( ref{10r}) reproduces the standard solution ( ref{1r}). In fact, say α=1 (α=2) the configuration of an absent (present) bond between spins i and j. The solution of Eq. ( ref{9r}) is W1=e-2Jβ and W2=1-e-2Jβ and e-β hat{H}(α,{Si})= δεijSiSj,1. From Eq. ( ref{10r}) it follows that the probability of a bond being present, P2, is given by Eq. ( ref{1r}) while the probability of it being absent is P1=1-P2.
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84931502406
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By standard solution we mean that Pα, written in terms of Wα, can be factorized in terms of P and 1-P with P given by Eq. ( ref{1r}). In this case the clusters coincide with the standard CK clusters.
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22
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84931502409
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Note that the condition ( ref{11r}) is satisfied only on a single unit. This does not imply that Eq. ( ref{7r}) is necessarily satisfied on the entire lattice [Eq. ( ref{7r})]. On the other hand, the larger the unit for which ( ref{11r}) holds, the better Eq. ( ref{7r}) is satisfied. However, for ferromagnetic or unfrustrated systems the solution which satisfies Eq. ( ref{11r}) on a single bond [Eq. ( ref{1r})] satisfies Eq. ( ref{11r}) on any larger units. In fact it satisfies Eq. ( ref{7r}) on the entire lattice for any i and j. This consideration gives a hint that in the more general case of frustrated systems it is enough to satisfy ( ref{11r}) on a smaller unit to ensure that Eq. ( ref{7r}) is well approximated.
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23
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84931502408
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In the general case the number of equations introduced by ( ref{9r}) and ( ref{11r}) may be different from the number of unknowns Wα. If the solution is not unique, we expect that any solution leads to improved dynamics and, going to larger and larger units, all of them will give the same result. On the other hand, when Eqs. ( ref{9r}) and ( ref{11r}) do not have solutions we can relax Eq. ( ref{11r}) by requiring only the equality of the moments sum rijk < γij > = sum rijk < SiSj > (rij is the distance between sites i and j) and choosing k in such a way that, at least, a solution can be obtained.
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