Spin correlations and the one-chain problem

Spin correlations and the one-chain problem Consider the spin-spin correlation function in zero field  

Note that the denominator of eq. (X.31) is equal to 1 (from eq. X.30). Then expand the exponential in the numerator. Again, for n = 0, the only graphs which contribute are self-avoiding paths, but here they are not closed loops because eq. (X.31) contains two extra spin factors St Sj. In fact what we have is a sum over all self-avoiding walks linking sites i and j (Fig. X.4). If the walk involves N steps, the resulting contribution to eq. (X.30) is simply 

All along the walk the component index a must be equal to the chosen value (a = 1). There is no summation to be carried on a. 

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