Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 35A / b / i / Solution

Insert $s_1{s}_i$ into the spin-basis expansion of the trace. Summing the spins along the first path from site $1$ to site $i$ produces $(M^{i-1}{)}_{st}$, while summing along the complementary path around the periodic chain produces $(M^{N-i+1}{)}_{ts}$. Therefore the two-point correlation function is