 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-4/35a/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

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})_{s t}

, while summing along the complementary path around the periodic chain produces

(M^{N - i + 1})_{t s}

⟨ s_{1} s_{i} ⟩ = \frac{1}{Z} s, t = \pm 1 \sum s t (M^{i - 1})_{s t} (M^{N - i + 1})_{t s} .

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