 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-81/1/d/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

The linear spin-wave approximation expands about the rotated all-up state, with

n_{n} = a_{n}^{†} a_{n}

S_{n}^{z} = S - n_{n}, S_{n}^{+} = 2 S a_{n} + O (S^{- 1/2}), S_{n}^{-} = 2 S a_{n}^{†} + O (S^{- 1/2}) .

In the longitudinal product,

S_{n}^{z} S_{n + 1}^{z} = S^{2} - S (n_{n} + n_{n + 1}) + O (S^{0})

. Each site belongs to two bonds, so the resulting quadratic Hamiltonian is

H_{2} = - N J S^{2} + J S n \sum [2 a_{n}^{†} a_{n} + a_{n} a_{n + 1} + a_{n}^{†} a_{n + 1}^{†}] .

The omitted terms are of order

S^{0}

at fixed small occupation number. The pair terms describe the quantum fluctuations that were missing from the classical Néel state.

 New to topics? Read the docs here!