 Solution

ID: past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2014/paper-1/1/b/solution

Rotate alternate sites by

π

about a transverse axis so all local reference spins point up. Choosing the transverse axes to give positive anomalous coupling, the linear spin-wave approximation yields

H = - J N S^{2} + J S \sum_{m} [2 a_{m}^{†} a_{m} + a_{m} a_{m + 1} + a_{m}^{†} a_{m + 1}^{†}] + O (S^{0}) .

Use lattice spacing one and

a_{m} = N^{- 1/2} \sum_{k} e^{ikm} a_{k}

, with all $N$ momenta

k = 2 πj / N

j = - N /2, \dots, N /2 - 1

, in

[- π, π)

. Then

γ_{k} = cos k

and

H = - J N S^{2} + J S \sum_{k} [2 a_{k}^{†} a_{k} + γ_{k} (a_{k} a_{- k} + a_{k}^{†} a_{- k}^{†})] .

Reordering

a_{- k} a_{- k}^{†} = a_{- k}^{†} a_{- k} + 1

gives the requested Nambu form with constant

- J NS (S + 1)

. Its extra

J SN

restores

- J N S^{2}

before diagonalization. This full-zone convention has no additional sublattice or half-zone factor.

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