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ID: past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2022/2/c/solution

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Past exam of the theoretical physics major option course of the University of Oxford / 2022 / 2 / c / Solution by

Codex  0 2026-10-25

Expand around the

+ z

polarized ground state. With

n_{j} = a_{j}^{†} a_{j}

fixed as

S \to \infty

, the Holstein–Primakoff transformation gives

S_{j}^{+} = 2 S a_{j} + O (S^{- 1/2})

,

S_{j}^{-} = 2 S a_{j}^{†} + O (S^{- 1/2})

and

S_{j}^{z} = S - n_{j}

. The longitudinal exchange contributes

- J (1 + u) \sum_{j} (S - n_{j}) (S - n_{j + 1}) = - L J (1 + u) S^{2} + 2 J S (1 + u) \sum_{j} n_{j} + O (S^{0}) .

Use

S_{j}^{x} S_{j + 1}^{x} + S_{j}^{y} S_{j + 1}^{y} (S_{j} \times S_{j + 1})_{z} = \frac{1}{2} (S_{j}^{+} S_{j + 1}^{-} + S_{j}^{-} S_{j + 1}^{+}), = \frac{i}{2} (S_{j}^{+} S_{j + 1}^{-} - S_{j}^{-} S_{j + 1}^{+}) .

The transverse exchange then gives

- J S (a_{j + 1}^{†} a_{j} + a_{j}^{†} a_{j + 1})

, and the Dzyaloshinskii-Moriya interaction gives

- i D S (a_{j + 1}^{†} a_{j} - a_{j}^{†} a_{j + 1})

. Discarding the constant

- L J (1 + u) S^{2}

and all

O (S^{0})

interactions yields

H_{LSW} = j \sum [A a_{j + 1}^{†} a_{j} + A^{*} a_{j}^{†} a_{j + 1} + B n_{j}], A = - S (J + i D), B = 2 J S (1 + u) .

Here periodic boundary conditions make each

n_{j}

occur twice in the longitudinal sum. The approximation is the linear spin-wave approximation, controlled by small local magnon occupation compared with

S

.

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