Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 337 / 2 / g

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- Solution g

Solution

Choose the orientation of spherical coordinates as

n = (sin θ cos ϕ, - sin θ sin ϕ, cos θ),

which differs from the opposite azimuth convention only by

ϕ \mapsto - ϕ

. With

B = B x

,

θ = π /2 - δ θ

, and

ϕ = δ ϕ

,

n = (1, - δ ϕ, δ θ) + O (δ^{2}) .

Substitution into the effective action gives

L_{e ff} = - α^{2} [(\nabla δ θ)^{2} + (\nabla δ ϕ)^{2}] + β^{2} [(\partial_{t} δ θ + B δ ϕ)^{2} + (\partial_{t} δ ϕ - B δ θ)^{2}],

where

α^{2} = \frac{J S ^{2}}{2}, β^{2} = \frac{1}{16 J a ^{2}} .

Let

ψ = δ θ + i δ ϕ

and define the zero-field spin-wave velocity

c = \frac{α}{β} = 22 J S a .

The linearized equation is

(\partial_{t} - i B)^{2} ψ - c^{2} \nabla^{2} ψ = 0.

For a plane wave, the two circular polarizations therefore obey

(ω \pm B)^{2} = c^{2} k^{2},

or, with signed-frequency branches,

ω = c k \pm B

and their negative-frequency partners. At

B = 0

these are the two degenerate, linearly dispersing antiferromagnetic spin waves. The field Zeeman-splits the two opposite circular polarizations by shifting their frequencies in opposite directions.

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