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 Incoming links: Uniform spin field and magnetization

Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2013 / Paper 1 / 3 / e / Solution 2026-10-20

Differentiate the uniform effective action:

0 = \frac{1}{β} \frac{\partial S _{eff}}{\partial M} = \frac{NM}{U} - \sum_{k, σ} σ n_{F} (ϵ_{k} - σ M) .

To reproduce the printed coefficient, define the spin-summed density of states per site by

ν (ϵ) = \frac{2}{N} \sum_{k} δ (ϵ - ϵ_{k}), \frac{1}{N} \sum_{k} f (ϵ_{k}) = \frac{1}{2} \int d ϵ ν (ϵ) f (ϵ) .

The saddle-point approximation then gives

M = \frac{U}{2} σ = \pm 1 \sum \int d ϵ ν (ϵ) σ n_{F} (ϵ - σ M) .

The uniform spin field and magnetization obey

M = U (n_{↑} - n_{↓}) = 2 U ⟨ S_{z} ⟩

for occupations per site. If a one-spin density of states were used instead, the factor

1/2

would be absent. The spin field is an energy, rather than directly a magnetic moment.

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