Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2013 / Paper 1 / 3 / d

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- Solution d

Solution

For the uniform mean-field approximation field

M = M \hat{e}_{z}

, the inverse fermion propagator is diagonal in momentum, frequency and spin:

- i ω_{l} + ξ_{k} - σ M

. The Grassmann Gaussian integral therefore gives

Z_{uniform} = \int d μ (M) e^{- βN M^{2} / (2 U)} \prod_{k, σ = \pm 1} [1 + e^{- β (ξ_{k} - σ M)}] .

Here

d μ (M)

denotes the residual uniform-field measure; integration over its orientation only affects a nonextensive prefactor. Hence

S_{eff} (M) = \frac{βN M ^{2}}{2 U} - k, σ = \pm 1 \sum ln [1 + e^{- β (ϵ_{k} - μ - σ M)}] .

The exchange field separates the two spin bands by

2 M

. Integrating out the fermions is exact for a specified auxiliary field; restricting that field to be constant is the mean-field approximation.

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