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

For the uniform mean-field approximation field $\mathbf{M}=M{\hat{\mathbf{e}}}_z$, the inverse fermion propagator is diagonal in momentum, frequency and spin: $-i{\omega }_l+{\xi }_k-\sigma M$. The Grassmann Gaussian integral therefore givesHere $d\mu (M)$ denotes the residual uniform-field measure; integration over its orientation only affects a nonextensive prefactor. HenceThe exchange field separates the two spin bands by $2M$. Integrating out the fermions is exact for a specified auxiliary field; restricting that field to be constant is the mean-field approximation.